3.576 \(\int \frac{(a+b \tan (c+d x))^3}{\tan ^{\frac{9}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=299 \[ \frac{2 a \left (a^2-3 b^2\right )}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}+\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d}+\frac{2 b \left (3 a^2-b^2\right )}{d \sqrt{\tan (c+d x)}}-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d}-\frac{2 a^2 (a+b \tan (c+d x))}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{32 a^2 b}{35 d \tan ^{\frac{5}{2}}(c+d x)} \]

[Out]

-(((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a - b)*(a^2 + 4*a*b +
b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[T
an[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]
+ Tan[c + d*x]])/(2*Sqrt[2]*d) - (32*a^2*b)/(35*d*Tan[c + d*x]^(5/2)) + (2*a*(a^2 - 3*b^2))/(3*d*Tan[c + d*x]^
(3/2)) + (2*b*(3*a^2 - b^2))/(d*Sqrt[Tan[c + d*x]]) - (2*a^2*(a + b*Tan[c + d*x]))/(7*d*Tan[c + d*x]^(7/2))

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Rubi [A]  time = 0.381528, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {3565, 3628, 3529, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac{2 a \left (a^2-3 b^2\right )}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}+\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d}+\frac{2 b \left (3 a^2-b^2\right )}{d \sqrt{\tan (c+d x)}}-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d}-\frac{2 a^2 (a+b \tan (c+d x))}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{32 a^2 b}{35 d \tan ^{\frac{5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*x])^3/Tan[c + d*x]^(9/2),x]

[Out]

-(((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a - b)*(a^2 + 4*a*b +
b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[T
an[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]
+ Tan[c + d*x]])/(2*Sqrt[2]*d) - (32*a^2*b)/(35*d*Tan[c + d*x]^(5/2)) + (2*a*(a^2 - 3*b^2))/(3*d*Tan[c + d*x]^
(3/2)) + (2*b*(3*a^2 - b^2))/(d*Sqrt[Tan[c + d*x]]) - (2*a^2*(a + b*Tan[c + d*x]))/(7*d*Tan[c + d*x]^(7/2))

Rule 3565

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[((b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3628

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 + b^2
)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e +
f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2
+ b^2, 0]

Rule 3529

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((
b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3534

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 1168

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

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 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

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]

