3.397 \(\int \frac{\sec (c+d x)}{(a+b \sin ^3(c+d x))^2} \, dx\)

Optimal. Leaf size=587 \[ \frac{b (a-\sin (c+d x) (b-a \sin (c+d x)))}{3 a d \left (a^2-b^2\right ) \left (a+b \sin ^3(c+d x)\right )}+\frac{\sqrt [3]{b} \left (2 a^{2/3} b^{4/3}+a^2+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} d \left (a^2-b^2\right )^2}+\frac{\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} d \left (a^2-b^2\right )}-\frac{2 a b \log \left (a+b \sin ^3(c+d x)\right )}{3 d \left (a^2-b^2\right )^2}-\frac{\sqrt [3]{b} \left (2 a^{2/3} b^{4/3}+a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} d \left (a^2-b^2\right )^2}-\frac{\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} d \left (a^2-b^2\right )}-\frac{\sqrt [3]{b} \left (-2 a^{2/3} b^{4/3}+a^2+b^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} d \left (a^2-b^2\right )^2}-\frac{\sqrt [3]{b} \left (a^{4/3}-2 b^{4/3}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} d \left (a^2-b^2\right )}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.687527, antiderivative size = 587, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.524, Rules used = {3223, 2074, 1854, 1860, 31, 634, 617, 204, 628, 1871, 260} \[ \frac{b (a-\sin (c+d x) (b-a \sin (c+d x)))}{3 a d \left (a^2-b^2\right ) \left (a+b \sin ^3(c+d x)\right )}+\frac{\sqrt [3]{b} \left (2 a^{2/3} b^{4/3}+a^2+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} d \left (a^2-b^2\right )^2}+\frac{\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} d \left (a^2-b^2\right )}-\frac{2 a b \log \left (a+b \sin ^3(c+d x)\right )}{3 d \left (a^2-b^2\right )^2}-\frac{\sqrt [3]{b} \left (2 a^{2/3} b^{4/3}+a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} d \left (a^2-b^2\right )^2}-\frac{\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} d \left (a^2-b^2\right )}-\frac{\sqrt [3]{b} \left (-2 a^{2/3} b^{4/3}+a^2+b^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} d \left (a^2-b^2\right )^2}-\frac{\sqrt [3]{b} \left (a^{4/3}-2 b^{4/3}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} d \left (a^2-b^2\right )}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^2} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

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

Rule 3223

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x]
, x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0
] || IGtQ[p, 0] || IntegersQ[m, p])

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

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]

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]

Rubi steps

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

Mathematica [C]  time = 4.36755, size = 503, normalized size = 0.86 \[ \frac{\frac{9 b \left (a^2+b^2\right ) \sin ^2(c+d x) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{b \sin ^3(c+d x)}{a}\right )}{a \left (a^2-b^2\right )^2}+\frac{9 b \sin ^2(c+d x) \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};-\frac{b \sin ^3(c+d x)}{a}\right )}{a^3-a b^2}-\frac{6 b^2 \sin (c+d x)}{a \left (a^2-b^2\right ) \left (a+b \sin ^3(c+d x)\right )}+\frac{6 b}{\left (a^2-b^2\right ) \left (a+b \sin ^3(c+d x)\right )}-\frac{12 a b \log \left (a+b \sin ^3(c+d x)\right )}{\left (a^2-b^2\right )^2}-\frac{12 \sqrt [3]{a} b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{\left (a^2-b^2\right )^2}+\frac{6 \sqrt [3]{a} b^{5/3} \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )\right )}{\left (a^2-b^2\right )^2}+\frac{2 b^{5/3} \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )\right )}{a^{5/3} \left (a^2-b^2\right )}-\frac{9 \log (1-\sin (c+d x))}{(a+b)^2}+\frac{9 \log (\sin (c+d x)+1)}{(a-b)^2}}{18 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

