Optimal. Leaf size=269 \[ \frac {a^2 e^{5/2} \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d}-\frac {a^2 e^{5/2} \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d}+\frac {\sqrt {2} a^2 e^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d}-\frac {\sqrt {2} a^2 e^{5/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{d}+\frac {4 a^2 e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {2 a^2 (e \cot (c+d x))^{7/2}}{7 d e}-\frac {4 a^2 (e \cot (c+d x))^{5/2}}{5 d} \]
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Rubi [A] time = 0.29, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 16, 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, 211, 1165, 628, 1162, 617, 204} \[ \frac {4 a^2 e^2 \sqrt {e \cot (c+d x)}}{d}+\frac {a^2 e^{5/2} \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d}-\frac {a^2 e^{5/2} \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d}+\frac {\sqrt {2} a^2 e^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d}-\frac {\sqrt {2} a^2 e^{5/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{d}-\frac {2 a^2 (e \cot (c+d x))^{7/2}}{7 d e}-\frac {4 a^2 (e \cot (c+d x))^{5/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 16
Rule 204
Rule 211
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))^{5/2} (a+a \cot (c+d x))^2 \, dx &=-\frac {2 a^2 (e \cot (c+d x))^{7/2}}{7 d e}+\int 2 a^2 \cot (c+d x) (e \cot (c+d x))^{5/2} \, dx\\ &=-\frac {2 a^2 (e \cot (c+d x))^{7/2}}{7 d e}+\left (2 a^2\right ) \int \cot (c+d x) (e \cot (c+d x))^{5/2} \, dx\\ &=-\frac {2 a^2 (e \cot (c+d x))^{7/2}}{7 d e}+\frac {\left (2 a^2\right ) \int (e \cot (c+d x))^{7/2} \, dx}{e}\\ &=-\frac {4 a^2 (e \cot (c+d x))^{5/2}}{5 d}-\frac {2 a^2 (e \cot (c+d x))^{7/2}}{7 d e}-\left (2 a^2 e\right ) \int (e \cot (c+d x))^{3/2} \, dx\\ &=\frac {4 a^2 e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {4 a^2 (e \cot (c+d x))^{5/2}}{5 d}-\frac {2 a^2 (e \cot (c+d x))^{7/2}}{7 d e}+\left (2 a^2 e^3\right ) \int \frac {1}{\sqrt {e \cot (c+d x)}} \, dx\\ &=\frac {4 a^2 e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {4 a^2 (e \cot (c+d x))^{5/2}}{5 d}-\frac {2 a^2 (e \cot (c+d x))^{7/2}}{7 d e}-\frac {\left (2 a^2 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (e^2+x^2\right )} \, dx,x,e \cot (c+d x)\right )}{d}\\ &=\frac {4 a^2 e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {4 a^2 (e \cot (c+d x))^{5/2}}{5 d}-\frac {2 a^2 (e \cot (c+d x))^{7/2}}{7 d e}-\frac {\left (4 a^2 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d}\\ &=\frac {4 a^2 e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {4 a^2 (e \cot (c+d x))^{5/2}}{5 d}-\frac {2 a^2 (e \cot (c+d x))^{7/2}}{7 d e}-\frac {\left (2 a^2 e^3\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^3\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^2 \sqrt {e \cot (c+d x)}}{d}-\frac {4 a^2 (e \cot (c+d x))^{5/2}}{5 d}-\frac {2 a^2 (e \cot (c+d x))^{7/2}}{7 d e}+\frac {\left (a^2 e^{5/2}\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^{5/2}\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\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^3\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^2 \sqrt {e \cot (c+d x)}}{d}-\frac {4 a^2 (e \cot (c+d x))^{5/2}}{5 d}-\frac {2 a^2 (e \cot (c+d x))^{7/2}}{7 d e}+\frac {a^2 e^{5/2} \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^{5/2} \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^{5/2}\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^{5/2}\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^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d}-\frac {\sqrt {2} a^2 e^{5/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {4 a^2 e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {4 a^2 (e \cot (c+d x))^{5/2}}{5 d}-\frac {2 a^2 (e \cot (c+d x))^{7/2}}{7 d e}+\frac {a^2 e^{5/2} \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^{5/2} \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 [A] time = 1.19, size = 187, normalized size = 0.70 \[ -\frac {a^2 (e \cot (c+d x))^{5/2} \left (20 \cot ^{\frac {7}{2}}(c+d x)+56 \cot ^{\frac {5}{2}}(c+d x)-280 \sqrt {\cot (c+d x)}-35 \sqrt {2} \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )+35 \sqrt {2} \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-70 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )+70 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )\right )}{70 d \cot ^{\frac {5}{2}}(c+d x)} \]
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 \[ \int {\left (a \cot \left (d x + c\right ) + a\right )}^{2} \left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.61, size = 234, normalized size = 0.87 \[ -\frac {2 a^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d e}-\frac {4 a^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5 d}+\frac {4 a^{2} e^{2} \sqrt {e \cot \left (d x +c \right )}}{d}+\frac {a^{2} e^{2} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d}-\frac {a^{2} e^{2} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{2 d}-\frac {a^{2} e^{2} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 232, normalized size = 0.86 \[ -\frac {{\left (35 \, {\left (2 \, \sqrt {2} e^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right ) + 2 \, \sqrt {2} e^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right ) + \sqrt {2} e^{\frac {3}{2}} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right ) - \sqrt {2} e^{\frac {3}{2}} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )\right )} a^{2} - \frac {4 \, {\left (70 \, a^{2} e^{3} \sqrt {\frac {e}{\tan \left (d x + c\right )}} - 14 \, a^{2} e \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {5}{2}} - 5 \, a^{2} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {7}{2}}\right )}}{e^{2}}\right )} e}{70 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.73, size = 125, normalized size = 0.46 \[ \frac {4\,a^2\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d}-\frac {4\,a^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}{5\,d}-\frac {2\,a^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{7/2}}{7\,d\,e}+\frac {{\left (-1\right )}^{1/4}\,a^2\,e^{5/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,2{}\mathrm {i}}{d}+\frac {2\,{\left (-1\right )}^{1/4}\,a^2\,e^{5/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,1{}\mathrm {i}}{\sqrt {e}}\right )}{d} \]
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 \[ a^{2} \left (\int \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx + \int 2 \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot {\left (c + d x \right )}\, dx + \int \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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