Optimal. Leaf size=188 \[ \frac{a^3 (27 A-56 C) \sin (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (A-8 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}+\frac{a^{5/2} (19 A+8 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{4 d}-\frac{a (3 A-4 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{6 d}+\frac{A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{5/2}}{2 d} \]
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Rubi [A] time = 0.598209, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4087, 4018, 4015, 3774, 203} \[ \frac{a^3 (27 A-56 C) \sin (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (A-8 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}+\frac{a^{5/2} (19 A+8 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{4 d}-\frac{a (3 A-4 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{6 d}+\frac{A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{5/2}}{2 d} \]
Antiderivative was successfully verified.
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Rule 4087
Rule 4018
Rule 4015
Rule 3774
Rule 203
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos (c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{2 d}+\frac{\int \cos (c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac{5 a A}{2}-\frac{1}{2} a (3 A-4 C) \sec (c+d x)\right ) \, dx}{2 a}\\ &=-\frac{a (3 A-4 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac{A \cos (c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{2 d}+\frac{\int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac{1}{4} a^2 (21 A-8 C)-\frac{3}{4} a^2 (A-8 C) \sec (c+d x)\right ) \, dx}{3 a}\\ &=-\frac{a^2 (A-8 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{2 d}-\frac{a (3 A-4 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac{A \cos (c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{2 d}+\frac{2 \int \cos (c+d x) \sqrt{a+a \sec (c+d x)} \left (\frac{1}{8} a^3 (27 A-56 C)+\frac{5}{8} a^3 (3 A+8 C) \sec (c+d x)\right ) \, dx}{3 a}\\ &=\frac{a^3 (27 A-56 C) \sin (c+d x)}{12 d \sqrt{a+a \sec (c+d x)}}-\frac{a^2 (A-8 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{2 d}-\frac{a (3 A-4 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac{A \cos (c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{2 d}+\frac{1}{8} \left (a^2 (19 A+8 C)\right ) \int \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^3 (27 A-56 C) \sin (c+d x)}{12 d \sqrt{a+a \sec (c+d x)}}-\frac{a^2 (A-8 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{2 d}-\frac{a (3 A-4 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac{A \cos (c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{2 d}-\frac{\left (a^3 (19 A+8 C)\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^{5/2} (19 A+8 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{4 d}+\frac{a^3 (27 A-56 C) \sin (c+d x)}{12 d \sqrt{a+a \sec (c+d x)}}-\frac{a^2 (A-8 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{2 d}-\frac{a (3 A-4 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac{A \cos (c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.885886, size = 137, normalized size = 0.73 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \sqrt{a (\sec (c+d x)+1)} \left (6 \sqrt{2} (19 A+8 C) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right ) \cos ^{\frac{3}{2}}(c+d x)+2 \sin \left (\frac{1}{2} (c+d x)\right ) ((9 A+128 C) \cos (c+d x)+33 A \cos (2 (c+d x))+3 A \cos (3 (c+d x))+33 A+16 C)\right )}{48 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.361, size = 402, normalized size = 2.1 \begin{align*} -{\frac{{a}^{2}}{48\,d\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( -57\,A\sqrt{2}\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 ) ^{3/2}-24\,C\cos \left ( dx+c \right ) \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 ) -57\,A\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 ) -24\,C{\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} \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{3/2}\sin \left ( dx+c \right ) +24\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+108\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}-132\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+256\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}-224\,C\cos \left ( dx+c \right ) -32\,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 = 0.671674, size = 1022, normalized size = 5.44 \begin{align*} \left [\frac{3 \,{\left ({\left (19 \, A + 8 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (19 \, A + 8 \, C\right )} a^{2} \cos \left (d x + c\right )\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 (6 \, A a^{2} \cos \left (d x + c\right )^{3} + 33 \, A a^{2} \cos \left (d x + c\right )^{2} + 64 \, C a^{2} \cos \left (d x + c\right ) + 8 \, 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 )}{24 \,{\left (d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )}}, -\frac{3 \,{\left ({\left (19 \, A + 8 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (19 \, A + 8 \, C\right )} a^{2} \cos \left (d x + c\right )\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 (6 \, A a^{2} \cos \left (d x + c\right )^{3} + 33 \, A a^{2} \cos \left (d x + c\right )^{2} + 64 \, C a^{2} \cos \left (d x + c\right ) + 8 \, 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 )}{12 \,{\left (d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\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 = 7.03085, size = 748, normalized size = 3.98 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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