Rubi steps

\begin{align*} \int \frac{(a+b \tan (c+d x))^3}{\tan ^{\frac{9}{2}}(c+d x)} \, dx &=-\frac{2 a^2 (a+b \tan (c+d x))}{7 d \tan ^{\frac{7}{2}}(c+d x)}+\frac{2}{7} \int \frac{8 a^2 b-\frac{7}{2} a \left (a^2-3 b^2\right ) \tan (c+d x)-\frac{1}{2} b \left (5 a^2-7 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac{7}{2}}(c+d x)} \, dx\\ &=-\frac{32 a^2 b}{35 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 a^2 (a+b \tan (c+d x))}{7 d \tan ^{\frac{7}{2}}(c+d x)}+\frac{2}{7} \int \frac{-\frac{7}{2} a \left (a^2-3 b^2\right )-\frac{7}{2} b \left (3 a^2-b^2\right ) \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x)} \, dx\\ &=-\frac{32 a^2 b}{35 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (a^2-3 b^2\right )}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 a^2 (a+b \tan (c+d x))}{7 d \tan ^{\frac{7}{2}}(c+d x)}+\frac{2}{7} \int \frac{-\frac{7}{2} b \left (3 a^2-b^2\right )+\frac{7}{2} a \left (a^2-3 b^2\right ) \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x)} \, dx\\ &=-\frac{32 a^2 b}{35 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (a^2-3 b^2\right )}{3 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (3 a^2-b^2\right )}{d \sqrt{\tan (c+d x)}}-\frac{2 a^2 (a+b \tan (c+d x))}{7 d \tan ^{\frac{7}{2}}(c+d x)}+\frac{2}{7} \int \frac{\frac{7}{2} a \left (a^2-3 b^2\right )+\frac{7}{2} b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt{\tan (c+d x)}} \, dx\\ &=-\frac{32 a^2 b}{35 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (a^2-3 b^2\right )}{3 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (3 a^2-b^2\right )}{d \sqrt{\tan (c+d x)}}-\frac{2 a^2 (a+b \tan (c+d x))}{7 d \tan ^{\frac{7}{2}}(c+d x)}+\frac{4 \operatorname{Subst}\left (\int \frac{\frac{7}{2} a \left (a^2-3 b^2\right )+\frac{7}{2} b \left (3 a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{7 d}\\ &=-\frac{32 a^2 b}{35 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (a^2-3 b^2\right )}{3 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (3 a^2-b^2\right )}{d \sqrt{\tan (c+d x)}}-\frac{2 a^2 (a+b \tan (c+d x))}{7 d \tan ^{\frac{7}{2}}(c+d x)}+\frac{\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}+\frac{\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=-\frac{32 a^2 b}{35 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (a^2-3 b^2\right )}{3 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (3 a^2-b^2\right )}{d \sqrt{\tan (c+d x)}}-\frac{2 a^2 (a+b \tan (c+d x))}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \sqrt{2} d}-\frac{\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \sqrt{2} d}+\frac{\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 d}+\frac{\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 d}\\ &=-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} d}+\frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} d}-\frac{32 a^2 b}{35 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (a^2-3 b^2\right )}{3 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (3 a^2-b^2\right )}{d \sqrt{\tan (c+d x)}}-\frac{2 a^2 (a+b \tan (c+d x))}{7 d \tan ^{\frac{7}{2}}(c+d x)}+\frac{\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}-\frac{\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}\\ &=-\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}+\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} d}+\frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} d}-\frac{32 a^2 b}{35 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (a^2-3 b^2\right )}{3 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (3 a^2-b^2\right )}{d \sqrt{\tan (c+d x)}}-\frac{2 a^2 (a+b \tan (c+d x))}{7 d \tan ^{\frac{7}{2}}(c+d x)}\\ \end{align*}

Mathematica [C]  time = 0.584387, size = 103, normalized size = 0.34 \[ -\frac{2 \left (5 a \left (a^2-3 b^2\right ) \, _2F_1\left (-\frac{7}{4},1;-\frac{3}{4};-\tan ^2(c+d x)\right )+b \left (7 \left (3 a^2-b^2\right ) \tan (c+d x) \, _2F_1\left (-\frac{5}{4},1;-\frac{1}{4};-\tan ^2(c+d x)\right )+b (15 a+7 b \tan (c+d x))\right )\right )}{35 d \tan ^{\frac{7}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[c + d*x])^3/Tan[c + d*x]^(9/2),x]

[Out]

(-2*(5*a*(a^2 - 3*b^2)*Hypergeometric2F1[-7/4, 1, -3/4, -Tan[c + d*x]^2] + b*(7*(3*a^2 - b^2)*Hypergeometric2F
1[-5/4, 1, -1/4, -Tan[c + d*x]^2]*Tan[c + d*x] + b*(15*a + 7*b*Tan[c + d*x]))))/(35*d*Tan[c + d*x]^(7/2))

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Maple [B]  time = 0.018, size = 539, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(d*x+c))^3/tan(d*x+c)^(9/2),x)

[Out]

-2/7/d*a^3/tan(d*x+c)^(7/2)+2/3/d*a^3/tan(d*x+c)^(3/2)-2/d*a/tan(d*x+c)^(3/2)*b^2-6/5*a^2*b/d/tan(d*x+c)^(5/2)
+6*a^2*b/d/tan(d*x+c)^(1/2)-2/d*b^3/tan(d*x+c)^(1/2)+1/2/d*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*a^3-3/2
/d*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*a*b^2+1/4/d*2^(1/2)*ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/
(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a^3-3/4/d*2^(1/2)*ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/
2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a*b^2+1/2/d*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*a^3-3/2/d*arctan(1+2^(
1/2)*tan(d*x+c)^(1/2))*2^(1/2)*a*b^2+3/4/d*ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1
/2)+tan(d*x+c)))*2^(1/2)*a^2*b-1/4/d*ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+ta
n(d*x+c)))*2^(1/2)*b^3+3/2/d*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*a^2*b-1/2/d*arctan(-1+2^(1/2)*tan(d*x
+c)^(1/2))*2^(1/2)*b^3+3/2/d*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*a^2*b-1/2/d*arctan(1+2^(1/2)*tan(d*x+c
)^(1/2))*2^(1/2)*b^3

