Optimal. Leaf size=165 \[ \frac {2 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {e} \cot (c+d x)+\sqrt {e}}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{9/2}}-\frac {4 a^3}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {4 a^3}{3 d e^3 (e \cot (c+d x))^{3/2}}+\frac {32 a^3}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{7 d e (e \cot (c+d x))^{7/2}} \]
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Rubi [A] time = 0.30, antiderivative size = 165, 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 = {3565, 3628, 3529, 3532, 208} \[ -\frac {4 a^3}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {4 a^3}{3 d e^3 (e \cot (c+d x))^{3/2}}+\frac {32 a^3}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {2 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {e} \cot (c+d x)+\sqrt {e}}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{9/2}}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{7 d e (e \cot (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 3529
Rule 3532
Rule 3565
Rule 3628
Rubi steps
\begin {align*} \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx &=\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int \frac {-8 a^3 e^2-7 a^3 e^2 \cot (c+d x)-a^3 e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{7/2}} \, dx}{7 e^3}\\ &=\frac {32 a^3}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int \frac {-7 a^3 e^3+7 a^3 e^3 \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx}{7 e^5}\\ &=\frac {32 a^3}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {4 a^3}{3 d e^3 (e \cot (c+d x))^{3/2}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int \frac {7 a^3 e^4+7 a^3 e^4 \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx}{7 e^7}\\ &=\frac {32 a^3}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {4 a^3}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {4 a^3}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int \frac {7 a^3 e^5-7 a^3 e^5 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{7 e^9}\\ &=\frac {32 a^3}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {4 a^3}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {4 a^3}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{7 d e (e \cot (c+d x))^{7/2}}+\frac {\left (28 a^6 e\right ) \operatorname {Subst}\left (\int \frac {1}{98 a^6 e^{10}-e x^2} \, dx,x,\frac {7 a^3 e^5+7 a^3 e^5 \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^{9/2}}+\frac {32 a^3}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {4 a^3}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {4 a^3}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{7 d e (e \cot (c+d x))^{7/2}}\\ \end {align*}
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Mathematica [C] time = 2.00, size = 174, normalized size = 1.05 \[ \frac {2 a^3 \cos (c+d x) (\cot (c+d x)+1)^3 \left (35 \cos ^2(c+d x) \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\cot ^2(c+d x)\right )+35 \cos ^2(c+d x) \cot (c+d x) \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2(c+d x)\right )+5 \sin ^2(c+d x) \, _2F_1\left (-\frac {7}{4},1;-\frac {3}{4};-\cot ^2(c+d x)\right )+\frac {21}{2} \sin (2 (c+d x)) \, _2F_1\left (-\frac {5}{4},1;-\frac {1}{4};-\cot ^2(c+d x)\right )\right )}{35 d (e \cot (c+d x))^{9/2} (\sin (c+d x)+\cos (c+d x))^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 514, normalized size = 3.12 \[ \left [\frac {\frac {105 \, \sqrt {2} {\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, 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 (55 \, a^{3} \cos \left (2 \, d x + 2 \, c\right )^{2} + 30 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) - 85 \, a^{3} + 21 \, {\left (13 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) + 7 \, a^{3}\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 e^{5} \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, d e^{5} \cos \left (2 \, d x + 2 \, c\right ) + d e^{5}\right )}}, -\frac {2 \, {\left (105 \, \sqrt {2} {\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, 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 (55 \, a^{3} \cos \left (2 \, d x + 2 \, c\right )^{2} + 30 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) - 85 \, a^{3} + 21 \, {\left (13 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) + 7 \, a^{3}\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 e^{5} \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, d e^{5} \cos \left (2 \, d x + 2 \, c\right ) + d e^{5}\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 \[ \int \frac {{\left (a \cot \left (d x + c\right ) + a\right )}^{3}}{\left (e \cot \left (d x + c\right )\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.54, size = 430, normalized size = 2.61 \[ \frac {a^{3} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{2 d \,e^{5}}+\frac {a^{3} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \,e^{5}}-\frac {a^{3} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \,e^{5}}-\frac {a^{3} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{2 d \,e^{4} \left (e^{2}\right )^{\frac {1}{4}}}-\frac {a^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \,e^{4} \left (e^{2}\right )^{\frac {1}{4}}}+\frac {a^{3} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \,e^{4} \left (e^{2}\right )^{\frac {1}{4}}}+\frac {2 a^{3}}{7 d e \left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}+\frac {6 a^{3}}{5 d \,e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {4 a^{3}}{d \,e^{4} \sqrt {e \cot \left (d x +c \right )}}+\frac {4 a^{3}}{3 d \,e^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 170, normalized size = 1.03 \[ \frac {e {\left (\frac {105 \, a^{3} {\left (\frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} - \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}\right )}}{e^{5}} + \frac {2 \, {\left (15 \, a^{3} e^{3} + \frac {63 \, a^{3} e^{3}}{\tan \left (d x + c\right )} + \frac {70 \, a^{3} e^{3}}{\tan \left (d x + c\right )^{2}} - \frac {210 \, a^{3} e^{3}}{\tan \left (d x + c\right )^{3}}\right )}}{e^{5} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {7}{2}}}\right )}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.92, size = 129, normalized size = 0.78 \[ \frac {-4\,e\,a^3\,{\mathrm {cot}\left (c+d\,x\right )}^3+\frac {4\,e\,a^3\,{\mathrm {cot}\left (c+d\,x\right )}^2}{3}+\frac {6\,e\,a^3\,\mathrm {cot}\left (c+d\,x\right )}{5}+\frac {2\,e\,a^3}{7}}{d\,e^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{7/2}}+\frac {2\,\sqrt {2}\,a^3\,\mathrm {atanh}\left (\frac {32\,\sqrt {2}\,a^6\,d\,e^{9/2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{32\,a^6\,d\,e^5+32\,a^6\,d\,e^5\,\mathrm {cot}\left (c+d\,x\right )}\right )}{d\,e^{9/2}} \]
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 \[ a^{3} \left (\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {9}{2}}}\, dx + \int \frac {3 \cot {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {9}{2}}}\, dx + \int \frac {3 \cot ^{2}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {9}{2}}}\, dx + \int \frac {\cot ^{3}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {9}{2}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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