Optimal. Leaf size=165 \[ -\frac{7 \sqrt{e \cot (c+d x)}}{8 a^3 d e (\cot (c+d x)+1)}-\frac{11 \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{8 a^3 d \sqrt{e}}-\frac{\tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d \sqrt{e}}-\frac{\sqrt{e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2} \]
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Rubi [A] time = 0.647756, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {3569, 3649, 3653, 3532, 205, 3634, 63} \[ -\frac{7 \sqrt{e \cot (c+d x)}}{8 a^3 d e (\cot (c+d x)+1)}-\frac{11 \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{8 a^3 d \sqrt{e}}-\frac{\tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d \sqrt{e}}-\frac{\sqrt{e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 3569
Rule 3649
Rule 3653
Rule 3532
Rule 205
Rule 3634
Rule 63
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^3} \, dx &=-\frac{\sqrt{e \cot (c+d x)}}{4 a d e (a+a \cot (c+d x))^2}-\frac{\int \frac{-\frac{7 a^2 e}{2}+2 a^2 e \cot (c+d x)-\frac{3}{2} a^2 e \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^2} \, dx}{4 a^3 e}\\ &=-\frac{7 \sqrt{e \cot (c+d x)}}{8 a^3 d e (1+\cot (c+d x))}-\frac{\sqrt{e \cot (c+d x)}}{4 a d e (a+a \cot (c+d x))^2}+\frac{\int \frac{\frac{7 a^4 e^2}{2}-4 a^4 e^2 \cot (c+d x)+\frac{7}{2} a^4 e^2 \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{8 a^6 e^2}\\ &=-\frac{7 \sqrt{e \cot (c+d x)}}{8 a^3 d e (1+\cot (c+d x))}-\frac{\sqrt{e \cot (c+d x)}}{4 a d e (a+a \cot (c+d x))^2}+\frac{11 \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{\int \frac{-4 a^5 e^2-4 a^5 e^2 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{16 a^8 e^2}\\ &=-\frac{7 \sqrt{e \cot (c+d x)}}{8 a^3 d e (1+\cot (c+d x))}-\frac{\sqrt{e \cot (c+d x)}}{4 a d e (a+a \cot (c+d x))^2}+\frac{11 \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^2\right ) \operatorname{Subst}\left (\int \frac{1}{-32 a^{10} e^4-e x^2} \, dx,x,\frac{-4 a^5 e^2+4 a^5 e^2 \cot (c+d x)}{\sqrt{e \cot (c+d x)}}\right )}{d}\\ &=-\frac{\tan ^{-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 \sqrt{e}}-\frac{7 \sqrt{e \cot (c+d x)}}{8 a^3 d e (1+\cot (c+d x))}-\frac{\sqrt{e \cot (c+d x)}}{4 a d e (a+a \cot (c+d x))^2}-\frac{11 \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 e}\\ &=-\frac{11 \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{8 a^3 d \sqrt{e}}-\frac{\tan ^{-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 \sqrt{e}}-\frac{7 \sqrt{e \cot (c+d x)}}{8 a^3 d e (1+\cot (c+d x))}-\frac{\sqrt{e \cot (c+d x)}}{4 a d e (a+a \cot (c+d x))^2}\\ \end{align*}
Mathematica [A] time = 1.22395, size = 217, normalized size = 1.32 \[ \frac{\sqrt{\cot (c+d x)} \left (-9 \sqrt{\cot (c+d x)}+9 \cos (2 (c+d x)) \sqrt{\cot (c+d x)}-7 \sin (2 (c+d x)) \sqrt{\cot (c+d x)}-22 \tan ^{-1}\left (\sqrt{\cot (c+d x)}\right )-22 \sin (2 (c+d x)) \tan ^{-1}\left (\sqrt{\cot (c+d x)}\right )-4 \sqrt{2} (\sin (c+d x)+\cos (c+d x))^2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )+4 \sqrt{2} (\sin (c+d x)+\cos (c+d x))^2 \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )}{16 a^3 d \sqrt{e \cot (c+d x)} (\sin (c+d x)+\cos (c+d x))^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.044, size = 426, normalized size = 2.6 \begin{align*}{\frac{\sqrt{2}}{16\,d{a}^{3}e}\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{\sqrt{2}}{8\,d{a}^{3}e}\sqrt [4]{{e}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }-{\frac{\sqrt{2}}{8\,d{a}^{3}e}\sqrt [4]{{e}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }+{\frac{\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{\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{\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{7}{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{9\,e}{8\,d{a}^{3} \left ( e\cot \left ( dx+c \right ) +e \right ) ^{2}}\sqrt{e\cot \left ( dx+c \right ) }}-{\frac{11}{8\,d{a}^{3}}\arctan \left ({\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt{e}}}} \right ){\frac{1}{\sqrt{e}}}} \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 = 1.68883, size = 1300, normalized size = 7.88 \begin{align*} \left [-\frac{2 \, \sqrt{2} \sqrt{-e}{\left (\sin \left (2 \, d x + 2 \, c\right ) + 1\right )} \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 ) + 11 \, \sqrt{-e}{\left (\sin \left (2 \, d x + 2 \, c\right ) + 1\right )} \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 ) - \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}{\left (9 \, \cos \left (2 \, d x + 2 \, c\right ) - 7 \, \sin \left (2 \, d x + 2 \, c\right ) - 9\right )}}{16 \,{\left (a^{3} d e \sin \left (2 \, d x + 2 \, c\right ) + a^{3} d e\right )}}, -\frac{4 \, \sqrt{2} \sqrt{e}{\left (\sin \left (2 \, d x + 2 \, c\right ) + 1\right )} \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 ) + 22 \, \sqrt{e}{\left (\sin \left (2 \, d x + 2 \, c\right ) + 1\right )} \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 ) - \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}{\left (9 \, \cos \left (2 \, d x + 2 \, c\right ) - 7 \, \sin \left (2 \, d x + 2 \, c\right ) - 9\right )}}{16 \,{\left (a^{3} d e \sin \left (2 \, d x + 2 \, c\right ) + a^{3} d e\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*} \frac{\int \frac{1}{\sqrt{e \cot{\left (c + d x \right )}} \cot ^{3}{\left (c + d x \right )} + 3 \sqrt{e \cot{\left (c + d x \right )}} \cot ^{2}{\left (c + d x \right )} + 3 \sqrt{e \cot{\left (c + d x \right )}} \cot{\left (c + d x \right )} + \sqrt{e \cot{\left (c + d x \right )}}}\, dx}{a^{3}} \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{1}{{\left (a \cot \left (d x + c\right ) + a\right )}^{3} \sqrt{e \cot \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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