Optimal. Leaf size=117 \[ -\frac {2 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {e} \cot (c+d x)+\sqrt {e}}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{5/2}}+\frac {16 a^3}{3 d e^2 \sqrt {e \cot (c+d x)}}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{3 d e (e \cot (c+d x))^{3/2}} \]
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Rubi [A] time = 0.19, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3565, 3628, 3532, 208} \[ \frac {16 a^3}{3 d e^2 \sqrt {e \cot (c+d x)}}-\frac {2 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {e} \cot (c+d x)+\sqrt {e}}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{5/2}}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{3 d e (e \cot (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 3532
Rule 3565
Rule 3628
Rubi steps
\begin {align*} \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{5/2}} \, dx &=\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{3 d e (e \cot (c+d x))^{3/2}}-\frac {2 \int \frac {-4 a^3 e^2-3 a^3 e^2 \cot (c+d x)-a^3 e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2}} \, dx}{3 e^3}\\ &=\frac {16 a^3}{3 d e^2 \sqrt {e \cot (c+d x)}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{3 d e (e \cot (c+d x))^{3/2}}-\frac {2 \int \frac {-3 a^3 e^3+3 a^3 e^3 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{3 e^5}\\ &=\frac {16 a^3}{3 d e^2 \sqrt {e \cot (c+d x)}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{3 d e (e \cot (c+d x))^{3/2}}+\frac {\left (12 a^6 e\right ) \operatorname {Subst}\left (\int \frac {1}{18 a^6 e^6-e x^2} \, dx,x,\frac {-3 a^3 e^3-3 a^3 e^3 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d}\\ &=-\frac {2 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{5/2}}+\frac {16 a^3}{3 d e^2 \sqrt {e \cot (c+d x)}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{3 d e (e \cot (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [C] time = 6.11, size = 417, normalized size = 3.56 \[ -\frac {2 \cos ^3(c+d x) \cot (c+d x) (a \cot (c+d x)+a)^3 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )}{3 d (e \cot (c+d x))^{5/2} (\sin (c+d x)+\cos (c+d x))^3}+\frac {6 \sin (c+d x) \cos ^2(c+d x) (a \cot (c+d x)+a)^3 \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2(c+d x)\right )}{d (e \cot (c+d x))^{5/2} (\sin (c+d x)+\cos (c+d x))^3}+\frac {2 \sin ^2(c+d x) \cos (c+d x) (a \cot (c+d x)+a)^3 \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\cot ^2(c+d x)\right )}{3 d (e \cot (c+d x))^{5/2} (\sin (c+d x)+\cos (c+d x))^3}+\frac {3 \sin ^3(c+d x) \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+a)^3 \left (\sqrt {2} \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-\sqrt {2} \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )+2 \left (\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-\sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )\right )\right )}{4 d (e \cot (c+d x))^{5/2} (\sin (c+d x)+\cos (c+d x))^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.54, size = 378, normalized size = 3.23 \[ \left [\frac {\frac {3 \, \sqrt {2} {\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right ) + a^{3} e\right )} \log \left (\frac {\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) - 1\right )}}{\sqrt {e}} + 2 \, \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}{\sqrt {e}} - 2 \, {\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) - 9 \, a^{3} \sin \left (2 \, d x + 2 \, c\right ) - a^{3}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{3 \, {\left (d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + d e^{3}\right )}}, \frac {2 \, {\left (3 \, \sqrt {2} {\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right ) + a^{3} e\right )} \sqrt {-\frac {1}{e}} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sqrt {-\frac {1}{e}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )}}\right ) - {\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) - 9 \, a^{3} \sin \left (2 \, d x + 2 \, c\right ) - a^{3}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right )}}{3 \, {\left (d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + d e^{3}\right )}}\right ] \]
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 \[ \int \frac {{\left (a \cot \left (d x + c\right ) + a\right )}^{3}}{\left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.50, size = 388, normalized size = 3.32 \[ -\frac {a^{3} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{2 d \,e^{3}}-\frac {a^{3} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \,e^{3}}+\frac {a^{3} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \,e^{3}}+\frac {a^{3} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{2 d \,e^{2} \left (e^{2}\right )^{\frac {1}{4}}}+\frac {a^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \,e^{2} \left (e^{2}\right )^{\frac {1}{4}}}-\frac {a^{3} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \,e^{2} \left (e^{2}\right )^{\frac {1}{4}}}+\frac {2 a^{3}}{3 d e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {6 a^{3}}{d \,e^{2} \sqrt {e \cot \left (d x +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 133, normalized size = 1.14 \[ -\frac {e {\left (\frac {3 \, a^{3} {\left (\frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} - \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}\right )}}{e^{3}} - \frac {2 \, {\left (a^{3} e + \frac {9 \, a^{3} e}{\tan \left (d x + c\right )}\right )}}{e^{3} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.71, size = 101, normalized size = 0.86 \[ \frac {\frac {2\,a^3\,e}{3}+6\,a^3\,e\,\mathrm {cot}\left (c+d\,x\right )}{d\,e^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}-\frac {2\,\sqrt {2}\,a^3\,\mathrm {atanh}\left (\frac {32\,\sqrt {2}\,a^6\,d\,e^{5/2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{32\,a^6\,d\,e^3+32\,a^6\,d\,e^3\,\mathrm {cot}\left (c+d\,x\right )}\right )}{d\,e^{5/2}} \]
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 \[ a^{3} \left (\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx + \int \frac {3 \cot {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx + \int \frac {3 \cot ^{2}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx + \int \frac {\cot ^{3}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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