3.379 \(\int \frac{1}{(a+b \tan ^3(c+d x))^2} \, dx\)

Optimal. Leaf size=558 \[ \frac{\sqrt [3]{b} \left (a^{4/3}-2 b^{4/3}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} d \left (a^2+b^2\right )}+\frac{\sqrt [3]{b} \left (-2 a^{2/3} b^{4/3}+a^2-b^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} d \left (a^2+b^2\right )^2}+\frac{b (\tan (c+d x) (b-a \tan (c+d x))+a)}{3 a d \left (a^2+b^2\right ) \left (a+b \tan ^3(c+d x)\right )}-\frac{\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{18 a^{5/3} d \left (a^2+b^2\right )}-\frac{\sqrt [3]{b} \left (2 a^{2/3} b^{4/3}+a^2-b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{6 \sqrt [3]{a} d \left (a^2+b^2\right )^2}+\frac{\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{9 a^{5/3} d \left (a^2+b^2\right )}+\frac{\sqrt [3]{b} \left (2 a^{2/3} b^{4/3}+a^2-b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 \sqrt [3]{a} d \left (a^2+b^2\right )^2}-\frac{2 a b \log \left (a \cos ^3(c+d x)+b \sin ^3(c+d x)\right )}{3 d \left (a^2+b^2\right )^2}+\frac{x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2} \]

[Out]

((a^2 - b^2)*x)/(a^2 + b^2)^2 + (b^(1/3)*(a^2 - 2*a^(2/3)*b^(4/3) - b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d
*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(1/3)*(a^2 + b^2)^2*d) + (b^(1/3)*(a^(4/3) - 2*b^(4/3))*ArcTan[(a^(1/3) -
2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*(a^2 + b^2)*d) - (2*a*b*Log[a*Cos[c + d*x]^3 +
b*Sin[c + d*x]^3])/(3*(a^2 + b^2)^2*d) + (b^(1/3)*(a^2 + 2*a^(2/3)*b^(4/3) - b^2)*Log[a^(1/3) + b^(1/3)*Tan[c
+ d*x]])/(3*a^(1/3)*(a^2 + b^2)^2*d) + (b^(1/3)*(a^(4/3) + 2*b^(4/3))*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]])/(9*
a^(5/3)*(a^2 + b^2)*d) - (b^(1/3)*(a^2 + 2*a^(2/3)*b^(4/3) - b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[c + d*x] +
b^(2/3)*Tan[c + d*x]^2])/(6*a^(1/3)*(a^2 + b^2)^2*d) - (b^(1/3)*(a^(4/3) + 2*b^(4/3))*Log[a^(2/3) - a^(1/3)*b
^(1/3)*Tan[c + d*x] + b^(2/3)*Tan[c + d*x]^2])/(18*a^(5/3)*(a^2 + b^2)*d) + (b*(a + Tan[c + d*x]*(b - a*Tan[c
+ d*x])))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]^3))

________________________________________________________________________________________

Rubi [A]  time = 0.727999, antiderivative size = 558, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 13, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.929, Rules used = {3661, 6725, 635, 203, 260, 1854, 1860, 31, 634, 617, 204, 628, 1871} \[ \frac{\sqrt [3]{b} \left (a^{4/3}-2 b^{4/3}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} d \left (a^2+b^2\right )}+\frac{\sqrt [3]{b} \left (-2 a^{2/3} b^{4/3}+a^2-b^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} d \left (a^2+b^2\right )^2}+\frac{b (\tan (c+d x) (b-a \tan (c+d x))+a)}{3 a d \left (a^2+b^2\right ) \left (a+b \tan ^3(c+d x)\right )}-\frac{\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{18 a^{5/3} d \left (a^2+b^2\right )}-\frac{\sqrt [3]{b} \left (2 a^{2/3} b^{4/3}+a^2-b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{6 \sqrt [3]{a} d \left (a^2+b^2\right )^2}+\frac{\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{9 a^{5/3} d \left (a^2+b^2\right )}+\frac{\sqrt [3]{b} \left (2 a^{2/3} b^{4/3}+a^2-b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 \sqrt [3]{a} d \left (a^2+b^2\right )^2}-\frac{2 a b \log \left (a \cos ^3(c+d x)+b \sin ^3(c+d x)\right )}{3 d \left (a^2+b^2\right )^2}+\frac{x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*x]^3)^(-2),x]

