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

Optimal. Leaf size=160 \[ -\frac {2 \sqrt {2} a^3 e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \cot (c+d x)+\sqrt {e}}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}-\frac {32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac {4 a^3 (e \cot (c+d x))^{3/2}}{3 d}+\frac {4 a^3 e \sqrt {e \cot (c+d x)}}{d}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{5/2}}{7 d e} \]

[Out]

-4/3*a^3*(e*cot(d*x+c))^(3/2)/d-32/35*a^3*(e*cot(d*x+c))^(5/2)/d/e-2/7*(e*cot(d*x+c))^(5/2)*(a^3+a^3*cot(d*x+c
))/d/e-2*a^3*e^(3/2)*arctanh(1/2*(e^(1/2)+cot(d*x+c)*e^(1/2))*2^(1/2)/(e*cot(d*x+c))^(1/2))*2^(1/2)/d+4*a^3*e*
(e*cot(d*x+c))^(1/2)/d

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Rubi [A]  time = 0.26, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3566, 3630, 3528, 3532, 208} \[ -\frac {2 \sqrt {2} a^3 e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \cot (c+d x)+\sqrt {e}}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}-\frac {32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac {4 a^3 (e \cot (c+d x))^{3/2}}{3 d}+\frac {4 a^3 e \sqrt {e \cot (c+d x)}}{d}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{5/2}}{7 d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2]*a^3*e^(3/2)*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d + (4*a^3*e
*Sqrt[e*Cot[c + d*x]])/d - (4*a^3*(e*Cot[c + d*x])^(3/2))/(3*d) - (32*a^3*(e*Cot[c + d*x])^(5/2))/(35*d*e) - (
2*(e*Cot[c + d*x])^(5/2)*(a^3 + a^3*Cot[c + d*x]))/(7*d*e)

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 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 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]

Rubi steps

\begin {align*} \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx &=-\frac {2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}-\frac {2 \int (e \cot (c+d x))^{3/2} \left (-a^3 e-7 a^3 e \cot (c+d x)-8 a^3 e \cot ^2(c+d x)\right ) \, dx}{7 e}\\ &=-\frac {32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac {2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}-\frac {2 \int (e \cot (c+d x))^{3/2} \left (7 a^3 e-7 a^3 e \cot (c+d x)\right ) \, dx}{7 e}\\ &=-\frac {4 a^3 (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac {2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}-\frac {2 \int \sqrt {e \cot (c+d x)} \left (7 a^3 e^2+7 a^3 e^2 \cot (c+d x)\right ) \, dx}{7 e}\\ &=\frac {4 a^3 e \sqrt {e \cot (c+d x)}}{d}-\frac {4 a^3 (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac {2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}-\frac {2 \int \frac {-7 a^3 e^3+7 a^3 e^3 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{7 e}\\ &=\frac {4 a^3 e \sqrt {e \cot (c+d x)}}{d}-\frac {4 a^3 (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac {2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}+\frac {\left (28 a^6 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{98 a^6 e^6-e x^2} \, dx,x,\frac {-7 a^3 e^3-7 a^3 e^3 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d}\\ &=-\frac {2 \sqrt {2} a^3 e^{3/2} \tanh ^{-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 \sqrt {e \cot (c+d x)}}{d}-\frac {4 a^3 (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac {2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}\\ \end {align*}

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Mathematica [C]  time = 2.81, size = 332, normalized size = 2.08 \[ \frac {a^3 \sin (c+d x) (\cot (c+d x)+1)^3 (e \cot (c+d x))^{3/2} \left (280 \sin ^2(c+d x) \cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )-60 \cos ^2(c+d x) \cot ^{\frac {3}{2}}(c+d x)-280 \sin ^2(c+d x) \cot ^{\frac {3}{2}}(c+d x)-126 \sin (2 (c+d x)) \cot ^{\frac {3}{2}}(c+d x)+840 \sin ^2(c+d x) \sqrt {\cot (c+d x)}+105 \sqrt {2} \sin ^2(c+d x) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-105 \sqrt {2} \sin ^2(c+d x) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )+210 \sqrt {2} \sin ^2(c+d x) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-210 \sqrt {2} \sin ^2(c+d x) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )\right )}{210 d \cot ^{\frac {3}{2}}(c+d x) (\sin (c+d x)+\cos (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(e*Cot[c + d*x])^(3/2)*(1 + Cot[c + d*x])^3*Sin[c + d*x]*(-60*Cos[c + d*x]^2*Cot[c + d*x]^(3/2) + 210*Sqr
t[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*Sin[c + d*x]^2 - 210*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]
]*Sin[c + d*x]^2 + 840*Sqrt[Cot[c + d*x]]*Sin[c + d*x]^2 - 280*Cot[c + d*x]^(3/2)*Sin[c + d*x]^2 + 280*Cot[c +
d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]*Sin[c + d*x]^2 + 105*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[
Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 - 105*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*
Sin[c + d*x]^2 - 126*Cot[c + d*x]^(3/2)*Sin[2*(c + d*x)]))/(210*d*Cot[c + d*x]^(3/2)*(Cos[c + d*x] + Sin[c + d
*x])^3)