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Maxima [A]  time = 1.59082, size = 350, normalized size = 1.17 \begin{align*} \frac{210 \, \sqrt{2}{\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right ) + 210 \, \sqrt{2}{\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right ) + 105 \, \sqrt{2}{\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - 105 \, \sqrt{2}{\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \frac{8 \,{\left (63 \, a^{2} b \tan \left (d x + c\right ) - 105 \,{\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{3} + 15 \, a^{3} - 35 \,{\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}\right )}}{\tan \left (d x + c\right )^{\frac{7}{2}}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(d*x+c))^3/tan(d*x+c)^(9/2),x, algorithm="maxima")

[Out]

1/420*(210*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 210*
sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) + 105*sqrt(2)*(a
^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - 105*sqrt(2)*(a^3 - 3*a^2*b
- 3*a*b^2 + b^3)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - 8*(63*a^2*b*tan(d*x + c) - 105*(3*a^2*b
- b^3)*tan(d*x + c)^3 + 15*a^3 - 35*(a^3 - 3*a*b^2)*tan(d*x + c)^2)/tan(d*x + c)^(7/2))/d

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Fricas [B]  time = 16.1975, size = 18336, normalized size = 61.32 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(d*x+c))^3/tan(d*x+c)^(9/2),x, algorithm="fricas")

[Out]