((-9*Log[1 - Sin[c + d*x]])/(a + b)^2 + (9*Log[1 + Sin[c + d*x]])/(a - b)^2 - (12*a^(1/3)*b^(5/3)*Log[a^(1/3)
+ b^(1/3)*Sin[c + d*x]])/(a^2 - b^2)^2 + (6*a^(1/3)*b^(5/3)*(2*Sqrt[3]*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x
])/(Sqrt[3]*a^(1/3))] + Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]))/(a^2 - b^2)^2 +
(2*b^(5/3)*(2*Sqrt[3]*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))] - 2*Log[a^(1/3) + b^(1/3)*
Sin[c + d*x]] + Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]))/(a^(5/3)*(a^2 - b^2)) -
(12*a*b*Log[a + b*Sin[c + d*x]^3])/(a^2 - b^2)^2 + (9*b*(a^2 + b^2)*Hypergeometric2F1[2/3, 1, 5/3, -((b*Sin[c
+ d*x]^3)/a)]*Sin[c + d*x]^2)/(a*(a^2 - b^2)^2) + (9*b*Hypergeometric2F1[2/3, 2, 5/3, -((b*Sin[c + d*x]^3)/a)
]*Sin[c + d*x]^2)/(a^3 - a*b^2) + (6*b)/((a^2 - b^2)*(a + b*Sin[c + d*x]^3)) - (6*b^2*Sin[c + d*x])/(a*(a^2 -
b^2)*(a + b*Sin[c + d*x]^3)))/(18*d)

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Maple [A]  time = 0.182, size = 934, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)/(a+b*sin(d*x+c)^3)^2,x)

[Out]

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

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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(sec(d*x+c)/(a+b*sin(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 22.6878, size = 22565, normalized size = 38.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/324*(216*a^3*b - 216*a*b^3 - 108*(a^3*b - a*b^3)*cos(d*x + c)^2 - 2*((a^6 - 2*a^4*b^2 + a^2*b^4)*d - ((a^5*b
- 2*a^3*b^3 + a*b^5)*d*cos(d*x + c)^2 - (a^5*b - 2*a^3*b^3 + a*b^5)*d)*sin(d*x + c))*(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(-56*a^5*b^2 + 20*a^3*b^4 + 1/324*(2*a^11 - 3*a^9*b^2 + a^5*b^6)*
(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 - 1/9*(12*a^8*b + 22*a^6*b^3 - 8*a^4*b
^5 + a^2*b^7)*(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 + 4*(8*a^6*b + 28*a^4*b^3 -
10*a^2*b^5 + b^7)*sin(d*x + c)) - (324*a^3*b - ((a^6 - 2*a^4*b^2 + a^2*b^4)*d - ((a^5*b - 2*a^3*b^3 + a*b^5)*d
*cos(d*x + c)^2 - (a^5*b - 2*a^3*b^3 + a*b^5)*d)*sin(d*x + c))*(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^6 - 2*a^4*b^2 + a^2*b^4)*d - ((a^5*b - 2*a^3*b^3 + a*b^5)*d*cos(d*x + c)^
2 - (a^5*b - 2*a^3*b^3 + a*b^5)*d)*sin(d*x + c))*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 - 1
0*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)) - 324*(a^2*b^2*cos(d*x + c)^2 - a^2*b^2)*sin(d*x + c
))*log(56*a^5*b^2 - 20*a^3*b^4 - 1/324*(2*a^11 - 3*a^9*b^2 + a^5*b^6)*(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 + 1/9*(12*a^8*b + 22*a^6*b^3 - 8*a^4*b^5 + a^2*b^7)*(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 - 1/108*sqrt(1/3)*((2*a^11 - 3*a^9*b^2 + a^5*b^6)*(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^2 - 36*(6*a^8*b - 13*a^6*b^3 + 8*a^4*b^5 - a^2*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*s
qrt(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*sqr
t(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)) + 8*(8*a^6*b + 28*a^4*b^3 - 10*a^2*b^5 + b^7)*sin(d*x + c)) - (324*a^3*b - ((a^6 - 2*a^4*b^2 + a^2*b^4)*d
- ((a^5*b - 2*a^3*b^3 + a*b^5)*d*cos(d*x + c)^2 - (a^5*b - 2*a^3*b^3 + a*b^5)*d)*sin(d*x + c))*(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*sqr
t(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d)) - 3*sqrt(1/3)*((a^6 - 2*a^4*b^2 + a^2*b^4)*d - ((a^5*b - 2*
a^3*b^3 + a*b^5)*d*cos(d*x + c)^2 - (a^5*b - 2*a^3*b^3 + a*b^5)*d)*sin(d*x + c))*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)) - 324*(a^2*b^2*cos(d
*x + c)^2 - a^2*b^2)*sin(d*x + c))*log(-56*a^5*b^2 + 20*a^3*b^4 + 1/324*(2*a^11 - 3*a^9*b^2 + a^5*b^6)*(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 - 1
0*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 - 1/9*(12*a^8*b + 22*a^6*b^3 - 8*a^4*b^5 + a^
2*b^7)*(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 - 1/108*sqrt(1/3)*((2*a^11 - 3*a^9*
b^2 + a^5*b^6)*(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^2 - 36*(6*a^8*b - 13*a^6*b^
3 + 8*a^4*b^5 - a^2*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)) - 4/72
9*(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) + 4/72
9*(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)) - 8*(8*a^6*b + 28*a^4*b^3 - 10*a^2*b^5 + b^7)*sin(d*x + c)) + 162*(a^4 +
2*a^3*b + a^2*b^2 + (a^3*b + 2*a^2*b^2 + a*b^3 - (a^3*b + 2*a^2*b^2 + a*b^3)*cos(d*x + c)^2)*sin(d*x + c))*log
(sin(d*x + c) + 1) - 162*(a^4 - 2*a^3*b + a^2*b^2 + (a^3*b - 2*a^2*b^2 + a*b^3 - (a^3*b - 2*a^2*b^2 + a*b^3)*c
os(d*x + c)^2)*sin(d*x + c))*log(-sin(d*x + c) + 1) - 108*(a^2*b^2 - b^4)*sin(d*x + c))/((a^6 - 2*a^4*b^2 + a^
2*b^4)*d - ((a^5*b - 2*a^3*b^3 + a*b^5)*d*cos(d*x + c)^2 - (a^5*b - 2*a^3*b^3 + a*b^5)*d)*sin(d*x + c))

