3.39 \(\int \frac{1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3} \, dx\)

Optimal. Leaf size=189 \[ \frac{31 \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{8 a^3 d e^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d e^{3/2}}+\frac{27}{8 a^3 d e \sqrt{e \cot (c+d x)}}-\frac{9}{8 a^3 d e (\cot (c+d x)+1) \sqrt{e \cot (c+d x)}}-\frac{1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt{e \cot (c+d x)}} \]

[Out]

(31*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(8*a^3*d*e^(3/2)) + ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2
]*Sqrt[e*Cot[c + d*x]])]/(2*Sqrt[2]*a^3*d*e^(3/2)) + 27/(8*a^3*d*e*Sqrt[e*Cot[c + d*x]]) - 9/(8*a^3*d*e*Sqrt[e
*Cot[c + d*x]]*(1 + Cot[c + d*x])) - 1/(4*a*d*e*Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])^2)

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Rubi [A]  time = 0.863312, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3569, 3649, 3654, 3532, 208, 3634, 63, 205} \[ \frac{31 \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{8 a^3 d e^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d e^{3/2}}+\frac{27}{8 a^3 d e \sqrt{e \cot (c+d x)}}-\frac{9}{8 a^3 d e (\cot (c+d x)+1) \sqrt{e \cot (c+d x)}}-\frac{1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt{e \cot (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/((e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x])^3),x]

[Out]

(31*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(8*a^3*d*e^(3/2)) + ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2
]*Sqrt[e*Cot[c + d*x]])]/(2*Sqrt[2]*a^3*d*e^(3/2)) + 27/(8*a^3*d*e*Sqrt[e*Cot[c + d*x]]) - 9/(8*a^3*d*e*Sqrt[e
*Cot[c + d*x]]*(1 + Cot[c + d*x])) - 1/(4*a*d*e*Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])^2)

Rule 3569

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 + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*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] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))

Rule 3649

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*(b*B - a*C))*(a + b*T
an[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3654

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*ta
n[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*x])^n*Simp[a*(A - C) - (A*b - b*
C)*Tan[e + f*x], x], x], x] + Dist[(A*b^2 + a^2*C)/(a^2 + b^2), Int[((c + d*Tan[e + f*x])^n*(1 + Tan[e + f*x]^
2))/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^
2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -1]

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 208

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

Rule 3634

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 \frac{1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3} \, dx &=-\frac{1}{4 a d e \sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^2}-\frac{\int \frac{-\frac{9 a^2 e}{2}+2 a^2 e \cot (c+d x)-\frac{5}{2} a^2 e \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2} \, dx}{4 a^3 e}\\ &=-\frac{9}{8 a^3 d e \sqrt{e \cot (c+d x)} (1+\cot (c+d x))}-\frac{1}{4 a d e \sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^2}+\frac{\int \frac{\frac{27 a^4 e^2}{2}-4 a^4 e^2 \cot (c+d x)+\frac{27}{2} a^4 e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))} \, dx}{8 a^6 e^2}\\ &=\frac{27}{8 a^3 d e \sqrt{e \cot (c+d x)}}-\frac{9}{8 a^3 d e \sqrt{e \cot (c+d x)} (1+\cot (c+d x))}-\frac{1}{4 a d e \sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^2}+\frac{\int \frac{-\frac{35}{4} a^5 e^4-\frac{27}{4} a^5 e^4 \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{4 a^7 e^5}\\ &=\frac{27}{8 a^3 d e \sqrt{e \cot (c+d x)}}-\frac{9}{8 a^3 d e \sqrt{e \cot (c+d x)} (1+\cot (c+d x))}-\frac{1}{4 a d e \sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^2}+\frac{\int \frac{-2 a^6 e^4+2 a^6 e^4 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{8 a^9 e^5}-\frac{31 \int \frac{1+\cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{16 a^2 e}\\ &=\frac{27}{8 a^3 d e \sqrt{e \cot (c+d x)}}-\frac{9}{8 a^3 d e \sqrt{e \cot (c+d x)} (1+\cot (c+d x))}-\frac{1}{4 a d e \sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^2}-\frac{31 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-e x} (a-a x)} \, dx,x,-\cot (c+d x)\right )}{16 a^2 d e}-\frac{\left (a^3 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{8 a^{12} e^8-e x^2} \, dx,x,\frac{-2 a^6 e^4-2 a^6 e^4 \cot (c+d x)}{\sqrt{e \cot (c+d x)}}\right )}{d}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{e}+\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d e^{3/2}}+\frac{27}{8 a^3 d e \sqrt{e \cot (c+d x)}}-\frac{9}{8 a^3 d e \sqrt{e \cot (c+d x)} (1+\cot (c+d x))}-\frac{1}{4 a d e \sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^2}+\frac{31 \operatorname{Subst}\left (\int \frac{1}{a+\frac{a x^2}{e}} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{8 a^2 d e^2}\\ &=\frac{31 \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{8 a^3 d e^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e}+\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d e^{3/2}}+\frac{27}{8 a^3 d e \sqrt{e \cot (c+d x)}}-\frac{9}{8 a^3 d e \sqrt{e \cot (c+d x)} (1+\cot (c+d x))}-\frac{1}{4 a d e \sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^2}\\ \end{align*}