-1/420*(420*sqrt(2)*(d^5*cos(d*x + c)^4 - 2*d^5*cos(d*x + c)^2 + d^5)*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 2
0*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12 + 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*d^2*sqrt((a^12 + 6*a^10*b^2 +
15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4))/(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^
6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12))*((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10
+ b^12)/d^4)^(3/4)*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d
^4)*arctan(((a^24 - 6*a^22*b^2 - 84*a^20*b^4 - 322*a^18*b^6 - 603*a^16*b^8 - 540*a^14*b^10 + 540*a^10*b^14 + 6
03*a^8*b^16 + 322*a^6*b^18 + 84*a^4*b^20 + 6*a^2*b^22 - b^24)*d^4*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^
6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^
8 - 30*a^2*b^10 + b^12)/d^4) - sqrt(2)*((a^3 - 3*a*b^2)*d^7*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6
+ 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30
*a^2*b^10 + b^12)/d^4) - (3*a^8*b + 8*a^6*b^3 + 6*a^4*b^5 - b^9)*d^5*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4 -
452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a
^4*b^8 + 6*a^2*b^10 + b^12 + 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*d^2*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*
a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4))/(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8
- 30*a^2*b^10 + b^12))*sqrt(((a^18 - 27*a^16*b^2 + 168*a^14*b^4 + 224*a^12*b^6 - 366*a^10*b^8 - 366*a^8*b^10 +
224*a^6*b^12 + 168*a^4*b^14 - 27*a^2*b^16 + b^18)*d^2*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*
a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)*cos(d*x + c) + sqrt(2)*((3*a^14*b - 91*a^12*b^3 + 795*a^10*b^5 - 1611*a^8*b^
7 + 1217*a^6*b^9 - 345*a^4*b^11 + 33*a^2*b^13 - b^15)*d^3*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 +
15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)*cos(d*x + c) - (a^21 - 30*a^19*b^2 + 249*a^17*b^4 - 280*a^15*b^6 - 1038*a
^13*b^8 + 732*a^11*b^10 + 1322*a^9*b^12 - 504*a^7*b^14 - 531*a^5*b^16 + 82*a^3*b^18 - 3*a*b^20)*d*cos(d*x + c)
)*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12 + 2*(3*a^5*b - 10*a^3*b^3
+ 3*a*b^5)*d^2*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4))/(a^1
2 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12))*sqrt(sin(d*x + c)/cos(d*x + c
))*((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)^(1/4) + (a^24 - 24*a^2
2*b^2 + 90*a^20*b^4 + 648*a^18*b^6 + 783*a^16*b^8 - 624*a^14*b^10 - 1748*a^12*b^12 - 624*a^10*b^14 + 783*a^8*b
^16 + 648*a^6*b^18 + 90*a^4*b^20 - 24*a^2*b^22 + b^24)*sin(d*x + c))/cos(d*x + c))*((a^12 + 6*a^10*b^2 + 15*a^
8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)^(3/4) - sqrt(2)*((a^15 - 15*a^13*b^2 + 9*a^11*b^4 +
81*a^9*b^6 + 27*a^7*b^8 - 69*a^5*b^10 - 37*a^3*b^12 + 3*a*b^14)*d^7*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*
a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*
b^8 - 30*a^2*b^10 + b^12)/d^4) - (3*a^20*b - 28*a^18*b^3 - 171*a^16*b^5 - 288*a^14*b^7 - 82*a^12*b^9 + 264*a^1
0*b^11 + 282*a^8*b^13 + 64*a^6*b^15 - 33*a^4*b^17 - 12*a^2*b^19 + b^21)*d^5*sqrt((a^12 - 30*a^10*b^2 + 255*a^8
*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6
+ 15*a^4*b^8 + 6*a^2*b^10 + b^12 + 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*d^2*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^
4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4))/(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a
^4*b^8 - 30*a^2*b^10 + b^12))*sqrt(sin(d*x + c)/cos(d*x + c))*((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 +
15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)^(3/4))/(a^36 - 18*a^34*b^2 - 39*a^32*b^4 + 848*a^30*b^6 + 5556*a^28*b^8 +
15240*a^26*b^10 + 20420*a^24*b^12 + 5424*a^22*b^14 - 25938*a^20*b^16 - 42988*a^18*b^18 - 25938*a^16*b^20 + 54
24*a^14*b^22 + 20420*a^12*b^24 + 15240*a^10*b^26 + 5556*a^8*b^28 + 848*a^6*b^30 - 39*a^4*b^32 - 18*a^2*b^34 +
b^36)) + 420*sqrt(2)*(d^5*cos(d*x + c)^4 - 2*d^5*cos(d*x + c)^2 + d^5)*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 +
20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12 + 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*d^2*sqrt((a^12 + 6*a^10*b^2 +
15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4))/(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b
^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12))*((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^1
0 + b^12)/d^4)^(3/4)*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/
d^4)*arctan(-((a^24 - 6*a^22*b^2 - 84*a^20*b^4 - 322*a^18*b^6 - 603*a^16*b^8 - 540*a^14*b^10 + 540*a^10*b^14 +