[Out]

((a^2 - b^2)*x)/(a^2 + b^2)^2 + (b^(1/3)*(a^2 - 2*a^(2/3)*b^(4/3) - b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d
*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(1/3)*(a^2 + b^2)^2*d) + (b^(1/3)*(a^(4/3) - 2*b^(4/3))*ArcTan[(a^(1/3) -
2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*(a^2 + b^2)*d) - (2*a*b*Log[a*Cos[c + d*x]^3 +
b*Sin[c + d*x]^3])/(3*(a^2 + b^2)^2*d) + (b^(1/3)*(a^2 + 2*a^(2/3)*b^(4/3) - b^2)*Log[a^(1/3) + b^(1/3)*Tan[c
+ d*x]])/(3*a^(1/3)*(a^2 + b^2)^2*d) + (b^(1/3)*(a^(4/3) + 2*b^(4/3))*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]])/(9*
a^(5/3)*(a^2 + b^2)*d) - (b^(1/3)*(a^2 + 2*a^(2/3)*b^(4/3) - b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[c + d*x] +
b^(2/3)*Tan[c + d*x]^2])/(6*a^(1/3)*(a^2 + b^2)^2*d) - (b^(1/3)*(a^(4/3) + 2*b^(4/3))*Log[a^(2/3) - a^(1/3)*b
^(1/3)*Tan[c + d*x] + b^(2/3)*Tan[c + d*x]^2])/(18*a^(5/3)*(a^2 + b^2)*d) + (b*(a + Tan[c + d*x]*(b - a*Tan[c
+ d*x])))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]^3))

Rule 3661

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[(c*ff)/f, Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 1854

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[((a*Coeff[Pq, x, q] -
b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q, x])*(a + b*x^n)^(p + 1))/(a*b*n*(p + 1)), x] + Dist[1/(a*n*(p + 1))
, Int[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^(p + 1), x], x] /; q == n - 1] /
; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1860