Mathematica [A]  time = 1.18744, size = 156, normalized size = 0.83 \[ \frac{\cot ^{\frac{3}{2}}(c+d x) \left (-2 \sqrt{2} \left (\log \left (-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right )-\log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )+62 \tan ^{-1}\left (\sqrt{\cot (c+d x)}\right )+\frac{45 \sin (2 (c+d x))+11 \cos (2 (c+d x))+43}{\sqrt{\cot (c+d x)} (\sin (c+d x)+\cos (c+d x))^2}\right )}{16 a^3 d (e \cot (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x])^3),x]

[Out]

(Cot[c + d*x]^(3/2)*(62*ArcTan[Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*(Log[-1 + Sqrt[2]*Sqrt[Cot[c + d*x]] - Cot[c +
d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]) + (43 + 11*Cos[2*(c + d*x)] + 45*Sin[2*(c + d*x)])
/(Sqrt[Cot[c + d*x]]*(Cos[c + d*x] + Sin[c + d*x])^2)))/(16*a^3*d*(e*Cot[c + d*x])^(3/2))

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Maple [B]  time = 0.043, size = 458, normalized size = 2.4 \begin{align*}{\frac{\sqrt{2}}{16\,d{a}^{3}{e}^{2}}\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{\sqrt{2}}{8\,d{a}^{3}{e}^{2}}\sqrt [4]{{e}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }-{\frac{\sqrt{2}}{8\,d{a}^{3}{e}^{2}}\sqrt [4]{{e}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }-{\frac{\sqrt{2}}{16\,d{a}^{3}e}\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{\sqrt{2}}{8\,d{a}^{3}e}\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{\sqrt{2}}{8\,d{a}^{3}e}\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}}}}}+2\,{\frac{1}{d{a}^{3}e\sqrt{e\cot \left ( dx+c \right ) }}}+{\frac{11}{8\,d{a}^{3}e \left ( e\cot \left ( dx+c \right ) +e \right ) ^{2}} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{13}{8\,d{a}^{3} \left ( e\cot \left ( dx+c \right ) +e \right ) ^{2}}\sqrt{e\cot \left ( dx+c \right ) }}+{\frac{31}{8\,d{a}^{3}}\arctan \left ({\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt{e}}}} \right ){e}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*cot(d*x+c))^(3/2)/(a+a*cot(d*x+c))^3,x)

[Out]

1/16/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*c
ot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+1/8/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)-1/8/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)-1/16/d/a^3/e/(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)))-1/8/d/a^3/e/(e^2)^(1/4)*2^(1/2)
*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)+1/8/d/a^3/e/(e^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(e^2)^(1/4
)*(e*cot(d*x+c))^(1/2)+1)+2/a^3/d/e/(e*cot(d*x+c))^(1/2)+11/8/d/a^3/e/(e*cot(d*x+c)+e)^2*(e*cot(d*x+c))^(3/2)+
13/8/d/a^3/(e*cot(d*x+c)+e)^2*(e*cot(d*x+c))^(1/2)+31/8*arctan((e*cot(d*x+c))^(1/2)/e^(1/2))/a^3/d/e^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.57934, size = 1775, normalized size = 9.39 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cot(d*x+c))^(3/2)/(a+a*cot(d*x+c))^3,x, algorithm="fricas")

[Out]

[-1/16*(4*sqrt(2)*((cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c) + cos(2*d*x + 2*c) + 1)*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)) + 31*((cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c) + cos(2*d*x + 2*c) + 1)*sqrt(-e)*log((e*cos(2*d*
x + 2*c) - e*sin(2*d*x + 2*c) - 2*sqrt(-e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*sin(2*d*x + 2*c) +
e)/(cos(2*d*x + 2*c) + sin(2*d*x + 2*c) + 1)) + (45*cos(2*d*x + 2*c)^2 - (11*cos(2*d*x + 2*c) + 43)*sin(2*d*x
+ 2*c) - 45)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(a^3*d*e^2*cos(2*d*x + 2*c) + a^3*d*e^2 + (a^3*d
*e^2*cos(2*d*x + 2*c) + a^3*d*e^2)*sin(2*d*x + 2*c)), 1/16*(2*sqrt(2)*((cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c)
+ cos(2*d*x + 2*c) + 1)*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) + 62*((cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c) +
cos(2*d*x + 2*c) + 1)*sqrt(e)*arctan(sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))/sqrt(e)) - (45*cos(2*d*x
+ 2*c)^2 - (11*cos(2*d*x + 2*c) + 43)*sin(2*d*x + 2*c) - 45)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/
(a^3*d*e^2*cos(2*d*x + 2*c) + a^3*d*e^2 + (a^3*d*e^2*cos(2*d*x + 2*c) + a^3*d*e^2)*sin(2*d*x + 2*c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}} \cot ^{3}{\left (c + d x \right )} + 3 \left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}} \cot ^{2}{\left (c + d x \right )} + 3 \left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}} \cot{\left (c + d x \right )} + \left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cot(d*x+c))**(3/2)/(a+a*cot(d*x+c))**3,x)

[Out]

Integral(1/((e*cot(c + d*x))**(3/2)*cot(c + d*x)**3 + 3*(e*cot(c + d*x))**(3/2)*cot(c + d*x)**2 + 3*(e*cot(c +
d*x))**(3/2)*cot(c + d*x) + (e*cot(c + d*x))**(3/2)), x)/a**3

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \cot \left (d x + c\right ) + a\right )}^{3} \left (e \cot \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cot(d*x+c))^(3/2)/(a+a*cot(d*x+c))^3,x, algorithm="giac")

[Out]

integrate(1/((a*cot(d*x + c) + a)^3*(e*cot(d*x + c))^(3/2)), x)