Optimal. Leaf size=106 \[ \frac{9 a^3 \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{19 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{4 d}+\frac{a^2 \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d} \]
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Rubi [A] time = 0.164392, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3813, 4015, 3774, 203} \[ \frac{9 a^3 \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{19 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{4 d}+\frac{a^2 \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d} \]
Antiderivative was successfully verified.
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Rule 3813
Rule 4015
Rule 3774
Rule 203
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \, dx &=\frac{a^2 \cos (c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{2 d}+\frac{1}{2} a \int \cos (c+d x) \sqrt{a+a \sec (c+d x)} \left (\frac{9 a}{2}+\frac{5}{2} a \sec (c+d x)\right ) \, dx\\ &=\frac{9 a^3 \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 \cos (c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{2 d}+\frac{1}{8} \left (19 a^2\right ) \int \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{9 a^3 \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 \cos (c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{2 d}-\frac{\left (19 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 )}{4 d}\\ &=\frac{19 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{4 d}+\frac{9 a^3 \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 \cos (c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [C] time = 0.522436, size = 150, normalized size = 1.42 \[ -\frac{a^2 \cos (c+d x) \sqrt{a (\sec (c+d x)+1)} \left (-32 \tan (c+d x) \sqrt{1-\sec (c+d x)} \text{Hypergeometric2F1}\left (\frac{1}{2},3,\frac{3}{2},1-\sec (c+d x)\right )+(\sin (c+d x)+3 \sin (2 (c+d x))) \sqrt{1-\sec (c+d x)}-7 \tan (c+d x) \tanh ^{-1}\left (\sqrt{1-\sec (c+d x)}\right )\right )}{4 d (\cos (c+d x)+1) \sqrt{1-\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.189, size = 224, normalized size = 2.1 \begin{align*}{\frac{{a}^{2}}{16\,d\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) } \left ( 19\,\sqrt{2} \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{3/2}{\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 ) +19\,\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 ) ^{3/2}\sin \left ( dx+c \right ) -8\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}-36\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+44\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \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 = 2.04913, size = 763, normalized size = 7.2 \begin{align*} \left [\frac{19 \,{\left (a^{2} \cos \left (d x + c\right ) + a^{2}\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 (2 \, a^{2} \cos \left (d x + c\right )^{2} + 11 \, 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 )}} \sin \left (d x + c\right )}{8 \,{\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac{19 \,{\left (a^{2} \cos \left (d x + c\right ) + a^{2}\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 (2 \, a^{2} \cos \left (d x + c\right )^{2} + 11 \, 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 )}} \sin \left (d x + c\right )}{4 \,{\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 [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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