Optimal. Leaf size=98 \[ \frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{14 a^3 \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d} \]
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Rubi [A] time = 0.102234, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3775, 3915, 3774, 203, 3792} \[ \frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{14 a^3 \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3775
Rule 3915
Rule 3774
Rule 203
Rule 3792
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^{5/2} \, dx &=\frac{2 a^2 \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{3 d}+\frac{1}{3} (2 a) \int \sqrt{a+a \sec (c+d x)} \left (\frac{3 a}{2}+\frac{7}{2} a \sec (c+d x)\right ) \, dx\\ &=\frac{2 a^2 \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{3 d}+a^2 \int \sqrt{a+a \sec (c+d x)} \, dx+\frac{1}{3} \left (7 a^2\right ) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{14 a^3 \tan (c+d x)}{3 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{3 d}-\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,-\frac{a \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{14 a^3 \tan (c+d x)}{3 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [C] time = 10.1788, size = 360, normalized size = 3.67 \[ \frac{\sqrt{\frac{1}{1-2 \sin ^2\left (\frac{1}{2} (c+d x)\right )}} \sqrt{1-2 \sin ^2\left (\frac{1}{2} (c+d x)\right )} \csc ^3\left (\frac{1}{2} (c+d x)\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right ) (a (\sec (c+d x)+1))^{5/2} \left (256 \sin ^6\left (\frac{1}{2} (c+d x)\right ) \cos ^4\left (\frac{1}{2} (c+d x)\right ) \text{HypergeometricPFQ}\left (\left \{\frac{3}{2},2,\frac{7}{2}\right \},\left \{1,\frac{9}{2}\right \},2 \sin ^2\left (\frac{1}{2} (c+d x)\right )\right )+512 \left (\sin ^4\left (\frac{1}{2} (c+d x)\right )-3 \sin ^2\left (\frac{1}{2} (c+d x)\right )+2\right ) \sin ^6\left (\frac{1}{2} (c+d x)\right ) \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{7}{2},\frac{9}{2},2 \sin ^2\left (\frac{1}{2} (c+d x)\right )\right )+\frac{21 \sqrt{2} \sin ^{-1}\left (\sqrt{2} \sqrt{\sin ^2\left (\frac{1}{2} (c+d x)\right )}\right ) \left (3 \sin ^4\left (\frac{1}{2} (c+d x)\right )-10 \sin ^2\left (\frac{1}{2} (c+d x)\right )+15\right )}{\sqrt{\sin ^2\left (\frac{1}{2} (c+d x)\right )}}-14 \sqrt{1-2 \sin ^2\left (\frac{1}{2} (c+d x)\right )} \left (12 \sin ^6\left (\frac{1}{2} (c+d x)\right )-31 \sin ^4\left (\frac{1}{2} (c+d x)\right )+30 \sin ^2\left (\frac{1}{2} (c+d x)\right )+45\right )\right )}{672 d \sec ^{\frac{5}{2}}(c+d x)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.164, size = 214, normalized size = 2.2 \begin{align*} -{\frac{{a}^{2}}{3\,d \left ( \cos \left ( dx+c \right ) +1 \right ) \cos \left ( dx+c \right ) } \left ( 3\,{\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 ) \sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{2}+3\,\cos \left ( dx+c \right ) \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 ) \sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}-16\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -2\,\sin \left ( dx+c \right ) \right ) \sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.44421, size = 1883, normalized size = 19.21 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1009, size = 792, normalized size = 8.08 \begin{align*} \left [\frac{3 \,{\left (a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \sqrt{-a} \log \left (\frac{2 \, a \cos \left (d x + c\right )^{2} - 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 ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \,{\left (8 \, a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3 \,{\left (d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )}}, -\frac{2 \,{\left (3 \,{\left (a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \sqrt{a} \arctan \left (\frac{\sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt{a} \sin \left (d x + c\right )}\right ) -{\left (8 \, a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{3 \,{\left (d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\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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