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

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

[Out]

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

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Rubi [A]  time = 0.442668, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {3673, 3565, 3628, 3529, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac{2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left (a^2-3 b^2\right )}{d \sqrt{\cot (c+d x)}}+\frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}-\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}+\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d}+\frac{2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac{7}{2}}(c+d x)}+\frac{32 a b^2}{35 d \cot ^{\frac{5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3673

Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_.))^(p_.), x_Symbol] :> Dist
[d^(n*p), Int[(d*Cot[e + f*x])^(m - n*p)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x
] &&  !IntegerQ[m] && IntegersQ[n, p]

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

Mathematica [C]  time = 0.613073, size = 101, normalized size = 0.34 \[ \frac{2 \left (5 b \left (b^2-3 a^2\right ) \, _2F_1\left (-\frac{7}{4},1;-\frac{3}{4};-\cot ^2(c+d x)\right )+a \left (a (7 a \cot (c+d x)+15 b)-7 \left (a^2-3 b^2\right ) \cot (c+d x) \, _2F_1\left (-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(c+d x)\right )\right )\right )}{35 d \cot ^{\frac{7}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.384, size = 2598, normalized size = 8.7 \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/cot(d*x+c)^(3/2),x)

[Out]

1/210/d*2^(1/2)*(cos(d*x+c)-1)*(210*2^(1/2)*cos(d*x+c)^4*a^3-126*cos(d*x+c)*2^(1/2)*a*b^2+105*((cos(d*x+c)-1)/
sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-(cos(d*x+c
)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*b^3+210
*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)*a^2*b-315*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*
x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+si
n(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*a*b^2+100*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)*b^3+630*((cos(d*
x+c)-1)/sin(d*x+c))^(1/2)*EllipticF((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))*(-(cos(d*x+c)-1
-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*a*b^2-315*
((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/
2))*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^3*si
n(d*x+c)*a^2*b-315*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),
1/2-1/2*I,1/2*2^(1/2))*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1
/2)*cos(d*x+c)^3*sin(d*x+c)*a*b^2-315*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))
/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x
+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*a^2*b+756*cos(d*x+c)^3*2^(1/2)*a*b^2-30*sin(d*x+c)*2^(1/2)*b^3-
210*cos(d*x+c)^2*sin(d*x+c)*2^(1/2)*a^2*b-210*cos(d*x+c)^3*2^(1/2)*a^3-756*2^(1/2)*cos(d*x+c)^4*a*b^2-100*2^(1
/2)*cos(d*x+c)^3*sin(d*x+c)*b^3+126*2^(1/2)*cos(d*x+c)^2*a*b^2+30*2^(1/2)*cos(d*x+c)*sin(d*x+c)*b^3+315*I*((co
s(d*x+c)-1)/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x
+c))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1
/2))*a*b^2+315*I*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-
1+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/
2),1/2-1/2*I,1/2*2^(1/2))*a^2*b-315*I*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c)
)^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*EllipticPi((-(cos(d*x+c)-1-sin(d*
x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*a*b^2-315*I*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-s
in(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*EllipticPi((
-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*a^2*b+105*((cos(d*x+c)-1)/sin(d*x+c))^(1/2
)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-(cos(d*x+c)-1-sin(d*x+c))/
sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*a^3+105*((cos(d*x+c)-1)
/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-(cos(d*x+
c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*b^3+10
5*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(
1/2))*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^3*
sin(d*x+c)*a^3-210*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticF((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1
/2*2^(1/2))*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x
+c)^3*sin(d*x+c)*a^3+105*I*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((c
os(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d
*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*a^3-105*I*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/si
n(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*EllipticPi((-(cos(d*x+c)-
1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*b^3-105*I*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*(-(cos(d*x+
c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*Ellipt
icPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*a^3+105*I*((cos(d*x+c)-1)/sin(d*x+c)
)^(1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^
3*sin(d*x+c)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*b^3)*(cos(d*x+c)+
1)^2/cos(d*x+c)^2/sin(d*x+c)^5/(cos(d*x+c)/sin(d*x+c))^(3/2)

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Maxima [A]  time = 1.71329, size = 358, normalized size = 1.2 \begin{align*} \frac{8 \,{\left (15 \, b^{3} + \frac{63 \, a b^{2}}{\tan \left (d x + c\right )} + \frac{35 \,{\left (3 \, a^{2} b - b^{3}\right )}}{\tan \left (d x + c\right )^{2}} + \frac{105 \,{\left (a^{3} - 3 \, a b^{2}\right )}}{\tan \left (d x + c\right )^{3}}\right )} \tan \left (d x + c\right )^{\frac{7}{2}} + 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} + \frac{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} - \frac{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 (\frac{\sqrt{2}}{\sqrt{\tan \left (d x + c\right )}} + \frac{1}{\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 (-\frac{\sqrt{2}}{\sqrt{\tan \left (d x + c\right )}} + \frac{1}{\tan \left (d x + c\right )} + 1\right )}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [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/cot(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \tan{\left (c + d x \right )}\right )^{3}}{\cot ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*tan(c + d*x))**3/cot(c + d*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \tan \left (d x + c\right ) + a\right )}^{3}}{\cot \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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