603*a^8*b^16 + 322*a^6*b^18 + 84*a^4*b^20 + 6*a^2*b^22 - b^24)*d^4*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*
a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*
b^8 - 30*a^2*b^10 + b^12)/d^4) + sqrt(2)*((a^3 - 3*a*b^2)*d^7*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^
6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 -
30*a^2*b^10 + b^12)/d^4) - (3*a^8*b + 8*a^6*b^3 + 6*a^4*b^5 - b^9)*d^5*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4
- 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15
*a^4*b^8 + 6*a^2*b^10 + b^12 + 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*d^2*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 2
0*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4))/(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^
8 - 30*a^2*b^10 + b^12))*sqrt(((a^18 - 27*a^16*b^2 + 168*a^14*b^4 + 224*a^12*b^6 - 366*a^10*b^8 - 366*a^8*b^10
+ 224*a^6*b^12 + 168*a^4*b^14 - 27*a^2*b^16 + b^18)*d^2*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 1
5*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)*cos(d*x + c) - sqrt(2)*((3*a^14*b - 91*a^12*b^3 + 795*a^10*b^5 - 1611*a^8*
b^7 + 1217*a^6*b^9 - 345*a^4*b^11 + 33*a^2*b^13 - b^15)*d^3*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6
+ 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)*cos(d*x + c) - (a^21 - 30*a^19*b^2 + 249*a^17*b^4 - 280*a^15*b^6 - 1038
*a^13*b^8 + 732*a^11*b^10 + 1322*a^9*b^12 - 504*a^7*b^14 - 531*a^5*b^16 + 82*a^3*b^18 - 3*a*b^20)*d*cos(d*x +
c))*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12 + 2*(3*a^5*b - 10*a^3*b
^3 + 3*a*b^5)*d^2*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4))/(a
^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12))*sqrt(sin(d*x + c)/cos(d*x +
c))*((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)^(1/4) + (a^24 - 24*a
^22*b^2 + 90*a^20*b^4 + 648*a^18*b^6 + 783*a^16*b^8 - 624*a^14*b^10 - 1748*a^12*b^12 - 624*a^10*b^14 + 783*a^8
*b^16 + 648*a^6*b^18 + 90*a^4*b^20 - 24*a^2*b^22 + b^24)*sin(d*x + c))/cos(d*x + c))*((a^12 + 6*a^10*b^2 + 15*
a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)^(3/4) + sqrt(2)*((a^15 - 15*a^13*b^2 + 9*a^11*b^4
+ 81*a^9*b^6 + 27*a^7*b^8 - 69*a^5*b^10 - 37*a^3*b^12 + 3*a*b^14)*d^7*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 2
0*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^
4*b^8 - 30*a^2*b^10 + b^12)/d^4) - (3*a^20*b - 28*a^18*b^3 - 171*a^16*b^5 - 288*a^14*b^7 - 82*a^12*b^9 + 264*a
^10*b^11 + 282*a^8*b^13 + 64*a^6*b^15 - 33*a^4*b^17 - 12*a^2*b^19 + b^21)*d^5*sqrt((a^12 - 30*a^10*b^2 + 255*a
^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b
^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12 + 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*d^2*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*
b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4))/(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255
*a^4*b^8 - 30*a^2*b^10 + b^12))*sqrt(sin(d*x + c)/cos(d*x + c))*((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6
+ 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)^(3/4))/(a^36 - 18*a^34*b^2 - 39*a^32*b^4 + 848*a^30*b^6 + 5556*a^28*b^8
+ 15240*a^26*b^10 + 20420*a^24*b^12 + 5424*a^22*b^14 - 25938*a^20*b^16 - 42988*a^18*b^18 - 25938*a^16*b^20 +
5424*a^14*b^22 + 20420*a^12*b^24 + 15240*a^10*b^26 + 5556*a^8*b^28 + 848*a^6*b^30 - 39*a^4*b^32 - 18*a^2*b^34
+ b^36)) + 105*sqrt(2)*((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d*cos(d
*x + c)^4 - 2*(a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d*cos(d*x + c)^2
+ (a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d - 2*((3*a^5*b - 10*a^3*b^3
+ 3*a*b^5)*d^3*cos(d*x + c)^4 - 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*d^3*cos(d*x + c)^2 + (3*a^5*b - 10*a^3*b^3
+ 3*a*b^5)*d^3)*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4))*sqrt
((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12 + 2*(3*a^5*b - 10*a^3*b^3 + 3*a
*b^5)*d^2*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4))/(a^12 - 30
*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12))*((a^12 + 6*a^10*b^2 + 15*a^8*b^4 +
20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)^(1/4)*log(((a^18 - 27*a^16*b^2 + 168*a^14*b^4 + 224*a^12*b^6
- 366*a^10*b^8 - 366*a^8*b^10 + 224*a^6*b^12 + 168*a^4*b^14 - 27*a^2*b^16 + b^18)*d^2*sqrt((a^12 + 6*a^10*b^2
+ 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)*cos(d*x + c) + sqrt(2)*((3*a^14*b - 91*a^12*
b^3 + 795*a^10*b^5 - 1611*a^8*b^7 + 1217*a^6*b^9 - 345*a^4*b^11 + 33*a^2*b^13 - b^15)*d^3*sqrt((a^12 + 6*a^10*
b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)*cos(d*x + c) - (a^21 - 30*a^19*b^2 + 249*
a^17*b^4 - 280*a^15*b^6 - 1038*a^13*b^8 + 732*a^11*b^10 + 1322*a^9*b^12 - 504*a^7*b^14 - 531*a^5*b^16 + 82*a^3