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1871

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \tan ^3(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \left (a+b x^3\right )^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2-b^2+2 a b x}{\left (a^2+b^2\right )^2 \left (1+x^2\right )}-\frac{b \left (-b+a x+b x^2\right )}{\left (a^2+b^2\right ) \left (a+b x^3\right )^2}+\frac{b \left (2 a b-\left (a^2-b^2\right ) x-2 a b x^2\right )}{\left (a^2+b^2\right )^2 \left (a+b x^3\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{a^2-b^2+2 a b x}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right )^2 d}+\frac{b \operatorname{Subst}\left (\int \frac{2 a b-\left (a^2-b^2\right ) x-2 a b x^2}{a+b x^3} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right )^2 d}-\frac{b \operatorname{Subst}\left (\int \frac{-b+a x+b x^2}{\left (a+b x^3\right )^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=\frac{b (a+\tan (c+d x) (b-a \tan (c+d x)))}{3 a \left (a^2+b^2\right ) d \left (a+b \tan ^3(c+d x)\right )}+\frac{b \operatorname{Subst}\left (\int \frac{2 a b+\left (-a^2+b^2\right ) x}{a+b x^3} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right )^2 d}+\frac{(2 a b) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right )^2 d}-\frac{\left (2 a b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^3} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right )^2 d}+\frac{\left (a^2-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right )^2 d}+\frac{b \operatorname{Subst}\left (\int \frac{2 b-a x}{a+b x^3} \, dx,x,\tan (c+d x)\right )}{3 a \left (a^2+b^2\right ) d}\\ &=\frac{\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac{2 a b \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac{2 a b \log \left (a+b \tan ^3(c+d x)\right )}{3 \left (a^2+b^2\right )^2 d}+\frac{b (a+\tan (c+d x) (b-a \tan (c+d x)))}{3 a \left (a^2+b^2\right ) d \left (a+b \tan ^3(c+d x)\right )}+\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{a} \left (4 a b^{4/3}+\sqrt [3]{a} \left (-a^2+b^2\right )\right )+\sqrt [3]{b} \left (-2 a b^{4/3}+\sqrt [3]{a} \left (-a^2+b^2\right )\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tan (c+d x)\right )}{3 a^{2/3} \left (a^2+b^2\right )^2 d}+\frac{\left (b^{2/3} \left (a^2+2 a^{2/3} b^{4/3}-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\tan (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2+b^2\right )^2 d}+\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{a} \left (-a^{4/3}+4 b^{4/3}\right )+\sqrt [3]{b} \left (-a^{4/3}-2 b^{4/3}\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tan (c+d x)\right )}{9 a^{5/3} \left (a^2+b^2\right ) d}+\frac{\left (b^{2/3} \left (a^{4/3}+2 b^{4/3}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\tan (c+d x)\right )}{9 a^{5/3} \left (a^2+b^2\right ) d}\\ &=\frac{\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac{2 a b \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac{\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}-b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2+b^2\right )^2 d}+\frac{\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{9 a^{5/3} \left (a^2+b^2\right ) d}-\frac{2 a b \log \left (a+b \tan ^3(c+d x)\right )}{3 \left (a^2+b^2\right )^2 d}+\frac{b (a+\tan (c+d x) (b-a \tan (c+d x)))}{3 a \left (a^2+b^2\right ) d \left (a+b \tan ^3(c+d x)\right )}-\frac{\left (b^{2/3} \left (a^2-2 a^{2/3} b^{4/3}-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}-\frac{\left (\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tan (c+d x)\right )}{6 \sqrt [3]{a} \left (a^2+b^2\right )^2 d}-\frac{\left (\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right )\right ) \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tan (c+d x)\right )}{18 a^{5/3} \left (a^2+b^2\right ) d}-\frac{\left (b^{2/3} \left (1-\frac{2 b^{4/3}}{a^{4/3}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tan (c+d x)\right )}{6 \left (a^2+b^2\right ) d}\\ &=\frac{\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac{2 a b \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac{\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}-b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2+b^2\right )^2 d}+\frac{\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{9 a^{5/3} \left (a^2+b^2\right ) d}-\frac{\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}-b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{6 \sqrt [3]{a} \left (a^2+b^2\right )^2 d}-\frac{\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{18 a^{5/3} \left (a^2+b^2\right ) d}-\frac{2 a b \log \left (a+b \tan ^3(c+d x)\right )}{3 \left (a^2+b^2\right )^2 d}+\frac{b (a+\tan (c+d x) (b-a \tan (c+d x)))}{3 a \left (a^2+b^2\right ) d \left (a+b \tan ^3(c+d x)\right )}-\frac{\left (\sqrt [3]{b} \left (a^2-2 a^{2/3} b^{4/3}-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \tan (c+d x)}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} \left (a^2+b^2\right )^2 d}-\frac{\left (\sqrt [3]{b} \left (a^{4/3}-2 b^{4/3}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \tan (c+d x)}{\sqrt [3]{a}}\right )}{3 a^{5/3} \left (a^2+b^2\right ) d}\\ &=\frac{\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac{\sqrt [3]{b} \left (a^2-2 a^{2/3} b^{4/3}-b^2\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \tan (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{a} \left (a^2+b^2\right )^2 d}+\frac{\sqrt [3]{b} \left (a^{4/3}-2 b^{4/3}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \tan (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{3 \sqrt{3} a^{5/3} \left (a^2+b^2\right ) d}-\frac{2 a b \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac{\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}-b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2+b^2\right )^2 d}+\frac{\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{9 a^{5/3} \left (a^2+b^2\right ) d}-\frac{\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}-b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{6 \sqrt [3]{a} \left (a^2+b^2\right )^2 d}-\frac{\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{18 a^{5/3} \left (a^2+b^2\right ) d}-\frac{2 a b \log \left (a+b \tan ^3(c+d x)\right )}{3 \left (a^2+b^2\right )^2 d}+\frac{b (a+\tan (c+d x) (b-a \tan (c+d x)))}{3 a \left (a^2+b^2\right ) d \left (a+b \tan ^3(c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 6.27911, size = 575, normalized size = 1.03 \[ -\frac{b (a-b) (a+b) \tan ^2(c+d x) \text{Hypergeometric2F1}\left (\frac{2}{3},1,\frac{5}{3},-\frac{b \tan ^3(c+d x)}{a}\right )}{2 a d \left (a^2+b^2\right )^2}-\frac{b \tan ^2(c+d x) \text{Hypergeometric2F1}\left (\frac{2}{3},2,\frac{5}{3},-\frac{b \tan ^3(c+d x)}{a}\right )}{2 a d \left (a^2+b^2\right )}+\frac{b^2 \tan (c+d x)}{3 a d \left (a^2+b^2\right ) \left (a+b \tan ^3(c+d x)\right )}+\frac{b}{3 d \left (a^2+b^2\right ) \left (a+b \tan ^3(c+d x)\right )}-\frac{2 a b \log \left (a+b \tan ^3(c+d x)\right )}{3 d \left (a^2+b^2\right )^2}-\frac{\sqrt [3]{a} \left (b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )+2 \sqrt{3} b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )\right )}{3 d \left (a^2+b^2\right )^2}+\frac{\frac{2 b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{a^{2/3}}-\frac{b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )+2 \sqrt{3} b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3}}}{9 a d \left (a^2+b^2\right )}+\frac{2 \sqrt [3]{a} b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 d \left (a^2+b^2\right )^2}-\frac{i \log (-\tan (c+d x)+i)}{2 d (a-i b)^2}+\frac{i \log (\tan (c+d x)+i)}{2 d (a+i b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[c + d*x]^3)^(-2),x]

