Optimal. Leaf size=96 \[ \frac{2 a^2 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}-\frac{2 a \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d} \]
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Rubi [A] time = 0.0760531, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3887, 325, 203} \[ \frac{2 a^2 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}-\frac{2 a \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3887
Rule 325
Rule 203
Rubi steps
\begin{align*} \int \cot ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx &=-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (1+a x^2\right )} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=-\frac{2 a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{3 d}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+a x^2\right )} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=\frac{2 a^2 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{d}-\frac{2 a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{3 d}-\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}+\frac{2 a^2 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{d}-\frac{2 a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{3 d}\\ \end{align*}
Mathematica [C] time = 0.229109, size = 81, normalized size = 0.84 \[ -\frac{2 \left (\frac{1}{\cos (c+d x)+1}\right )^{3/2} \cot ^3(c+d x) (a (\sec (c+d x)+1))^{5/2} \text{Hypergeometric2F1}\left (-\frac{3}{2},-\frac{3}{2},-\frac{1}{2},2 \sin ^2\left (\frac{1}{2} (c+d x)\right )\right )}{3 d \sqrt{\frac{1}{\sec (c+d x)+1}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.279, size = 214, normalized size = 2.2 \begin{align*} -{\frac{{a}^{2}}{3\,d\sin \left ( dx+c \right ) \left ( -1+\cos \left ( dx+c \right ) \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 3\,\sqrt{2}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}} \right ) -3\,\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}} \right ) \sin \left ( dx+c \right ) -8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+6\,\cos \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 1.99897, size = 888, normalized size = 9.25 \begin{align*} \left [\frac{3 \,{\left (a^{2} \cos \left (d x + c\right ) - a^{2}\right )} \sqrt{-a} \log \left (-\frac{8 \, a \cos \left (d x + c\right )^{3} - 4 \,{\left (2 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right )\right )} \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) + 4 \,{\left (4 \, a^{2} \cos \left (d x + c\right )^{2} - 3 \, a^{2} \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{6 \,{\left (d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )}, \frac{3 \,{\left (a^{2} \cos \left (d x + c\right ) - a^{2}\right )} \sqrt{a} \arctan \left (\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a}\right ) \sin \left (d x + c\right ) + 2 \,{\left (4 \, a^{2} \cos \left (d x + c\right )^{2} - 3 \, a^{2} \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{3 \,{\left (d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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