Optimal. Leaf size=160 \[ -\frac{2 \sqrt{2} a^3 e^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d}-\frac{32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac{4 a^3 (e \cot (c+d x))^{3/2}}{3 d}+\frac{4 a^3 e \sqrt{e \cot (c+d x)}}{d}-\frac{2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{5/2}}{7 d e} \]
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Rubi [A] time = 0.258923, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3566, 3630, 3528, 3532, 208} \[ -\frac{2 \sqrt{2} a^3 e^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d}-\frac{32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac{4 a^3 (e \cot (c+d x))^{3/2}}{3 d}+\frac{4 a^3 e \sqrt{e \cot (c+d x)}}{d}-\frac{2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{5/2}}{7 d e} \]
Antiderivative was successfully verified.
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Rule 3566
Rule 3630
Rule 3528
Rule 3532
Rule 208
Rubi steps
\begin{align*} \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx &=-\frac{2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}-\frac{2 \int (e \cot (c+d x))^{3/2} \left (-a^3 e-7 a^3 e \cot (c+d x)-8 a^3 e \cot ^2(c+d x)\right ) \, dx}{7 e}\\ &=-\frac{32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac{2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}-\frac{2 \int (e \cot (c+d x))^{3/2} \left (7 a^3 e-7 a^3 e \cot (c+d x)\right ) \, dx}{7 e}\\ &=-\frac{4 a^3 (e \cot (c+d x))^{3/2}}{3 d}-\frac{32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac{2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}-\frac{2 \int \sqrt{e \cot (c+d x)} \left (7 a^3 e^2+7 a^3 e^2 \cot (c+d x)\right ) \, dx}{7 e}\\ &=\frac{4 a^3 e \sqrt{e \cot (c+d x)}}{d}-\frac{4 a^3 (e \cot (c+d x))^{3/2}}{3 d}-\frac{32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac{2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}-\frac{2 \int \frac{-7 a^3 e^3+7 a^3 e^3 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{7 e}\\ &=\frac{4 a^3 e \sqrt{e \cot (c+d x)}}{d}-\frac{4 a^3 (e \cot (c+d x))^{3/2}}{3 d}-\frac{32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac{2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}+\frac{\left (28 a^6 e^5\right ) \operatorname{Subst}\left (\int \frac{1}{98 a^6 e^6-e x^2} \, dx,x,\frac{-7 a^3 e^3-7 a^3 e^3 \cot (c+d x)}{\sqrt{e \cot (c+d x)}}\right )}{d}\\ &=-\frac{2 \sqrt{2} a^3 e^{3/2} \tanh ^{-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 \sqrt{e \cot (c+d x)}}{d}-\frac{4 a^3 (e \cot (c+d x))^{3/2}}{3 d}-\frac{32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac{2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}\\ \end{align*}
Mathematica [C] time = 2.8406, size = 332, normalized size = 2.08 \[ \frac{a^3 \sin (c+d x) (\cot (c+d x)+1)^3 (e \cot (c+d x))^{3/2} \left (280 \sin ^2(c+d x) \cot ^{\frac{3}{2}}(c+d x) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(c+d x)\right )-60 \cos ^2(c+d x) \cot ^{\frac{3}{2}}(c+d x)-280 \sin ^2(c+d x) \cot ^{\frac{3}{2}}(c+d x)-126 \sin (2 (c+d x)) \cot ^{\frac{3}{2}}(c+d x)+840 \sin ^2(c+d x) \sqrt{\cot (c+d x)}+105 \sqrt{2} \sin ^2(c+d x) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-105 \sqrt{2} \sin ^2(c+d x) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+210 \sqrt{2} \sin ^2(c+d x) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-210 \sqrt{2} \sin ^2(c+d x) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )}{210 d \cot ^{\frac{3}{2}}(c+d x) (\sin (c+d x)+\cos (c+d x))^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.024, size = 419, normalized size = 2.6 \begin{align*} -{\frac{2\,{a}^{3}}{7\,d{e}^{2}} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{7}{2}}}}-{\frac{6\,{a}^{3}}{5\,de} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{4\,{a}^{3}}{3\,d} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}+4\,{\frac{{a}^{3}e\sqrt{e\cot \left ( dx+c \right ) }}{d}}-{\frac{{a}^{3}e\sqrt{2}}{2\,d}\sqrt [4]{{e}^{2}}\ln \left ({ \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ) }-{\frac{{a}^{3}e\sqrt{2}}{d}\sqrt [4]{{e}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }+{\frac{{a}^{3}e\sqrt{2}}{d}\sqrt [4]{{e}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }+{\frac{{a}^{3}{e}^{2}\sqrt{2}}{2\,d}\ln \left ({ \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}+{\frac{{a}^{3}{e}^{2}\sqrt{2}}{d}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}-{\frac{{a}^{3}{e}^{2}\sqrt{2}}{d}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72703, size = 1206, normalized size = 7.54 \begin{align*} \left [\frac{105 \, \sqrt{2}{\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - a^{3} e\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 ) \sin \left (2 \, d x + 2 \, c\right ) - 2 \,{\left (55 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right )^{2} - 30 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - 85 \, a^{3} e - 21 \,{\left (13 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - 7 \, a^{3} e\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 \cos \left (2 \, d x + 2 \, c\right ) - d\right )} \sin \left (2 \, d x + 2 \, c\right )}, \frac{2 \,{\left (105 \, \sqrt{2}{\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - a^{3} e\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 ) \sin \left (2 \, d x + 2 \, c\right ) -{\left (55 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right )^{2} - 30 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - 85 \, a^{3} e - 21 \,{\left (13 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - 7 \, a^{3} e\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 \cos \left (2 \, d x + 2 \, c\right ) - d\right )} \sin \left (2 \, d x + 2 \, c\right )}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int \left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}}\, dx + \int 3 \left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}} \cot{\left (c + d x \right )}\, dx + \int 3 \left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}} \cot ^{2}{\left (c + d x \right )}\, dx + \int \left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}} \cot ^{3}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \cot \left (d x + c\right ) + a\right )}^{3} \left (e \cot \left (d x + c\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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