*b^18 - 3*a*b^20)*d*cos(d*x + c))*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10
+ b^12 + 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*d^2*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8
+ 6*a^2*b^10 + b^12)/d^4))/(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12
))*sqrt(sin(d*x + c)/cos(d*x + c))*((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b
^12)/d^4)^(1/4) + (a^24 - 24*a^22*b^2 + 90*a^20*b^4 + 648*a^18*b^6 + 783*a^16*b^8 - 624*a^14*b^10 - 1748*a^12*
b^12 - 624*a^10*b^14 + 783*a^8*b^16 + 648*a^6*b^18 + 90*a^4*b^20 - 24*a^2*b^22 + b^24)*sin(d*x + c))/cos(d*x +
c)) - 105*sqrt(2)*((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d*cos(d*x +
c)^4 - 2*(a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d*cos(d*x + c)^2 + (a
^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d - 2*((3*a^5*b - 10*a^3*b^3 + 3*
a*b^5)*d^3*cos(d*x + c)^4 - 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*d^3*cos(d*x + c)^2 + (3*a^5*b - 10*a^3*b^3 + 3*
a*b^5)*d^3)*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4))*sqrt((a^
12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12 + 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5
)*d^2*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4))/(a^12 - 30*a^1
0*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12))*((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a
^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)^(1/4)*log(((a^18 - 27*a^16*b^2 + 168*a^14*b^4 + 224*a^12*b^6 - 3
66*a^10*b^8 - 366*a^8*b^10 + 224*a^6*b^12 + 168*a^4*b^14 - 27*a^2*b^16 + b^18)*d^2*sqrt((a^12 + 6*a^10*b^2 + 1
5*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)*cos(d*x + c) - sqrt(2)*((3*a^14*b - 91*a^12*b^3
+ 795*a^10*b^5 - 1611*a^8*b^7 + 1217*a^6*b^9 - 345*a^4*b^11 + 33*a^2*b^13 - b^15)*d^3*sqrt((a^12 + 6*a^10*b^2
+ 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)/d^4)*cos(d*x + c) - (a^21 - 30*a^19*b^2 + 249*a^17
*b^4 - 280*a^15*b^6 - 1038*a^13*b^8 + 732*a^11*b^10 + 1322*a^9*b^12 - 504*a^7*b^14 - 531*a^5*b^16 + 82*a^3*b^1
8 - 3*a*b^20)*d*cos(d*x + c))*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^
12 + 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*d^2*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6
*a^2*b^10 + b^12)/d^4))/(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12))*s
qrt(sin(d*x + c)/cos(d*x + c))*((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)
/d^4)^(1/4) + (a^24 - 24*a^22*b^2 + 90*a^20*b^4 + 648*a^18*b^6 + 783*a^16*b^8 - 624*a^14*b^10 - 1748*a^12*b^12
- 624*a^10*b^14 + 783*a^8*b^16 + 648*a^6*b^18 + 90*a^4*b^20 - 24*a^2*b^22 + b^24)*sin(d*x + c))/cos(d*x + c))
+ 8*(5*(10*a^15 + 39*a^13*b^2 + 24*a^11*b^4 - 115*a^9*b^6 - 270*a^7*b^8 - 255*a^5*b^10 - 116*a^3*b^12 - 21*a*
b^14)*cos(d*x + c)^4 - 35*(a^15 + 3*a^13*b^2 - 3*a^11*b^4 - 25*a^9*b^6 - 45*a^7*b^8 - 39*a^5*b^10 - 17*a^3*b^1
2 - 3*a*b^14)*cos(d*x + c)^2 + 21*((18*a^14*b + 103*a^12*b^3 + 240*a^10*b^5 + 285*a^8*b^7 + 170*a^6*b^9 + 33*a
^4*b^11 - 12*a^2*b^13 - 5*b^15)*cos(d*x + c)^3 - 5*(3*a^14*b + 17*a^12*b^3 + 39*a^10*b^5 + 45*a^8*b^7 + 25*a^6
*b^9 + 3*a^4*b^11 - 3*a^2*b^13 - b^15)*cos(d*x + c))*sin(d*x + c))*sqrt(sin(d*x + c)/cos(d*x + c)))/((a^12 + 6
*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d*cos(d*x + c)^4 - 2*(a^12 + 6*a^10*b^2
+ 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d*cos(d*x + c)^2 + (a^12 + 6*a^10*b^2 + 15*a^8*b^4
+ 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*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((a+b*tan(d*x+c))**3/tan(d*x+c)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.98245, size = 436, normalized size = 1.46 \begin{align*} \frac{{\left (\sqrt{2} a^{3} + 3 \, \sqrt{2} a^{2} b - 3 \, \sqrt{2} a b^{2} - \sqrt{2} b^{3}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right )}{2 \, d} + \frac{{\left (\sqrt{2} a^{3} + 3 \, \sqrt{2} a^{2} b - 3 \, \sqrt{2} a b^{2} - \sqrt{2} b^{3}\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right )}{2 \, d} + \frac{{\left (\sqrt{2} a^{3} - 3 \, \sqrt{2} a^{2} b - 3 \, \sqrt{2} a b^{2} + \sqrt{2} b^{3}\right )} \log \left (\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{4 \, d} - \frac{{\left (\sqrt{2} a^{3} - 3 \, \sqrt{2} a^{2} b - 3 \, \sqrt{2} a b^{2} + \sqrt{2} b^{3}\right )} \log \left (-\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{4 \, d} + \frac{2 \,{\left (315 \, a^{2} b \tan \left (d x + c\right )^{3} - 105 \, b^{3} \tan \left (d x + c\right )^{3} + 35 \, a^{3} \tan \left (d x + c\right )^{2} - 105 \, a b^{2} \tan \left (d x + c\right )^{2} - 63 \, a^{2} b \tan \left (d x + c\right ) - 15 \, a^{3}\right )}}{105 \, d \tan \left (d x + c\right )^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(d*x+c))^3/tan(d*x+c)^(9/2),x, algorithm="giac")

