Optimal. Leaf size=246 \[ \frac {a^2 e^{3/2} \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d}-\frac {a^2 e^{3/2} \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d}-\frac {\sqrt {2} a^2 e^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {\sqrt {2} a^2 e^{3/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{d}-\frac {2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}-\frac {4 a^2 (e \cot (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 0.23, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3543, 12, 16, 3473, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {a^2 e^{3/2} \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d}-\frac {a^2 e^{3/2} \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d}-\frac {\sqrt {2} a^2 e^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {\sqrt {2} a^2 e^{3/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{d}-\frac {2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}-\frac {4 a^2 (e \cot (c+d x))^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 16
Rule 204
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3473
Rule 3476
Rule 3543
Rubi steps
\begin {align*} \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2 \, dx &=-\frac {2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}+\int 2 a^2 \cot (c+d x) (e \cot (c+d x))^{3/2} \, dx\\ &=-\frac {2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}+\left (2 a^2\right ) \int \cot (c+d x) (e \cot (c+d x))^{3/2} \, dx\\ &=-\frac {2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}+\frac {\left (2 a^2\right ) \int (e \cot (c+d x))^{5/2} \, dx}{e}\\ &=-\frac {4 a^2 (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}-\left (2 a^2 e\right ) \int \sqrt {e \cot (c+d x)} \, dx\\ &=-\frac {4 a^2 (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}+\frac {\left (2 a^2 e^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{e^2+x^2} \, dx,x,e \cot (c+d x)\right )}{d}\\ &=-\frac {4 a^2 (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}+\frac {\left (4 a^2 e^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d}\\ &=-\frac {4 a^2 (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}-\frac {\left (2 a^2 e^2\right ) \operatorname {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d}+\frac {\left (2 a^2 e^2\right ) \operatorname {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d}\\ &=-\frac {4 a^2 (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}+\frac {\left (a^2 e^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2 e^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d}+\frac {\left (a^2 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d}\\ &=-\frac {4 a^2 (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}+\frac {a^2 e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d}-\frac {a^2 e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (\sqrt {2} a^2 e^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d}-\frac {\left (\sqrt {2} a^2 e^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d}\\ &=-\frac {\sqrt {2} a^2 e^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {\sqrt {2} a^2 e^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d}-\frac {4 a^2 (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}+\frac {a^2 e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d}-\frac {a^2 e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d}\\ \end {align*}
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Mathematica [C] time = 0.39, size = 52, normalized size = 0.21 \[ -\frac {2 a^2 (e \cot (c+d x))^{3/2} \left (-10 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )+3 \cot (c+d x)+10\right )}{15 d} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \cot \left (d x + c\right ) + a\right )}^{2} \left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.61, size = 213, normalized size = 0.87 \[ -\frac {2 a^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5 d e}-\frac {4 a^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}+\frac {a^{2} e^{2} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{2 d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {a^{2} e^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {a^{2} e^{2} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \left (e^{2}\right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 213, normalized size = 0.87 \[ \frac {{\left (15 \, a^{2} e {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}\right )} - \frac {4 \, {\left (10 \, a^{2} e \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}} + 3 \, a^{2} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {5}{2}}\right )}}{e^{2}}\right )} e}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.95, size = 104, normalized size = 0.42 \[ \frac {2\,{\left (-1\right )}^{1/4}\,a^2\,e^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{d}-\frac {2\,a^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}{5\,d\,e}-\frac {4\,a^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d}+\frac {{\left (-1\right )}^{1/4}\,a^2\,e^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,1{}\mathrm {i}}{\sqrt {e}}\right )\,2{}\mathrm {i}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} \left (\int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx + \int 2 \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot {\left (c + d x \right )}\, dx + \int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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