[Out]

((-I/2)*Log[I - Tan[c + d*x]])/((a - I*b)^2*d) + ((I/2)*Log[I + Tan[c + d*x]])/((a + I*b)^2*d) + (2*a^(1/3)*b^
(5/3)*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]])/(3*(a^2 + b^2)^2*d) - (a^(1/3)*(2*Sqrt[3]*b^(5/3)*ArcTan[(a^(1/3) -
2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1/3))] + b^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[c + d*x] + b^(2/3)*Tan
[c + d*x]^2]))/(3*(a^2 + b^2)^2*d) + ((2*b^(5/3)*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]])/a^(2/3) - (2*Sqrt[3]*b^(
5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1/3))] + b^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[
c + d*x] + b^(2/3)*Tan[c + d*x]^2])/a^(2/3))/(9*a*(a^2 + b^2)*d) - (2*a*b*Log[a + b*Tan[c + d*x]^3])/(3*(a^2 +
b^2)^2*d) - ((a - b)*b*(a + b)*Hypergeometric2F1[2/3, 1, 5/3, -((b*Tan[c + d*x]^3)/a)]*Tan[c + d*x]^2)/(2*a*(
a^2 + b^2)^2*d) - (b*Hypergeometric2F1[2/3, 2, 5/3, -((b*Tan[c + d*x]^3)/a)]*Tan[c + d*x]^2)/(2*a*(a^2 + b^2)*
d) + b/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x]^3)) + (b^2*Tan[c + d*x])/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]^3)
)

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Maple [B]  time = 0.036, size = 1086, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*tan(d*x+c)^3)^2,x)