[Out]

1/2*(sqrt(2)*a^3 + 3*sqrt(2)*a^2*b - 3*sqrt(2)*a*b^2 - sqrt(2)*b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d
*x + c))))/d + 1/2*(sqrt(2)*a^3 + 3*sqrt(2)*a^2*b - 3*sqrt(2)*a*b^2 - sqrt(2)*b^3)*arctan(-1/2*sqrt(2)*(sqrt(2
) - 2*sqrt(tan(d*x + c))))/d + 1/4*(sqrt(2)*a^3 - 3*sqrt(2)*a^2*b - 3*sqrt(2)*a*b^2 + sqrt(2)*b^3)*log(sqrt(2)
*sqrt(tan(d*x + c)) + tan(d*x + c) + 1)/d - 1/4*(sqrt(2)*a^3 - 3*sqrt(2)*a^2*b - 3*sqrt(2)*a*b^2 + sqrt(2)*b^3
)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1)/d + 2/105*(315*a^2*b*tan(d*x + c)^3 - 105*b^3*tan(d*x +
c)^3 + 35*a^3*tan(d*x + c)^2 - 105*a*b^2*tan(d*x + c)^2 - 63*a^2*b*tan(d*x + c) - 15*a^3)/(d*tan(d*x + c)^(7/2
))