Optimal. Leaf size=144 \[ \frac{11 a^2 \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{8 d}+\frac{a^2 \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^2 \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.186515, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3813, 21, 3805, 3774, 203} \[ \frac{11 a^2 \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{8 d}+\frac{a^2 \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^2 \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3813
Rule 21
Rule 3805
Rule 3774
Rule 203
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \, dx &=\frac{a^2 \cos ^2(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \sec (c+d x)}}+\frac{1}{3} a \int \frac{\cos ^2(c+d x) \left (\frac{11 a}{2}+\frac{11}{2} a \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx\\ &=\frac{a^2 \cos ^2(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \sec (c+d x)}}+\frac{1}{6} (11 a) \int \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{11 a^2 \cos (c+d x) \sin (c+d x)}{12 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 \cos ^2(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \sec (c+d x)}}+\frac{1}{8} (11 a) \int \cos (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{11 a^2 \sin (c+d x)}{8 d \sqrt{a+a \sec (c+d x)}}+\frac{11 a^2 \cos (c+d x) \sin (c+d x)}{12 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 \cos ^2(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \sec (c+d x)}}+\frac{1}{16} (11 a) \int \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{11 a^2 \sin (c+d x)}{8 d \sqrt{a+a \sec (c+d x)}}+\frac{11 a^2 \cos (c+d x) \sin (c+d x)}{12 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 \cos ^2(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \sec (c+d x)}}-\frac{\left (11 a^2\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 )}{8 d}\\ &=\frac{11 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{8 d}+\frac{11 a^2 \sin (c+d x)}{8 d \sqrt{a+a \sec (c+d x)}}+\frac{11 a^2 \cos (c+d x) \sin (c+d x)}{12 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 \cos ^2(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.517337, size = 120, normalized size = 0.83 \[ \frac{a \cos (c+d x) \sqrt{a (\sec (c+d x)+1)} \left ((35 \sin (c+d x)+11 \sin (2 (c+d x))+2 \sin (3 (c+d x))) \sqrt{1-\sec (c+d x)}+33 \tan (c+d x) \tanh ^{-1}\left (\sqrt{1-\sec (c+d x)}\right )\right )}{24 d (\cos (c+d x)+1) \sqrt{1-\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.227, size = 311, normalized size = 2.2 \begin{align*} -{\frac{a}{192\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}} \left ( 33\, \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{5/2}\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 ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+66\, \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{5/2}\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 ) \cos \left ( dx+c \right ) +33\,\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 ) ^{5/2}\sin \left ( dx+c \right ) +64\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}+112\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}+88\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}-264\, \left ( \cos \left ( dx+c \right ) \right ) ^{3} \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 [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 = 2.09422, size = 803, normalized size = 5.58 \begin{align*} \left [\frac{33 \,{\left (a \cos \left (d x + c\right ) + a\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 \cos \left (d x + c\right )^{3} + 22 \, a \cos \left (d x + c\right )^{2} + 33 \, a \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{48 \,{\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac{33 \,{\left (a \cos \left (d x + c\right ) + a\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 \cos \left (d x + c\right )^{3} + 22 \, a \cos \left (d x + c\right )^{2} + 33 \, a \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{24 \,{\left (d \cos \left (d x + c\right ) + 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 [B] time = 7.36423, size = 702, normalized size = 4.88 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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