Optimal. Leaf size=201 \[ \frac{(11 A+3 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(19 A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \sec (c+d x)+a}}+\frac{(7 A+3 C) \sin (c+d x)}{6 a d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.514951, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {4085, 4022, 4013, 3808, 206} \[ \frac{(11 A+3 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(19 A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \sec (c+d x)+a}}+\frac{(7 A+3 C) \sin (c+d x)}{6 a d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4085
Rule 4022
Rule 4013
Rule 3808
Rule 206
Rubi steps
\begin{align*} \int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx &=-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}}-\frac{\int \frac{-\frac{1}{2} a (7 A+3 C)+2 a A \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}}+\frac{(7 A+3 C) \sin (c+d x)}{6 a d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}-\frac{\int \frac{\frac{1}{4} a^2 (19 A+3 C)-\frac{1}{2} a^2 (7 A+3 C) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx}{3 a^3}\\ &=-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}}+\frac{(7 A+3 C) \sin (c+d x)}{6 a d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}-\frac{(19 A+3 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{6 a d \sqrt{a+a \sec (c+d x)}}+\frac{(11 A+3 C) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+a \sec (c+d x)}} \, dx}{4 a}\\ &=-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}}+\frac{(7 A+3 C) \sin (c+d x)}{6 a d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}-\frac{(19 A+3 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{6 a d \sqrt{a+a \sec (c+d x)}}-\frac{(11 A+3 C) \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{(11 A+3 C) \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+C) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}}+\frac{(7 A+3 C) \sin (c+d x)}{6 a d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}-\frac{(19 A+3 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{6 a d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 3.00724, size = 316, normalized size = 1.57 \[ \frac{(\sec (c+d x)+1)^{3/2} \left (A+C \sec ^2(c+d x)\right ) \left (\frac{2 \left (\sin \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{3}{2} (c+d x)\right )\right ) \sqrt{\sec (c+d x)+1} \sec ^3\left (\frac{1}{2} (c+d x)\right ) (12 A \cos (c+d x)-2 A \cos (2 (c+d x))+17 A+3 C)}{\sec ^{\frac{3}{2}}(c+d x)}+3 \sqrt{2} (11 A+3 C) \cos ^2(c+d x) \sqrt{\tan ^2(c+d x)} \cot (c+d x) \left (\log \left (-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right )-\log \left (-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right )\right )\right )}{24 d (a (\sec (c+d x)+1))^{3/2} (A \cos (2 (c+d x))+A+2 C)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.381, size = 306, normalized size = 1.5 \begin{align*}{\frac{ \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{12\,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 ( 33\,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}}+9\,C\cos \left ( dx+c \right ) \sin \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}}+33\,\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 ) +8\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+9\,C\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) \sin \left ( dx+c \right ) -32\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}-14\,A\cos \left ( dx+c \right ) -6\,C\cos \left ( dx+c \right ) +38\,A+6\,C \right ) \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{3}{2}}}} \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 = 0.538732, size = 1197, normalized size = 5.96 \begin{align*} \left [\frac{3 \, \sqrt{2}{\left ({\left (11 \, A + 3 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (11 \, A + 3 \, C\right )} \cos \left (d x + c\right ) + 11 \, A + 3 \, C\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 (4 \, A \cos \left (d x + c\right )^{3} - 12 \, A \cos \left (d x + c\right )^{2} -{\left (19 \, A + 3 \, C\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 )}}}{24 \,{\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{3 \, \sqrt{2}{\left ({\left (11 \, A + 3 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (11 \, A + 3 \, C\right )} \cos \left (d x + c\right ) + 11 \, A + 3 \, C\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 (4 \, A \cos \left (d x + c\right )^{3} - 12 \, A \cos \left (d x + c\right )^{2} -{\left (19 \, A + 3 \, C\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 )}}}{12 \,{\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{C \sec \left (d x + c\right )^{2} + A}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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