[Out]

-1/3/d*b/(a^4+2*a^2*b^2+b^4)/(a+b*tan(d*x+c)^3)*tan(d*x+c)^2*a^2-1/3/d*b^3/(a^4+2*a^2*b^2+b^4)/(a+b*tan(d*x+c)
^3)*tan(d*x+c)^2+1/3/d*b^2/(a^4+2*a^2*b^2+b^4)/(a+b*tan(d*x+c)^3)*a*tan(d*x+c)+1/3/d*b^4/(a^4+2*a^2*b^2+b^4)/(
a+b*tan(d*x+c)^3)/a*tan(d*x+c)+1/3/d*b/(a^4+2*a^2*b^2+b^4)/(a+b*tan(d*x+c)^3)*a^2+1/3/d*b^3/(a^4+2*a^2*b^2+b^4
)/(a+b*tan(d*x+c)^3)+8/9/d*b/(a^4+2*a^2*b^2+b^4)*a/(a/b)^(2/3)*ln(tan(d*x+c)+(a/b)^(1/3))+2/9/d*b^3/(a^4+2*a^2
*b^2+b^4)/a/(a/b)^(2/3)*ln(tan(d*x+c)+(a/b)^(1/3))-4/9/d*b/(a^4+2*a^2*b^2+b^4)*a/(a/b)^(2/3)*ln(tan(d*x+c)^2-(
a/b)^(1/3)*tan(d*x+c)+(a/b)^(2/3))-1/9/d*b^3/(a^4+2*a^2*b^2+b^4)/a/(a/b)^(2/3)*ln(tan(d*x+c)^2-(a/b)^(1/3)*tan
(d*x+c)+(a/b)^(2/3))+8/9/d*b/(a^4+2*a^2*b^2+b^4)*a/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*tan(d
*x+c)-1))+2/9/d*b^3/(a^4+2*a^2*b^2+b^4)/a/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*tan(d*x+c)-1))
+4/9/d/(a^4+2*a^2*b^2+b^4)*a^2/(a/b)^(1/3)*ln(tan(d*x+c)+(a/b)^(1/3))-2/9/d*b^2/(a^4+2*a^2*b^2+b^4)/(a/b)^(1/3
)*ln(tan(d*x+c)+(a/b)^(1/3))-2/9/d/(a^4+2*a^2*b^2+b^4)*a^2/(a/b)^(1/3)*ln(tan(d*x+c)^2-(a/b)^(1/3)*tan(d*x+c)+
(a/b)^(2/3))+1/9/d*b^2/(a^4+2*a^2*b^2+b^4)/(a/b)^(1/3)*ln(tan(d*x+c)^2-(a/b)^(1/3)*tan(d*x+c)+(a/b)^(2/3))-4/9
/d/(a^4+2*a^2*b^2+b^4)*a^2*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*tan(d*x+c)-1))+2/9/d*b^2/(a^4
+2*a^2*b^2+b^4)*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*tan(d*x+c)-1))-2/3/d*b/(a^4+2*a^2*b^2+b^
4)*a*ln(a+b*tan(d*x+c)^3)+1/d/(a^4+2*a^2*b^2+b^4)*a*b*ln(tan(d*x+c)^2+1)+1/d/(a^4+2*a^2*b^2+b^4)*arctan(tan(d*
x+c))*a^2-1/d/(a^4+2*a^2*b^2+b^4)*arctan(tan(d*x+c))*b^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 13.9013, size = 24260, normalized size = 43.48 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

