Optimal. Leaf size=250 \[ -\frac{(15 A-11 B) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A-5 B) \sin (c+d x)}{10 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{(A-B) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}+\frac{(147 A-95 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{30 a d \sqrt{a \sec (c+d x)+a}}-\frac{(39 A-35 B) \sin (c+d x)}{30 a d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.734469, antiderivative size = 250, 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 = {4020, 4022, 4013, 3808, 206} \[ -\frac{(15 A-11 B) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A-5 B) \sin (c+d x)}{10 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{(A-B) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}+\frac{(147 A-95 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{30 a d \sqrt{a \sec (c+d x)+a}}-\frac{(39 A-35 B) \sin (c+d x)}{30 a d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 4020
Rule 4022
Rule 4013
Rule 3808
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx &=-\frac{(A-B) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac{\int \frac{\frac{1}{2} a (9 A-5 B)-3 a (A-B) \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{(A-B) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac{(9 A-5 B) \sin (c+d x)}{10 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{\int \frac{-\frac{1}{4} a^2 (39 A-35 B)+a^2 (9 A-5 B) \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx}{5 a^3}\\ &=-\frac{(A-B) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac{(9 A-5 B) \sin (c+d x)}{10 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}-\frac{(39 A-35 B) \sin (c+d x)}{30 a d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{2 \int \frac{\frac{1}{8} a^3 (147 A-95 B)-\frac{1}{4} a^3 (39 A-35 B) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx}{15 a^4}\\ &=-\frac{(A-B) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac{(9 A-5 B) \sin (c+d x)}{10 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}-\frac{(39 A-35 B) \sin (c+d x)}{30 a d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{(147 A-95 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{30 a d \sqrt{a+a \sec (c+d x)}}-\frac{(15 A-11 B) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+a \sec (c+d x)}} \, dx}{4 a}\\ &=-\frac{(A-B) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac{(9 A-5 B) \sin (c+d x)}{10 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}-\frac{(39 A-35 B) \sin (c+d x)}{30 a d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{(147 A-95 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{30 a d \sqrt{a+a \sec (c+d x)}}+\frac{(15 A-11 B) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,-\frac{a \sqrt{\sec (c+d x)} \sin (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{2 a d}\\ &=-\frac{(15 A-11 B) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{\sec (c+d x)} \sin (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac{(9 A-5 B) \sin (c+d x)}{10 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}-\frac{(39 A-35 B) \sin (c+d x)}{30 a d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{(147 A-95 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{30 a d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.40641, size = 171, normalized size = 0.68 \[ \frac{\sec (c+d x) \left (\frac{15 \sqrt{2} (15 A-11 B) \cos ^2\left (\frac{1}{2} (c+d x)\right ) \tan (c+d x) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right )}{\sqrt{1-\sec (c+d x)}}+\sin (c+d x) \sqrt{\sec (c+d x)} (3 (39 A-20 B) \cos (c+d x)+(10 B-6 A) \cos (2 (c+d x))+3 A \cos (3 (c+d x))+141 A-85 B)\right )}{30 d (a (\sec (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.324, size = 339, normalized size = 1.4 \begin{align*}{\frac{ \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{60\,d{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 24\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}-225\,A\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) \arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}+165\,B\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) \arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}-48\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}-225\,\arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}A\sin \left ( dx+c \right ) +40\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+165\,\arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}B\sin \left ( dx+c \right ) +240\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}-160\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+78\,A\cos \left ( dx+c \right ) -70\,B\cos \left ( dx+c \right ) -294\,A+190\,B \right ) \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.543179, size = 1328, normalized size = 5.31 \begin{align*} \left [-\frac{15 \, \sqrt{2}{\left ({\left (15 \, A - 11 \, B\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (15 \, A - 11 \, B\right )} \cos \left (d x + c\right ) + 15 \, A - 11 \, B\right )} \sqrt{a} \log \left (-\frac{a \cos \left (d x + c\right )^{2} - 2 \, \sqrt{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 ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) - \frac{4 \,{\left (12 \, A \cos \left (d x + c\right )^{4} - 4 \,{\left (3 \, A - 5 \, B\right )} \cos \left (d x + c\right )^{3} + 12 \,{\left (9 \, A - 5 \, B\right )} \cos \left (d x + c\right )^{2} +{\left (147 \, A - 95 \, B\right )} \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 )}{\sqrt{\cos \left (d x + c\right )}}}{120 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}, \frac{15 \, \sqrt{2}{\left ({\left (15 \, A - 11 \, B\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (15 \, A - 11 \, B\right )} \cos \left (d x + c\right ) + 15 \, A - 11 \, B\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{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 )}}{a \sin \left (d x + c\right )}\right ) + \frac{2 \,{\left (12 \, A \cos \left (d x + c\right )^{4} - 4 \,{\left (3 \, A - 5 \, B\right )} \cos \left (d x + c\right )^{3} + 12 \,{\left (9 \, A - 5 \, B\right )} \cos \left (d x + c\right )^{2} +{\left (147 \, A - 95 \, B\right )} \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 )}{\sqrt{\cos \left (d x + c\right )}}}{60 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \sec \left (d x + c\right ) + A}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \sec \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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