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

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

[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) + (2*b*(3*a^2 - b^2)*Sqrt[Cot[c + d*x]])/d + (2*a*(a^2 -
3*b^2)*Cot[c + d*x]^(3/2))/(3*d) - (32*a^2*b*Cot[c + d*x]^(5/2))/(35*d) - (2*a^2*Cot[c + d*x]^(5/2)*(b + a*Co
t[c + d*x]))/(7*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) + ((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 = 0.438223, 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, 3566, 3630, 3528, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac{2 a \left (a^2-3 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 b \left (3 a^2-b^2\right ) \sqrt{\cot (c+d x)}}{d}-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}-\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d}-\frac{2 a^2 \cot ^{\frac{5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}-\frac{32 a^2 b \cot ^{\frac{5}{2}}(c+d x)}{35 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^3,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) + (2*b*(3*a^2 - b^2)*Sqrt[Cot[c + d*x]])/d + (2*a*(a^2 -
3*b^2)*Cot[c + d*x]^(3/2))/(3*d) - (32*a^2*b*Cot[c + d*x]^(5/2))/(35*d) - (2*a^2*Cot[c + d*x]^(5/2)*(b + a*Co
t[c + d*x]))/(7*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) + ((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 3566

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n - 1)), x] + Dist[1/(d*(m + n -
1)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0]
&& NeQ[a, 0])))

Rule 3630

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

Rule 3528

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

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 \cot ^{\frac{9}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx &=\int \cot ^{\frac{3}{2}}(c+d x) (b+a \cot (c+d x))^3 \, dx\\ &=-\frac{2 a^2 \cot ^{\frac{5}{2}}(c+d x) (b+a \cot (c+d x))}{7 d}-\frac{2}{7} \int \cot ^{\frac{3}{2}}(c+d x) \left (\frac{1}{2} b \left (5 a^2-7 b^2\right )+\frac{7}{2} a \left (a^2-3 b^2\right ) \cot (c+d x)-8 a^2 b \cot ^2(c+d x)\right ) \, dx\\ &=-\frac{32 a^2 b \cot ^{\frac{5}{2}}(c+d x)}{35 d}-\frac{2 a^2 \cot ^{\frac{5}{2}}(c+d x) (b+a \cot (c+d x))}{7 d}-\frac{2}{7} \int \cot ^{\frac{3}{2}}(c+d x) \left (\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)\right ) \, dx\\ &=\frac{2 a \left (a^2-3 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{32 a^2 b \cot ^{\frac{5}{2}}(c+d x)}{35 d}-\frac{2 a^2 \cot ^{\frac{5}{2}}(c+d x) (b+a \cot (c+d x))}{7 d}-\frac{2}{7} \int \sqrt{\cot (c+d x)} \left (-\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)\right ) \, dx\\ &=\frac{2 b \left (3 a^2-b^2\right ) \sqrt{\cot (c+d x)}}{d}+\frac{2 a \left (a^2-3 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{32 a^2 b \cot ^{\frac{5}{2}}(c+d x)}{35 d}-\frac{2 a^2 \cot ^{\frac{5}{2}}(c+d x) (b+a \cot (c+d x))}{7 d}-\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{2 b \left (3 a^2-b^2\right ) \sqrt{\cot (c+d x)}}{d}+\frac{2 a \left (a^2-3 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{32 a^2 b \cot ^{\frac{5}{2}}(c+d x)}{35 d}-\frac{2 a^2 \cot ^{\frac{5}{2}}(c+d x) (b+a \cot (c+d x))}{7 d}-\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{2 b \left (3 a^2-b^2\right ) \sqrt{\cot (c+d x)}}{d}+\frac{2 a \left (a^2-3 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{32 a^2 b \cot ^{\frac{5}{2}}(c+d x)}{35 d}-\frac{2 a^2 \cot ^{\frac{5}{2}}(c+d x) (b+a \cot (c+d x))}{7 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{\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{2 b \left (3 a^2-b^2\right ) \sqrt{\cot (c+d x)}}{d}+\frac{2 a \left (a^2-3 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{32 a^2 b \cot ^{\frac{5}{2}}(c+d x)}{35 d}-\frac{2 a^2 \cot ^{\frac{5}{2}}(c+d x) (b+a \cot (c+d x))}{7 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{\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{2 b \left (3 a^2-b^2\right ) \sqrt{\cot (c+d x)}}{d}+\frac{2 a \left (a^2-3 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{32 a^2 b \cot ^{\frac{5}{2}}(c+d x)}{35 d}-\frac{2 a^2 \cot ^{\frac{5}{2}}(c+d x) (b+a \cot (c+d x))}{7 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{(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{2 b \left (3 a^2-b^2\right ) \sqrt{\cot (c+d x)}}{d}+\frac{2 a \left (a^2-3 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{32 a^2 b \cot ^{\frac{5}{2}}(c+d x)}{35 d}-\frac{2 a^2 \cot ^{\frac{5}{2}}(c+d x) (b+a \cot (c+d x))}{7 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{(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 = 2.12684, size = 229, normalized size = 0.77 \[ -\frac{\frac{2}{3} a \left (a^2-3 b^2\right ) \cot ^{\frac{3}{2}}(c+d x) \left (\, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right )-1\right )+\frac{1}{4} b \left (b^2-3 a^2\right ) \left (8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-\sqrt{2} \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )+\frac{6}{5} a^2 b \cot ^{\frac{5}{2}}(c+d x)+\frac{2}{7} a^3 \cot ^{\frac{7}{2}}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((6*a^2*b*Cot[c + d*x]^(5/2))/5 + (2*a^3*Cot[c + d*x]^(7/2))/7 + (2*a*(a^2 - 3*b^2)*Cot[c + d*x]^(3/2)*(-1 +
Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/3 + (b*(-3*a^2 + b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Co
t[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[
2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/4)/d)

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Maple [C]  time = 0.55, size = 9273, normalized size = 31. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [A]  time = 1.61052, size = 354, normalized size = 1.18 \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} + \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 ) + \frac{504 \, a^{2} b}{\tan \left (d x + c\right )^{\frac{5}{2}}} - \frac{840 \,{\left (3 \, a^{2} b - b^{3}\right )}}{\sqrt{\tan \left (d x + c\right )}} + \frac{120 \, a^{3}}{\tan \left (d x + c\right )^{\frac{7}{2}}} - \frac{280 \,{\left (a^{3} - 3 \, a b^{2}\right )}}{\tan \left (d x + c\right )^{\frac{3}{2}}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^3,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)) + 1/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)) + 1/tan(d*x + c) + 1) + 504*a^2*b/tan(d*x + c)^(5/2) - 840
*(3*a^2*b - b^3)/sqrt(tan(d*x + c)) + 120*a^3/tan(d*x + c)^(7/2) - 280*(a^3 - 3*a*b^2)/tan(d*x + c)^(3/2))/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(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac{9}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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