3.15 \(\int (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3 \, dx\)

Optimal. Leaf size=186 \[ \frac{4 a^3 e^2 \sqrt{e \cot (c+d x)}}{d}+\frac{2 \sqrt{2} a^3 e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d}-\frac{2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{7/2}}{9 d e}-\frac{40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac{4 a^3 (e \cot (c+d x))^{5/2}}{5 d}+\frac{4 a^3 e (e \cot (c+d x))^{3/2}}{3 d} \]

[Out]

(2*Sqrt[2]*a^3*e^(5/2)*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d + (4*a^3*e^2
*Sqrt[e*Cot[c + d*x]])/d + (4*a^3*e*(e*Cot[c + d*x])^(3/2))/(3*d) - (4*a^3*(e*Cot[c + d*x])^(5/2))/(5*d) - (40
*a^3*(e*Cot[c + d*x])^(7/2))/(63*d*e) - (2*(e*Cot[c + d*x])^(7/2)*(a^3 + a^3*Cot[c + d*x]))/(9*d*e)

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Rubi [A]  time = 0.300442, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3566, 3630, 3528, 3532, 205} \[ \frac{4 a^3 e^2 \sqrt{e \cot (c+d x)}}{d}+\frac{2 \sqrt{2} a^3 e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d}-\frac{2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{7/2}}{9 d e}-\frac{40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac{4 a^3 (e \cot (c+d x))^{5/2}}{5 d}+\frac{4 a^3 e (e \cot (c+d x))^{3/2}}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3,x]

[Out]

(2*Sqrt[2]*a^3*e^(5/2)*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d + (4*a^3*e^2
*Sqrt[e*Cot[c + d*x]])/d + (4*a^3*e*(e*Cot[c + d*x])^(3/2))/(3*d) - (4*a^3*(e*Cot[c + d*x])^(5/2))/(5*d) - (40
*a^3*(e*Cot[c + d*x])^(7/2))/(63*d*e) - (2*(e*Cot[c + d*x])^(7/2)*(a^3 + a^3*Cot[c + d*x]))/(9*d*e)

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 3532

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(-2*d^2)/f,
Subst[Int[1/(2*c*d + b*x^2), x], x, (c - d*Tan[e + f*x])/Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x
] && EqQ[c^2 - d^2, 0]

