Optimal. Leaf size=170 \[ \frac{2 a^3 (49 A+32 C) \tan (c+d x)}{21 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (7 A+8 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 a^{5/2} A \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{2 a C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d} \]
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Rubi [A] time = 0.308362, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4055, 3917, 3915, 3774, 203, 3792} \[ \frac{2 a^3 (49 A+32 C) \tan (c+d x)}{21 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (7 A+8 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 a^{5/2} A \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{2 a C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 4055
Rule 3917
Rule 3915
Rule 3774
Rule 203
Rule 3792
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac{2 \int (a+a \sec (c+d x))^{5/2} \left (\frac{7 a A}{2}+\frac{5}{2} a C \sec (c+d x)\right ) \, dx}{7 a}\\ &=\frac{2 a C (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac{2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac{4 \int (a+a \sec (c+d x))^{3/2} \left (\frac{35 a^2 A}{4}+\frac{5}{4} a^2 (7 A+8 C) \sec (c+d x)\right ) \, dx}{35 a}\\ &=\frac{2 a^2 (7 A+8 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{21 d}+\frac{2 a C (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac{2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac{8 \int \sqrt{a+a \sec (c+d x)} \left (\frac{105 a^3 A}{8}+\frac{5}{8} a^3 (49 A+32 C) \sec (c+d x)\right ) \, dx}{105 a}\\ &=\frac{2 a^2 (7 A+8 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{21 d}+\frac{2 a C (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac{2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\left (a^2 A\right ) \int \sqrt{a+a \sec (c+d x)} \, dx+\frac{1}{21} \left (a^2 (49 A+32 C)\right ) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{2 a^3 (49 A+32 C) \tan (c+d x)}{21 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (7 A+8 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{21 d}+\frac{2 a C (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac{2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}-\frac{\left (2 a^3 A\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{2 a^{5/2} A \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}+\frac{2 a^3 (49 A+32 C) \tan (c+d x)}{21 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (7 A+8 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{21 d}+\frac{2 a C (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac{2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 1.83551, size = 151, normalized size = 0.89 \[ \frac{a^2 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^3(c+d x) \sqrt{a (\sec (c+d x)+1)} \left (\sqrt{\sec (c+d x)-1} ((84 A+93 C) \cos (c+d x)+(7 A+23 C) \cos (2 (c+d x))+28 A \cos (3 (c+d x))+7 A+23 C \cos (3 (c+d x))+29 C)+42 A \cos ^3(c+d x) \tan ^{-1}\left (\sqrt{\sec (c+d x)-1}\right )\right )}{21 d \sqrt{\sec (c+d x)-1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.314, size = 434, normalized size = 2.6 \begin{align*}{\frac{{a}^{2}}{168\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 21\,A\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}\sqrt{2}+63\,A\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}\sqrt{2}+63\,A\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}\sqrt{2}+21\,A{\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}\sqrt{2}\sin \left ( dx+c \right ) -896\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}-736\,C \left ( \cos \left ( dx+c \right ) \right ) ^{4}+784\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+368\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+112\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+176\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+144\,C\cos \left ( dx+c \right ) +48\,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 = 0.582656, size = 1030, normalized size = 6.06 \begin{align*} \left [\frac{21 \,{\left (A a^{2} \cos \left (d x + c\right )^{4} + A 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 (2 \,{\left (28 \, A + 23 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} +{\left (7 \, A + 23 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 12 \, C a^{2} \cos \left (d x + c\right ) + 3 \, C 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 a^{2} \cos \left (d x + c\right )^{4} + A 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 (2 \,{\left (28 \, A + 23 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} +{\left (7 \, A + 23 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 12 \, C a^{2} \cos \left (d x + c\right ) + 3 \, C 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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