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fricas [A]  time = 0.72, size = 487, normalized size = 3.04 \[ \left [\frac {105 \, \sqrt {2} {\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - a^{3} e\right )} \sqrt {e} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} + 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) \sin \left (2 \, d x + 2 \, c\right ) - 2 \, {\left (55 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right )^{2} - 30 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - 85 \, a^{3} e - 21 \, {\left (13 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - 7 \, a^{3} e\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{105 \, {\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )} \sin \left (2 \, d x + 2 \, c\right )}, \frac {2 \, {\left (105 \, \sqrt {2} {\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - a^{3} e\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {2} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) \sin \left (2 \, d x + 2 \, c\right ) - {\left (55 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right )^{2} - 30 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - 85 \, a^{3} e - 21 \, {\left (13 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - 7 \, a^{3} e\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right )}}{105 \, {\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )} \sin \left (2 \, d x + 2 \, c\right )}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/105*(105*sqrt(2)*(a^3*e*cos(2*d*x + 2*c) - a^3*e)*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)*sin(2*d*x + 2*c) - 2*
(55*a^3*e*cos(2*d*x + 2*c)^2 - 30*a^3*e*cos(2*d*x + 2*c) - 85*a^3*e - 21*(13*a^3*e*cos(2*d*x + 2*c) - 7*a^3*e)
*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) - d)*sin(2*d*x + 2*c)
), 2/105*(105*sqrt(2)*(a^3*e*cos(2*d*x + 2*c) - a^3*e)*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))*sin(2*d*x +
2*c) - (55*a^3*e*cos(2*d*x + 2*c)^2 - 30*a^3*e*cos(2*d*x + 2*c) - 85*a^3*e - 21*(13*a^3*e*cos(2*d*x + 2*c) - 7
*a^3*e)*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) - d)*sin(2*d*x
+ 2*c))]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \cot \left (d x + c\right ) + a\right )}^{3} \left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.87, size = 419, normalized size = 2.62 \[ -\frac {2 a^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d \,e^{2}}-\frac {6 a^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5 d e}-\frac {4 a^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}+\frac {4 a^{3} e \sqrt {e \cot \left (d x +c \right )}}{d}-\frac {a^{3} e \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{2 d}-\frac {a^{3} e \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d}+\frac {a^{3} e \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d}+\frac {a^{3} e^{2} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{2 d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {a^{3} e^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {a^{3} e^{2} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \left (e^{2}\right )^{\frac {1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.47, size = 175, normalized size = 1.09 \[ -\frac {{\left (105 \, a^{3} e {\left (\frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} - \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}\right )} - \frac {2 \, {\left (210 \, a^{3} e^{3} \sqrt {\frac {e}{\tan \left (d x + c\right )}} - 70 \, a^{3} e^{2} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}} - 63 \, a^{3} e \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {5}{2}} - 15 \, a^{3} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {7}{2}}\right )}}{e^{3}}\right )} e}{105 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(105*a^3*e*(sqrt(2)*log(sqrt(2)*sqrt(e)*sqrt(e/tan(d*x + c)) + e + e/tan(d*x + c))/sqrt(e) - sqrt(2)*lo
g(-sqrt(2)*sqrt(e)*sqrt(e/tan(d*x + c)) + e + e/tan(d*x + c))/sqrt(e)) - 2*(210*a^3*e^3*sqrt(e/tan(d*x + c)) -
70*a^3*e^2*(e/tan(d*x + c))^(3/2) - 63*a^3*e*(e/tan(d*x + c))^(5/2) - 15*a^3*(e/tan(d*x + c))^(7/2))/e^3)*e/d

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mupad [B]  time = 1.62, size = 143, normalized size = 0.89 \[ \frac {4\,a^3\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d}-\frac {6\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}{5\,d\,e}-\frac {2\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{7/2}}{7\,d\,e^2}-\frac {4\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d}+\frac {\sqrt {2}\,a^3\,e^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,a^6\,e^{9/2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,32{}\mathrm {i}}{32\,a^6\,e^5+32\,a^6\,e^5\,\mathrm {cot}\left (c+d\,x\right )}\right )\,2{}\mathrm {i}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*a^3*e*(e*cot(c + d*x))^(1/2))/d - (6*a^3*(e*cot(c + d*x))^(5/2))/(5*d*e) - (2*a^3*(e*cot(c + d*x))^(7/2))/(
7*d*e^2) - (4*a^3*(e*cot(c + d*x))^(3/2))/(3*d) + (2^(1/2)*a^3*e^(3/2)*atan((2^(1/2)*a^6*e^(9/2)*(e*cot(c + d*
x))^(1/2)*32i)/(32*a^6*e^5 + 32*a^6*e^5*cot(c + d*x)))*2i)/d

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a^{3} \left (\int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx + \int 3 \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot {\left (c + d x \right )}\, dx + \int 3 \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot ^{2}{\left (c + d x \right )}\, dx + \int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot ^{3}{\left (c + d x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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