Rule 205

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

Rubi steps

\begin{align*} \int (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3 \, dx &=-\frac{2 (e \cot (c+d x))^{7/2} \left (a^3+a^3 \cot (c+d x)\right )}{9 d e}-\frac{2 \int (e \cot (c+d x))^{5/2} \left (-a^3 e-9 a^3 e \cot (c+d x)-10 a^3 e \cot ^2(c+d x)\right ) \, dx}{9 e}\\ &=-\frac{40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac{2 (e \cot (c+d x))^{7/2} \left (a^3+a^3 \cot (c+d x)\right )}{9 d e}-\frac{2 \int (e \cot (c+d x))^{5/2} \left (9 a^3 e-9 a^3 e \cot (c+d x)\right ) \, dx}{9 e}\\ &=-\frac{4 a^3 (e \cot (c+d x))^{5/2}}{5 d}-\frac{40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac{2 (e \cot (c+d x))^{7/2} \left (a^3+a^3 \cot (c+d x)\right )}{9 d e}-\frac{2 \int (e \cot (c+d x))^{3/2} \left (9 a^3 e^2+9 a^3 e^2 \cot (c+d x)\right ) \, dx}{9 e}\\ &=\frac{4 a^3 e (e \cot (c+d x))^{3/2}}{3 d}-\frac{4 a^3 (e \cot (c+d x))^{5/2}}{5 d}-\frac{40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac{2 (e \cot (c+d x))^{7/2} \left (a^3+a^3 \cot (c+d x)\right )}{9 d e}-\frac{2 \int \sqrt{e \cot (c+d x)} \left (-9 a^3 e^3+9 a^3 e^3 \cot (c+d x)\right ) \, dx}{9 e}\\ &=\frac{4 a^3 e^2 \sqrt{e \cot (c+d x)}}{d}+\frac{4 a^3 e (e \cot (c+d x))^{3/2}}{3 d}-\frac{4 a^3 (e \cot (c+d x))^{5/2}}{5 d}-\frac{40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac{2 (e \cot (c+d x))^{7/2} \left (a^3+a^3 \cot (c+d x)\right )}{9 d e}-\frac{2 \int \frac{-9 a^3 e^4-9 a^3 e^4 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{9 e}\\ &=\frac{4 a^3 e^2 \sqrt{e \cot (c+d x)}}{d}+\frac{4 a^3 e (e \cot (c+d x))^{3/2}}{3 d}-\frac{4 a^3 (e \cot (c+d x))^{5/2}}{5 d}-\frac{40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac{2 (e \cot (c+d x))^{7/2} \left (a^3+a^3 \cot (c+d x)\right )}{9 d e}+\frac{\left (36 a^6 e^7\right ) \operatorname{Subst}\left (\int \frac{1}{-162 a^6 e^8-e x^2} \, dx,x,\frac{-9 a^3 e^4+9 a^3 e^4 \cot (c+d x)}{\sqrt{e \cot (c+d x)}}\right )}{d}\\ &=\frac{2 \sqrt{2} a^3 e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d}+\frac{4 a^3 e^2 \sqrt{e \cot (c+d x)}}{d}+\frac{4 a^3 e (e \cot (c+d x))^{3/2}}{3 d}-\frac{4 a^3 (e \cot (c+d x))^{5/2}}{5 d}-\frac{40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac{2 (e \cot (c+d x))^{7/2} \left (a^3+a^3 \cot (c+d x)\right )}{9 d e}\\ \end{align*}

Mathematica [C]  time = 6.09588, size = 729, normalized size = 3.92 \[ -\frac{4 \sin ^3(c+d x) \tan (c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2} \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(c+d x)\right )}{3 d (\sin (c+d x)+\cos (c+d x))^3}-\frac{2 \sin (c+d x) \cos ^2(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2}}{9 d (\sin (c+d x)+\cos (c+d x))^3}-\frac{4 \sin ^3(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2}}{5 d (\sin (c+d x)+\cos (c+d x))^3}-\frac{6 \sin ^2(c+d x) \cos (c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2}}{7 d (\sin (c+d x)+\cos (c+d x))^3}+\frac{\sin ^3(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2} \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d \cot ^{\frac{5}{2}}(c+d x) (\sin (c+d x)+\cos (c+d x))^3}-\frac{\sin ^3(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2} \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d \cot ^{\frac{5}{2}}(c+d x) (\sin (c+d x)+\cos (c+d x))^3}+\frac{\sqrt{2} \sin ^3(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{d \cot ^{\frac{5}{2}}(c+d x) (\sin (c+d x)+\cos (c+d x))^3}-\frac{\sqrt{2} \sin ^3(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{d \cot ^{\frac{5}{2}}(c+d x) (\sin (c+d x)+\cos (c+d x))^3}+\frac{4 \sin ^3(c+d x) \tan ^2(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2}}{d (\sin (c+d x)+\cos (c+d x))^3}+\frac{4 \sin ^3(c+d x) \tan (c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2}}{3 d (\sin (c+d x)+\cos (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3,x]

[Out]