-1/648*(216*a^3*b - 432*a*b^3 + 216*(2*a^2*b^2 - b^4 - 3*(a^3*b - a*b^3)*d*x)*tan(d*x + c)^3 - 648*(a^4 - a^2*
b^2)*d*x + 2*((a^5*b + 2*a^3*b^3 + a*b^5)*d*tan(d*x + c)^3 + (a^6 + 2*a^4*b^2 + a^2*b^4)*d)*(4*(9*a^2*b^2/(a^4
*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^
4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*
d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 -
b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*
d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2
*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3
) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))*log(1/324*(10368*a^6 - 12960*a^4*b^2 - 3888*a^2*b^4 + ((2*a^10
+ 5*a^8*b^2 + 4*a^6*b^4 + a^4*b^6)*d^2*tan(d*x + c)^2 - 4*(a^9*b + 2*a^7*b^3 + a^5*b^5)*d^2*tan(d*x + c) - (2
*a^10 + 5*a^8*b^2 + 4*a^6*b^4 + a^4*b^6)*d^2)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2
*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^
6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b
^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-
8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*
a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^
2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d)
)^2 - 1296*(18*a^4*b^2 + 7*a^2*b^4 + b^6)*tan(d*x + c)^2 - 36*((8*a^7*b - 2*a^5*b^3 - a^3*b^5)*d*tan(d*x + c)^
2 + 2*(4*a^8 - 20*a^6*b^2 - 7*a^4*b^4 - a^2*b^6)*d*tan(d*x + c) - (8*a^7*b - 2*a^5*b^3 - a^3*b^5)*d)*(4*(9*a^2
*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^
3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2
*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*
a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b
^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 +
2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*
(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d)) + 2592*(4*a^5*b + 2*a^3*b^3 + a*b^5)*tan(d*x + c))/(t
an(d*x + c)^2 + 1)) + 216*(a^3*b + a*b^3)*tan(d*x + c)^2 + (324*a^2*b^2*tan(d*x + c)^3 + 324*a^3*b - ((a^5*b +
2*a^3*b^3 + a*b^5)*d*tan(d*x + c)^3 + (a^6 + 2*a^4*b^2 + a^2*b^4)*d)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4
*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^
4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b +
b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4
*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 +
a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) -
4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*
d + 2*a^2*b^2*d + b^4*d)) + 3*sqrt(1/3)*((a^5*b + 2*a^3*b^3 + a*b^5)*d*tan(d*x + c)^3 + (a^6 + 2*a^4*b^2 + a^2
*b^4)*d)*sqrt((29808*a^4*b^2 - 10368*a^2*b^4 - 5184*b^6 - (a^10 + 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 + a^2*b^8)
*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)
/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d +
2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4
*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3
+ 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(
a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^
3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))^2*d^2 + 216*(a^7*b + 2*a^5*b^3 + a^3*b^5)*(
4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(
-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2
*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b
^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 +
4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^
9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3)
)^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))*d)/((a^10 + 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6
+ a^2*b^8)*d^2)))*log(1/324*(20736*a^8 - 106272*a^6*b^2 - 22032*a^4*b^4 - ((2*a^12 + 7*a^10*b^2 + 9*a^8*b^4 +
5*a^6*b^6 + a^4*b^8)*d^2*tan(d*x + c)^2 - 4*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*d^2*tan(d*x + c) - (2*
a^12 + 7*a^10*b^2 + 9*a^8*b^4 + 5*a^6*b^6 + a^4*b^8)*d^2)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/
(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/8
1*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d
^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(
1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)
*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6
- 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^
2*d + b^4*d))^2 + 1296*(42*a^6*b^2 - 59*a^4*b^4 - 22*a^2*b^6 - 2*b^8)*tan(d*x + c)^2 + 36*((8*a^9*b + 6*a^7*b^
3 - 3*a^5*b^5 - a^3*b^7)*d*tan(d*x + c)^2 + 2*(4*a^10 - 16*a^8*b^2 - 27*a^6*b^4 - 8*a^4*b^6 - a^2*b^8)*d*tan(d
*x + c) - (8*a^9*b + 6*a^7*b^3 - 3*a^5*b^5 - a^3*b^7)*d)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(
a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81
*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^
3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1
/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*
(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6
- 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2
*d + b^4*d)) + 3*sqrt(1/3)*(36*(10*a^9*b + 21*a^7*b^3 + 12*a^5*b^5 + a^3*b^7)*d*tan(d*x + c)^2 - 72*(4*a^10 +
2*a^8*b^2 - 9*a^6*b^4 - 8*a^4*b^6 - a^2*b^8)*d*tan(d*x + c) - ((2*a^12 + 7*a^10*b^2 + 9*a^8*b^4 + 5*a^6*b^6 +
a^4*b^8)*d^2*tan(d*x + c)^2 - 4*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*d^2*tan(d*x + c) - (2*a^12 + 7*a^10
*b^2 + 9*a^8*b^4 + 5*a^6*b^6 + a^4*b^8)*d^2)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*
a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6
*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^
2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8
/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a
^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2
- 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))
- 36*(10*a^9*b + 21*a^7*b^3 + 12*a^5*b^5 + a^3*b^7)*d)*sqrt((29808*a^4*b^2 - 10368*a^2*b^4 - 5184*b^6 - (a^10
+ 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 + a^2*b^8)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 +
2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((
a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7
