Optimal. Leaf size=164 \[ -\frac{5 e^2 \sqrt{e \cot (c+d x)}}{8 a^3 d (\cot (c+d x)+1)}-\frac{e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{8 a^3 d}+\frac{e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d}+\frac{e^2 \sqrt{e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2} \]
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Rubi [A] time = 0.617429, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3565, 3649, 3654, 3532, 208, 3634, 63, 205} \[ -\frac{5 e^2 \sqrt{e \cot (c+d x)}}{8 a^3 d (\cot (c+d x)+1)}-\frac{e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{8 a^3 d}+\frac{e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d}+\frac{e^2 \sqrt{e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 3565
Rule 3649
Rule 3654
Rule 3532
Rule 208
Rule 3634
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{(e \cot (c+d x))^{5/2}}{(a+a \cot (c+d x))^3} \, dx &=\frac{e^2 \sqrt{e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}-\frac{\int \frac{-\frac{1}{2} a^2 e^3+2 a^2 e^3 \cot (c+d x)-\frac{5}{2} a^2 e^3 \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^2} \, dx}{4 a^3}\\ &=-\frac{5 e^2 \sqrt{e \cot (c+d x)}}{8 a^3 d (1+\cot (c+d x))}+\frac{e^2 \sqrt{e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}+\frac{\int \frac{-\frac{3}{2} a^4 e^4+\frac{5}{2} a^4 e^4 \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{8 a^6 e}\\ &=-\frac{5 e^2 \sqrt{e \cot (c+d x)}}{8 a^3 d (1+\cot (c+d x))}+\frac{e^2 \sqrt{e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}+\frac{\int \frac{-4 a^5 e^4+4 a^5 e^4 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{16 a^8 e}+\frac{e^3 \int \frac{1+\cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{16 a^2}\\ &=-\frac{5 e^2 \sqrt{e \cot (c+d x)}}{8 a^3 d (1+\cot (c+d x))}+\frac{e^2 \sqrt{e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}+\frac{e^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-e x} (a-a x)} \, dx,x,-\cot (c+d x)\right )}{16 a^2 d}-\frac{\left (2 a^2 e^7\right ) \operatorname{Subst}\left (\int \frac{1}{32 a^{10} e^8-e x^2} \, dx,x,\frac{-4 a^5 e^4-4 a^5 e^4 \cot (c+d x)}{\sqrt{e \cot (c+d x)}}\right )}{d}\\ &=\frac{e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e}+\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d}-\frac{5 e^2 \sqrt{e \cot (c+d x)}}{8 a^3 d (1+\cot (c+d x))}+\frac{e^2 \sqrt{e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}-\frac{e^2 \operatorname{Subst}\left (\int \frac{1}{a+\frac{a x^2}{e}} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{8 a^2 d}\\ &=-\frac{e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{8 a^3 d}+\frac{e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e}+\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d}-\frac{5 e^2 \sqrt{e \cot (c+d x)}}{8 a^3 d (1+\cot (c+d x))}+\frac{e^2 \sqrt{e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}\\ \end{align*}
Mathematica [A] time = 2.07205, size = 192, normalized size = 1.17 \[ \frac{\csc (c+d x) (e \cot (c+d x))^{5/2} (\sin (c+d x)+\cos (c+d x))^3 \left (\frac{\sec ^4(c+d x) (-5 \sin (2 (c+d x))+3 \cos (2 (c+d x))-3)}{(\tan (c+d x)+1)^2}-\frac{2 \csc (c+d x) \sec (c+d x) \left (\sqrt{2} \left (\log \left (-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right )-\log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )+\tan ^{-1}\left (\sqrt{\cot (c+d x)}\right )\right )}{\cot ^{\frac{3}{2}}(c+d x)}\right )}{16 a^3 d (\cot (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 440, normalized size = 2.7 \begin{align*}{\frac{{e}^{2}\sqrt{2}}{16\,d{a}^{3}}\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{{e}^{2}\sqrt{2}}{8\,d{a}^{3}}\sqrt [4]{{e}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }-{\frac{{e}^{2}\sqrt{2}}{8\,d{a}^{3}}\sqrt [4]{{e}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }-{\frac{{e}^{3}\sqrt{2}}{16\,d{a}^{3}}\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{{e}^{3}\sqrt{2}}{8\,d{a}^{3}}\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{{e}^{3}\sqrt{2}}{8\,d{a}^{3}}\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{5\,{e}^{3}}{8\,d{a}^{3} \left ( e\cot \left ( dx+c \right ) +e \right ) ^{2}} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{e}^{4}}{8\,d{a}^{3} \left ( e\cot \left ( dx+c \right ) +e \right ) ^{2}}\sqrt{e\cot \left ( dx+c \right ) }}-{\frac{1}{8\,d{a}^{3}}{e}^{{\frac{5}{2}}}\arctan \left ({\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt{e}}}} \right ) } \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 = 2.10143, size = 1407, normalized size = 8.58 \begin{align*} \left [-\frac{4 \,{\left (\sqrt{2} e^{2} \sin \left (2 \, d x + 2 \, c\right ) + \sqrt{2} e^{2}\right )} \sqrt{-e} \arctan \left (\frac{{\left (\sqrt{2} \cos \left (2 \, d x + 2 \, c\right ) + \sqrt{2} \sin \left (2 \, d x + 2 \, c\right ) + \sqrt{2}\right )} \sqrt{-e} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{2 \,{\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) -{\left (e^{2} \sin \left (2 \, d x + 2 \, c\right ) + e^{2}\right )} \sqrt{-e} \log \left (\frac{e \cos \left (2 \, d x + 2 \, c\right ) - e \sin \left (2 \, d x + 2 \, c\right ) - 2 \, \sqrt{-e} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) + e}{\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1}\right ) -{\left (3 \, e^{2} \cos \left (2 \, d x + 2 \, c\right ) - 5 \, e^{2} \sin \left (2 \, d x + 2 \, c\right ) - 3 \, e^{2}\right )} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{16 \,{\left (a^{3} d \sin \left (2 \, d x + 2 \, c\right ) + a^{3} d\right )}}, -\frac{2 \,{\left (e^{2} \sin \left (2 \, d x + 2 \, c\right ) + e^{2}\right )} \sqrt{e} \arctan \left (\frac{\sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{\sqrt{e}}\right ) - 2 \,{\left (\sqrt{2} e^{2} \sin \left (2 \, d x + 2 \, c\right ) + \sqrt{2} e^{2}\right )} \sqrt{e} \log \left (-{\left (\sqrt{2} \cos \left (2 \, d x + 2 \, c\right ) - \sqrt{2} \sin \left (2 \, d x + 2 \, c\right ) - \sqrt{2}\right )} \sqrt{e} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} + 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) -{\left (3 \, e^{2} \cos \left (2 \, d x + 2 \, c\right ) - 5 \, e^{2} \sin \left (2 \, d x + 2 \, c\right ) - 3 \, e^{2}\right )} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{16 \,{\left (a^{3} d \sin \left (2 \, d x + 2 \, c\right ) + a^{3} d\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{\left (e \cot \left (d x + c\right )\right )^{\frac{5}{2}}}{{\left (a \cot \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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