(-2*Cos[c + d*x]^2*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Sin[c + d*x])/(9*d*(Cos[c + d*x] + Sin[c + d*
x])^3) - (6*Cos[c + d*x]*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Sin[c + d*x]^2)/(7*d*(Cos[c + d*x] + Si
n[c + d*x])^3) - (4*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Sin[c + d*x]^3)/(5*d*(Cos[c + d*x] + Sin[c +
d*x])^3) + (Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Sin[
c + d*x]^3)/(d*Cot[c + d*x]^(5/2)*(Cos[c + d*x] + Sin[c + d*x])^3) - (Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c +
d*x]]]*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Sin[c + d*x]^3)/(d*Cot[c + d*x]^(5/2)*(Cos[c + d*x] + Sin
[c + d*x])^3) + ((e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*
x]]*Sin[c + d*x]^3)/(Sqrt[2]*d*Cot[c + d*x]^(5/2)*(Cos[c + d*x] + Sin[c + d*x])^3) - ((e*Cot[c + d*x])^(5/2)*(
a + a*Cot[c + d*x])^3*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^3)/(Sqrt[2]*d*Cot[c + d*
x]^(5/2)*(Cos[c + d*x] + Sin[c + d*x])^3) + (4*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Sin[c + d*x]^3*Ta
n[c + d*x])/(3*d*(Cos[c + d*x] + Sin[c + d*x])^3) - (4*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Hypergeom
etric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]*Sin[c + d*x]^3*Tan[c + d*x])/(3*d*(Cos[c + d*x] + Sin[c + d*x])^3) + (4
*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Sin[c + d*x]^3*Tan[c + d*x]^2)/(d*(Cos[c + d*x] + Sin[c + d*x])
^3)

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Maple [B]  time = 0.025, size = 446, normalized size = 2.4 \begin{align*} -{\frac{2\,{a}^{3}}{9\,d{e}^{2}} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{9}{2}}}}-{\frac{6\,{a}^{3}}{7\,de} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{7}{2}}}}-{\frac{4\,{a}^{3}}{5\,d} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{4\,{a}^{3}e}{3\,d} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}+4\,{\frac{{a}^{3}{e}^{2}\sqrt{e\cot \left ( dx+c \right ) }}{d}}-{\frac{{a}^{3}{e}^{2}\sqrt{2}}{2\,d}\sqrt [4]{{e}^{2}}\ln \left ({ \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ) }-{\frac{{a}^{3}{e}^{2}\sqrt{2}}{d}\sqrt [4]{{e}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }+{\frac{{a}^{3}{e}^{2}\sqrt{2}}{d}\sqrt [4]{{e}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }-{\frac{{a}^{3}{e}^{3}\sqrt{2}}{2\,d}\ln \left ({ \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}-{\frac{{a}^{3}{e}^{3}\sqrt{2}}{d}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}+{\frac{{a}^{3}{e}^{3}\sqrt{2}}{d}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cot(d*x+c))^(5/2)*(a+a*cot(d*x+c))^3,x)

[Out]

-2/9/d*a^3/e^2*(e*cot(d*x+c))^(9/2)-6/7*a^3*(e*cot(d*x+c))^(7/2)/d/e-4/5*a^3*(e*cot(d*x+c))^(5/2)/d+4/3*a^3*e*
(e*cot(d*x+c))^(3/2)/d+4*a^3*e^2*(e*cot(d*x+c))^(1/2)/d-1/2/d*a^3*e^2*(e^2)^(1/4)*2^(1/2)*ln((e*cot(d*x+c)+(e^
2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2
)^(1/2)))-1/d*a^3*e^2*(e^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)+1/d*a^3*e^2*(e^2)
^(1/4)*2^(1/2)*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-1/2/d*a^3*e^3/(e^2)^(1/4)*2^(1/2)*ln((e*cot
(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2
^(1/2)+(e^2)^(1/2)))-1/d*a^3*e^3/(e^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)+1/d*a^
3*e^3/(e^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)

