Optimal. Leaf size=180 \[ \frac{a^3 (4 A-9 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (4 A+7 B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{a^{5/2} (20 A+19 B) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{4 d}+\frac{a B \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}{2 d} \]
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Rubi [A] time = 0.504055, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {4018, 4015, 3801, 215} \[ \frac{a^3 (4 A-9 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (4 A+7 B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{a^{5/2} (20 A+19 B) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{4 d}+\frac{a B \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}{2 d} \]
Antiderivative was successfully verified.
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Rule 4018
Rule 4015
Rule 3801
Rule 215
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx &=\frac{a B \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{2 d}+\frac{1}{2} \int \frac{(a+a \sec (c+d x))^{3/2} \left (\frac{1}{2} a (4 A-B)+\frac{1}{2} a (4 A+7 B) \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{a^2 (4 A+7 B) \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{a B \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{2 d}+\frac{1}{2} \int \frac{\sqrt{a+a \sec (c+d x)} \left (\frac{1}{4} a^2 (4 A-9 B)+\frac{1}{4} a^2 (20 A+19 B) \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{a^3 (4 A-9 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (4 A+7 B) \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{a B \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{2 d}+\frac{1}{8} \left (a^2 (20 A+19 B)\right ) \int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^3 (4 A-9 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (4 A+7 B) \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{a B \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{2 d}-\frac{\left (a^2 (20 A+19 B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a}}} \, dx,x,-\frac{a \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{4 d}\\ &=\frac{a^{5/2} (20 A+19 B) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{4 d}+\frac{a^3 (4 A-9 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (4 A+7 B) \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{a B \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 1.81061, size = 137, normalized size = 0.76 \[ \frac{a^3 \left (\sqrt{-(\sec (c+d x)-1) \sec (c+d x)} (\tan (c+d x) (4 A+2 B \sec (c+d x)+11 B)+8 A \sin (c+d x))+20 A \tan (c+d x) \sin ^{-1}\left (\sqrt{1-\sec (c+d x)}\right )-19 B \tan (c+d x) \sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right )}{4 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.367, size = 386, normalized size = 2.1 \begin{align*} -{\frac{{a}^{2}}{16\,d\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 20\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) \sqrt{2}\arctan \left ( 1/4\,\sqrt{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \left ( -\cos \left ( dx+c \right ) -1+\sin \left ( dx+c \right ) \right ) \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}+20\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) \sqrt{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\arctan \left ( 1/4\,\sqrt{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \left ( \cos \left ( dx+c \right ) +1+\sin \left ( dx+c \right ) \right ) \right ) +19\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) \sqrt{2}\arctan \left ( 1/4\,\sqrt{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \left ( -\cos \left ( dx+c \right ) -1+\sin \left ( dx+c \right ) \right ) \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}+19\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) \sqrt{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\arctan \left ( 1/4\,\sqrt{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \left ( \cos \left ( dx+c \right ) +1+\sin \left ( dx+c \right ) \right ) \right ) +32\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}-16\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+44\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}-16\,A\cos \left ( dx+c \right ) -36\,B\cos \left ( dx+c \right ) -8\,B \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}}} \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 = 0.768991, size = 1169, normalized size = 6.49 \begin{align*} \left [\frac{{\left ({\left (20 \, A + 19 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (20 \, A + 19 \, B\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt{a} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - \frac{4 \,{\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right )\right )} \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + \frac{4 \,{\left (8 \, A a^{2} \cos \left (d x + c\right )^{2} +{\left (4 \, A + 11 \, B\right )} a^{2} \cos \left (d x + c\right ) + 2 \, B 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 )}{\sqrt{\cos \left (d x + c\right )}}}{16 \,{\left (d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )}}, \frac{{\left ({\left (20 \, A + 19 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (20 \, A + 19 \, B\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - 2 \, a}\right ) + \frac{2 \,{\left (8 \, A a^{2} \cos \left (d x + c\right )^{2} +{\left (4 \, A + 11 \, B\right )} a^{2} \cos \left (d x + c\right ) + 2 \, B 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 )}{\sqrt{\cos \left (d x + c\right )}}}{8 \,{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \sec \left (d x + c\right ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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