Optimal. Leaf size=160 \[ \frac{2 a^6 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^5 \tan ^5(c+d x)}{d (a \sec (c+d x)+a)^{5/2}}+\frac{14 a^4 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{2 a^3 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.0987673, antiderivative size = 160, 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, 461, 203} \[ \frac{2 a^6 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^5 \tan ^5(c+d x)}{d (a \sec (c+d x)+a)^{5/2}}+\frac{14 a^4 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{2 a^3 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3887
Rule 461
Rule 203
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^{5/2} \tan ^2(c+d x) \, dx &=-\frac{\left (2 a^4\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (2+a x^2\right )^3}{1+a x^2} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=-\frac{\left (2 a^4\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a}+7 x^2+5 a x^4+a^2 x^6-\frac{1}{a \left (1+a x^2\right )}\right ) \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=\frac{2 a^3 \tan (c+d x)}{d \sqrt{a+a \sec (c+d x)}}+\frac{14 a^4 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac{2 a^5 \tan ^5(c+d x)}{d (a+a \sec (c+d x))^{5/2}}+\frac{2 a^6 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\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^3 \tan (c+d x)}{d \sqrt{a+a \sec (c+d x)}}+\frac{14 a^4 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac{2 a^5 \tan ^5(c+d x)}{d (a+a \sec (c+d x))^{5/2}}+\frac{2 a^6 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}\\ \end{align*}
Mathematica [A] time = 5.75748, size = 125, normalized size = 0.78 \[ -\frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sec ^3(c+d x) \sqrt{a (\sec (c+d x)+1)} \left (-35 \sin \left (\frac{1}{2} (c+d x)\right )+7 \sin \left (\frac{3}{2} (c+d x)\right )-21 \sin \left (\frac{5}{2} (c+d x)\right )+5 \sin \left (\frac{7}{2} (c+d x)\right )+42 \sqrt{2} \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right ) \cos ^{\frac{7}{2}}(c+d x)\right )}{42 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.189, size = 391, normalized size = 2.4 \begin{align*} -{\frac{{a}^{2}}{168\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 21\,\sqrt{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{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 ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{7/2}+63\,\sqrt{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{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 ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{7/2}+63\,\sqrt{2}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ){\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 ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{7/2}+21\,\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 ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{7/2}\sin \left ( dx+c \right ) -160\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+416\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-64\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-144\,\cos \left ( dx+c \right ) -48 \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 = 1.82668, size = 945, normalized size = 5.91 \begin{align*} \left [\frac{21 \,{\left (a^{2} \cos \left (d x + c\right )^{4} + a^{2} \cos \left (d x + c\right )^{3}\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 (10 \, a^{2} \cos \left (d x + c\right )^{3} - 16 \, a^{2} \cos \left (d x + c\right )^{2} - 12 \, a^{2} \cos \left (d x + c\right ) - 3 \, 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 )}{21 \,{\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}}, \frac{2 \,{\left (21 \,{\left (a^{2} \cos \left (d x + c\right )^{4} + a^{2} \cos \left (d x + c\right )^{3}\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 (10 \, a^{2} \cos \left (d x + c\right )^{3} - 16 \, a^{2} \cos \left (d x + c\right )^{2} - 12 \, a^{2} \cos \left (d x + c\right ) - 3 \, 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 )}}{21 \,{\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\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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