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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(sec(d*x+c)/(a+b*sin(d*x+c)**3)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.26537, size = 760, normalized size = 1.29 \begin{align*} -\frac{\frac{12 \, a b \log \left ({\left | b \sin \left (d x + c\right )^{3} + a \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac{4 \,{\left (2 \, a^{8} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 3 \, a^{6} b^{4} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{2} b^{8} \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 4 \, a^{7} b^{3} + 9 \, a^{5} b^{5} - 6 \, a^{3} b^{7} + a b^{9}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | -\left (-\frac{a}{b}\right )^{\frac{1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{a^{11} b - 4 \, a^{9} b^{3} + 6 \, a^{7} b^{5} - 4 \, a^{5} b^{7} + a^{3} b^{9}} + \frac{12 \,{\left ({\left (2 \, a^{3} + a b^{2}\right )} \left (-a b^{2}\right )^{\frac{2}{3}} +{\left (4 \, a^{2} b^{2} - b^{4}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )} \arctan \left (\frac{\sqrt{3}{\left (\left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} a^{6} b - 2 \, \sqrt{3} a^{4} b^{3} + \sqrt{3} a^{2} b^{5}} - \frac{2 \,{\left ({\left (2 \, a^{3} + a b^{2}\right )} \left (-a b^{2}\right )^{\frac{2}{3}} -{\left (4 \, a^{2} b^{2} - b^{4}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac{a}{b}\right )^{\frac{1}{3}} \sin \left (d x + c\right ) + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}} - \frac{9 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} + \frac{9 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2} + 2 \, a b + b^{2}} - \frac{6 \,{\left (2 \, a^{2} b^{2} \sin \left (d x + c\right )^{3} + a^{3} b \sin \left (d x + c\right )^{2} - a b^{3} \sin \left (d x + c\right )^{2} - a^{2} b^{2} \sin \left (d x + c\right ) + b^{4} \sin \left (d x + c\right ) + 3 \, a^{3} b - a b^{3}\right )}}{{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )}{\left (b \sin \left (d x + c\right )^{3} + a\right )}}}{18 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(12*a*b*log(abs(b*sin(d*x + c)^3 + a))/(a^4 - 2*a^2*b^2 + b^4) + 4*(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) + sin(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) + 12*((2*a^3 + a*b^2)*(-a*b^2)^(2
/3) + (4*a^2*b^2 - b^4)*(-a*b^2)^(1/3))*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(sqrt
(3)*a^6*b - 2*sqrt(3)*a^4*b^3 + sqrt(3)*a^2*b^5) - 2*((2*a^3 + a*b^2)*(-a*b^2)^(2/3) - (4*a^2*b^2 - b^4)*(-a*b
^2)^(1/3))*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a^6*b - 2*a^4*b^3 + a^2*b^5) - 9*lo
g(abs(sin(d*x + c) + 1))/(a^2 - 2*a*b + b^2) + 9*log(abs(sin(d*x + c) - 1))/(a^2 + 2*a*b + b^2) - 6*(2*a^2*b^2
*sin(d*x + c)^3 + a^3*b*sin(d*x + c)^2 - a*b^3*sin(d*x + c)^2 - a^2*b^2*sin(d*x + c) + b^4*sin(d*x + c) + 3*a^
3*b - a*b^3)/((a^5 - 2*a^3*b^2 + a*b^4)*(b*sin(d*x + c)^3 + a)))/d