Optimal. Leaf size=65 \[ \frac{a^2 \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{3 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d} \]
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Rubi [A] time = 0.118184, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3814, 21, 3805, 3774, 203} \[ \frac{a^2 \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{3 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3814
Rule 21
Rule 3805
Rule 3774
Rule 203
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \, dx &=-\frac{2 a^2 \sin (c+d x)}{d \sqrt{a+a \sec (c+d x)}}-(2 a) \int \frac{\cos (c+d x) \left (-\frac{3 a}{2}-\frac{3}{2} a \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx\\ &=-\frac{2 a^2 \sin (c+d x)}{d \sqrt{a+a \sec (c+d x)}}+(3 a) \int \cos (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^2 \sin (c+d x)}{d \sqrt{a+a \sec (c+d x)}}+\frac{1}{2} (3 a) \int \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^2 \sin (c+d x)}{d \sqrt{a+a \sec (c+d x)}}-\frac{\left (3 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 )}{d}\\ &=\frac{3 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}+\frac{a^2 \sin (c+d x)}{d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.192966, size = 89, normalized size = 1.37 \[ \frac{a \sqrt{\cos (c+d x)} \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} \left (3 \sqrt{2} \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )+2 \sin \left (\frac{1}{2} (c+d x)\right ) \sqrt{\cos (c+d x)}\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.162, size = 125, normalized size = 1.9 \begin{align*} -{\frac{a}{2\,d\sin \left ( dx+c \right ) } \left ( 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 ) +2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-2\,\cos \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 = 2.04867, size = 1084, normalized size = 16.68 \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 = 1.73447, size = 662, normalized size = 10.18 \begin{align*} \left [\frac{2 \, 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 ) + 3 \,{\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 (d \cos \left (d x + c\right ) + d\right )}}, \frac{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 ) - 3 \,{\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 )}{d \cos \left (d x + c\right ) + d}\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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