*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*
(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d +
2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*
b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*
d))^2*d^2 + 216*(a^7*b + 2*a^5*b^3 + a^3*b^5)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2
*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^
6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b
^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-
8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*
a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^
2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d)
)*d)/((a^10 + 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 + a^2*b^8)*d^2)) - 2592*(28*a^7*b - 78*a^5*b^3 - 27*a^3*b^5 -
2*a*b^7)*tan(d*x + c))/(tan(d*x + c)^2 + 1)) + (324*a^2*b^2*tan(d*x + c)^3 + 324*a^3*b - ((a^5*b + 2*a^3*b^3 +
a*b^5)*d*tan(d*x + c)^3 + (a^6 + 2*a^4*b^2 + a^2*b^4)*d)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/
(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/8
1*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d
^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(
1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)
*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6
- 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^
2*d + b^4*d)) - 3*sqrt(1/3)*((a^5*b + 2*a^3*b^3 + a*b^5)*d*tan(d*x + c)^3 + (a^6 + 2*a^4*b^2 + a^2*b^4)*d)*sqr
t((29808*a^4*b^2 - 10368*a^2*b^4 - 5184*b^6 - (a^10 + 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 + a^2*b^8)*(4*(9*a^2*b
^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*
b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d
+ b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^
2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3
/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*
a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I
*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))^2*d^2 + 216*(a^7*b + 2*a^5*b^3 + a^3*b^5)*(4*(9*a^2*b^2
/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^
3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d +
b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*
b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/(
(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^
7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*s
qrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))*d)/((a^10 + 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 + a^2*b^8)*
d^2)))*log(-1/324*(20736*a^8 - 106272*a^6*b^2 - 22032*a^4*b^4 - ((2*a^12 + 7*a^10*b^2 + 9*a^8*b^4 + 5*a^6*b^6
+ a^4*b^8)*d^2*tan(d*x + c)^2 - 4*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*d^2*tan(d*x + c) - (2*a^12 + 7*a^
10*b^2 + 9*a^8*b^4 + 5*a^6*b^6 + a^4*b^8)*d^2)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 +
2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a
^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*
b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(
-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2
*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b
^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d
))^2 + 1296*(42*a^6*b^2 - 59*a^4*b^4 - 22*a^2*b^6 - 2*b^8)*tan(d*x + c)^2 + 36*((8*a^9*b + 6*a^7*b^3 - 3*a^5*b
^5 - a^3*b^7)*d*tan(d*x + c)^2 + 2*(4*a^10 - 16*a^8*b^2 - 27*a^6*b^4 - 8*a^4*b^6 - a^2*b^8)*d*tan(d*x + c) - (
8*a^9*b + 6*a^7*b^3 - 3*a^5*b^5 - a^3*b^7)*d)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2
*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^
6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b
^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-
8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*
a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^
2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d)
) - 3*sqrt(1/3)*(36*(10*a^9*b + 21*a^7*b^3 + 12*a^5*b^5 + a^3*b^7)*d*tan(d*x + c)^2 - 72*(4*a^10 + 2*a^8*b^2 -
9*a^6*b^4 - 8*a^4*b^6 - a^2*b^8)*d*tan(d*x + c) - ((2*a^12 + 7*a^10*b^2 + 9*a^8*b^4 + 5*a^6*b^6 + a^4*b^8)*d^
2*tan(d*x + c)^2 - 4*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*d^2*tan(d*x + c) - (2*a^12 + 7*a^10*b^2 + 9*a^
8*b^4 + 5*a^6*b^6 + a^4*b^8)*d^2)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2
+ a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^
4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5
*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3
/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d +
b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b
^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d)) - 36*(10*a
^9*b + 21*a^7*b^3 + 12*a^5*b^5 + a^3*b^7)*d)*sqrt((29808*a^4*b^2 - 10368*a^2*b^4 - 5184*b^6 - (a^10 + 4*a^8*b^
2 + 6*a^6*b^4 + 4*a^4*b^6 + a^2*b^8)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*
d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2
*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 +
a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*
b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d
+ b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^
2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))^2*d^2 +
216*(a^7*b + 2*a^5*b^3 + a^3*b^5)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^
2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a
^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^
5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^
3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d +
b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*
b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))*d)/((a^10
+ 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 + a^2*b^8)*d^2)) - 2592*(28*a^7*b - 78*a^5*b^3 - 27*a^3*b^5 - 2*a*b^7)*ta
n(d*x + c))/(tan(d*x + c)^2 + 1)) - 216*(a^2*b^2 + b^4)*tan(d*x + c))/((a^5*b + 2*a^3*b^3 + a*b^5)*d*tan(d*x +
c)^3 + (a^6 + 2*a^4*b^2 + a^2*b^4)*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c)**3)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.6547, size = 811, normalized size = 1.45 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c)^3)^2,x, algorithm="giac")

