Optimal. Leaf size=119 \[ \frac{a^2 (5 A+4 B) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a^{3/2} (7 A+12 B) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{4 d}+\frac{a A \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d} \]
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Rubi [A] time = 0.272987, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {4017, 4015, 3774, 203} \[ \frac{a^2 (5 A+4 B) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a^{3/2} (7 A+12 B) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{4 d}+\frac{a A \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d} \]
Antiderivative was successfully verified.
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Rule 4017
Rule 4015
Rule 3774
Rule 203
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx &=\frac{a A \cos (c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{2 d}+\frac{1}{2} \int \cos (c+d x) \sqrt{a+a \sec (c+d x)} \left (\frac{1}{2} a (5 A+4 B)+\frac{1}{2} a (A+4 B) \sec (c+d x)\right ) \, dx\\ &=\frac{a^2 (5 A+4 B) \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{a A \cos (c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{2 d}+\frac{1}{8} (a (7 A+12 B)) \int \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^2 (5 A+4 B) \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{a A \cos (c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{2 d}-\frac{\left (a^2 (7 A+12 B)\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{a^{3/2} (7 A+12 B) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{4 d}+\frac{a^2 (5 A+4 B) \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{a A \cos (c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.597141, size = 111, normalized size = 0.93 \[ \frac{a \sqrt{\cos (c+d x)} \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} \left (\sqrt{2} (7 A+12 B) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )+2 \sin \left (\frac{1}{2} (c+d x)\right ) \sqrt{\cos (c+d x)} (2 A \cos (c+d x)+7 A+4 B)\right )}{8 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.296, size = 399, normalized size = 3.4 \begin{align*}{\frac{a}{16\,d\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) } \left ( 7\,A\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) \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 ) \sqrt{2}+12\,B\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) \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 ) \sqrt{2}+7\,A \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 ) \sqrt{2}\sin \left ( dx+c \right ) +12\,B \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 ) \sqrt{2}\sin \left ( dx+c \right ) -8\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}-20\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}-16\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+28\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+16\,B \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 = 0.616904, size = 833, normalized size = 7. \begin{align*} \left [\frac{{\left ({\left (7 \, A + 12 \, B\right )} a \cos \left (d x + c\right ) +{\left (7 \, A + 12 \, B\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 (2 \, A a \cos \left (d x + c\right )^{2} +{\left (7 \, A + 4 \, B\right )} 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 )}{8 \,{\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac{{\left ({\left (7 \, A + 12 \, B\right )} a \cos \left (d x + c\right ) +{\left (7 \, A + 12 \, B\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 (2 \, A a \cos \left (d x + c\right )^{2} +{\left (7 \, A + 4 \, B\right )} 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 )}{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 [B] time = 7.10897, size = 863, normalized size = 7.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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