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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((e*cot(d*x+c))^(5/2)*(a+a*cot(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.85677, size = 1311, normalized size = 7.05 \begin{align*} \left [\frac{315 \, \sqrt{2}{\left (a^{3} e^{2} \cos \left (2 \, d x + 2 \, c\right )^{2} - 2 \, a^{3} e^{2} \cos \left (2 \, d x + 2 \, c\right ) + a^{3} e^{2}\right )} \sqrt{-e} \log \left (-\sqrt{2} \sqrt{-e} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}{\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) + 2 \,{\left (721 \, a^{3} e^{2} \cos \left (2 \, d x + 2 \, c\right )^{2} - 1330 \, a^{3} e^{2} \cos \left (2 \, d x + 2 \, c\right ) + 469 \, a^{3} e^{2} - 15 \,{\left (23 \, a^{3} e^{2} \cos \left (2 \, d x + 2 \, c\right ) - 5 \, a^{3} e^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{315 \,{\left (d \cos \left (2 \, d x + 2 \, c\right )^{2} - 2 \, d \cos \left (2 \, d x + 2 \, c\right ) + d\right )}}, \frac{2 \,{\left (315 \, \sqrt{2}{\left (a^{3} e^{2} \cos \left (2 \, d x + 2 \, c\right )^{2} - 2 \, a^{3} e^{2} \cos \left (2 \, d x + 2 \, c\right ) + a^{3} e^{2}\right )} \sqrt{e} \arctan \left (-\frac{\sqrt{2} \sqrt{e} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}{\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \,{\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) +{\left (721 \, a^{3} e^{2} \cos \left (2 \, d x + 2 \, c\right )^{2} - 1330 \, a^{3} e^{2} \cos \left (2 \, d x + 2 \, c\right ) + 469 \, a^{3} e^{2} - 15 \,{\left (23 \, a^{3} e^{2} \cos \left (2 \, d x + 2 \, c\right ) - 5 \, a^{3} e^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right )}}{315 \,{\left (d \cos \left (2 \, d x + 2 \, c\right )^{2} - 2 \, d \cos \left (2 \, d x + 2 \, c\right ) + d\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cot(d*x+c))^(5/2)*(a+a*cot(d*x+c))^3,x, algorithm="fricas")

[Out]

[1/315*(315*sqrt(2)*(a^3*e^2*cos(2*d*x + 2*c)^2 - 2*a^3*e^2*cos(2*d*x + 2*c) + a^3*e^2)*sqrt(-e)*log(-sqrt(2)*
sqrt(-e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*(cos(2*d*x + 2*c) + sin(2*d*x + 2*c) - 1) - 2*e*sin(2
*d*x + 2*c) + e) + 2*(721*a^3*e^2*cos(2*d*x + 2*c)^2 - 1330*a^3*e^2*cos(2*d*x + 2*c) + 469*a^3*e^2 - 15*(23*a^
3*e^2*cos(2*d*x + 2*c) - 5*a^3*e^2)*sin(2*d*x + 2*c))*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(d*cos(
2*d*x + 2*c)^2 - 2*d*cos(2*d*x + 2*c) + d), 2/315*(315*sqrt(2)*(a^3*e^2*cos(2*d*x + 2*c)^2 - 2*a^3*e^2*cos(2*d
*x + 2*c) + a^3*e^2)*sqrt(e)*arctan(-1/2*sqrt(2)*sqrt(e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*(cos(
2*d*x + 2*c) - sin(2*d*x + 2*c) + 1)/(e*cos(2*d*x + 2*c) + e)) + (721*a^3*e^2*cos(2*d*x + 2*c)^2 - 1330*a^3*e^
2*cos(2*d*x + 2*c) + 469*a^3*e^2 - 15*(23*a^3*e^2*cos(2*d*x + 2*c) - 5*a^3*e^2)*sin(2*d*x + 2*c))*sqrt((e*cos(
2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(d*cos(2*d*x + 2*c)^2 - 2*d*cos(2*d*x + 2*c) + 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((e*cot(d*x+c))**(5/2)*(a+a*cot(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 (a \cot \left (d x + c\right ) + a\right )}^{3} \left (e \cot \left (d x + c\right )\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cot(d*x+c))^(5/2)*(a+a*cot(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((a*cot(d*x + c) + a)^3*(e*cot(d*x + c))^(5/2), x)