[Out]

1/9*(9*a*b*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - 6*a*b*log(abs(b*tan(d*x + c)^3 + a))/(a^4 + 2*a^2
*b^2 + b^4) + 2*(2*a^8*b^2*(-a/b)^(1/3) + 3*a^6*b^4*(-a/b)^(1/3) - a^2*b^8*(-a/b)^(1/3) - 4*a^7*b^3 - 9*a^5*b^
5 - 6*a^3*b^7 - a*b^9)*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + tan(d*x + c)))/(a^11*b + 4*a^9*b^3 + 6*a^7*b^5 + 4
*a^5*b^7 + a^3*b^9) + 9*(a^2 - b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) + 6*(pi*floor((d*x + c)/pi + 1/2)*sgn((-
a/b)^(1/3)) + arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*tan(d*x + c))/(-a/b)^(1/3)))*((2*a^3 - a*b^2)*(-a*b^2)^(2/3
) + (4*a^2*b^2 + b^4)*(-a*b^2)^(1/3))/(sqrt(3)*a^6*b + 2*sqrt(3)*a^4*b^3 + sqrt(3)*a^2*b^5) - ((2*a^3 - a*b^2)
*(-a*b^2)^(2/3) - (4*a^2*b^2 + b^4)*(-a*b^2)^(1/3))*log(tan(d*x + c)^2 + (-a/b)^(1/3)*tan(d*x + c) + (-a/b)^(2
/3))/(a^6*b + 2*a^4*b^3 + a^2*b^5) + 3*(2*a^2*b^2*tan(d*x + c)^3 - a^3*b*tan(d*x + c)^2 - a*b^3*tan(d*x + c)^2
+ a^2*b^2*tan(d*x + c) + b^4*tan(d*x + c) + 3*a^3*b + a*b^3)/((a^5 + 2*a^3*b^2 + a*b^4)*(b*tan(d*x + c)^3 + a
)))/d