Optimal. Leaf size=186 \[ \frac {2 \sqrt {2} a^3 e^{5/2} \tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}+\frac {4 a^3 e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{7/2}}{9 d e}-\frac {40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac {4 a^3 (e \cot (c+d x))^{5/2}}{5 d}+\frac {4 a^3 e (e \cot (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 0.30, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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.
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Rule 205
Rule 3528
Rule 3532
Rule 3566
Rule 3630
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*}
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Mathematica [C] time = 6.10, size = 729, normalized size = 3.92 \[ -\frac {4 \sin ^3(c+d x) \tan (c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )}{3 d (\sin (c+d x)+\cos (c+d x))^3}-\frac {2 \sin (c+d x) \cos ^2(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2}}{9 d (\sin (c+d x)+\cos (c+d x))^3}-\frac {4 \sin ^3(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2}}{5 d (\sin (c+d x)+\cos (c+d x))^3}-\frac {6 \sin ^2(c+d x) \cos (c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2}}{7 d (\sin (c+d x)+\cos (c+d x))^3}+\frac {\sin ^3(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2} \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \cot ^{\frac {5}{2}}(c+d x) (\sin (c+d x)+\cos (c+d x))^3}-\frac {\sin ^3(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2} \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \cot ^{\frac {5}{2}}(c+d x) (\sin (c+d x)+\cos (c+d x))^3}+\frac {\sqrt {2} \sin ^3(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{d \cot ^{\frac {5}{2}}(c+d x) (\sin (c+d x)+\cos (c+d x))^3}-\frac {\sqrt {2} \sin ^3(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2} \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{d \cot ^{\frac {5}{2}}(c+d x) (\sin (c+d x)+\cos (c+d x))^3}+\frac {4 \sin ^3(c+d x) \tan ^2(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2}}{d (\sin (c+d x)+\cos (c+d x))^3}+\frac {4 \sin ^3(c+d x) \tan (c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2}}{3 d (\sin (c+d x)+\cos (c+d x))^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 535, normalized size = 2.88 \[ \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 ] \]
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 \[ \int {\left (a \cot \left (d x + c\right ) + a\right )}^{3} \left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.79, size = 446, normalized size = 2.40 \[ -\frac {2 a^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {9}{2}}}{9 d \,e^{2}}-\frac {6 a^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d e}-\frac {4 a^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5 d}+\frac {4 a^{3} e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}+\frac {4 a^{3} e^{2} \sqrt {e \cot \left (d x +c \right )}}{d}-\frac {a^{3} e^{2} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{2 d}-\frac {a^{3} e^{2} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d}+\frac {a^{3} e^{2} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d}-\frac {a^{3} e^{3} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{2 d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {a^{3} e^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {a^{3} e^{3} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \left (e^{2}\right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 193, normalized size = 1.04 \[ -\frac {2 \, {\left (315 \, a^{3} e^{2} {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}}\right )} - \frac {630 \, a^{3} e^{4} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + 210 \, a^{3} e^{3} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}} - 126 \, a^{3} e^{2} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {5}{2}} - 135 \, a^{3} e \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {7}{2}} - 35 \, a^{3} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {9}{2}}}{e^{3}}\right )} e}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.44, size = 177, normalized size = 0.95 \[ \frac {4\,a^3\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d}-\frac {4\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}{5\,d}-\frac {6\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{7/2}}{7\,d\,e}-\frac {2\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{9/2}}{9\,d\,e^2}+\frac {4\,a^3\,e\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d}-\frac {\sqrt {2}\,a^3\,e^{5/2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}+\frac {\sqrt {2}\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{2\,e^{3/2}}\right )\right )}{d} \]
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 \[ a^{3} \left (\int \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx + \int 3 \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot {\left (c + d x \right )}\, dx + \int 3 \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot ^{2}{\left (c + d x \right )}\, dx + \int \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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