Optimal. Leaf size=217 \[ \frac{(19 A+8 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{4 a^{3/2} d}-\frac{(13 A+5 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(7 A+2 C) \tan (c+d x)}{4 a d \sqrt{a \cos (c+d x)+a}}+\frac{(2 A+C) \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.709254, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {3042, 2984, 2985, 2649, 206, 2773} \[ \frac{(19 A+8 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{4 a^{3/2} d}-\frac{(13 A+5 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(7 A+2 C) \tan (c+d x)}{4 a d \sqrt{a \cos (c+d x)+a}}+\frac{(2 A+C) \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3042
Rule 2984
Rule 2985
Rule 2649
Rule 206
Rule 2773
Rubi steps
\begin{align*} \int \frac{\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx &=-\frac{(A+C) \sec (c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{\int \frac{\left (2 a (2 A+C)-\frac{1}{2} a (5 A+C) \cos (c+d x)\right ) \sec ^3(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{(A+C) \sec (c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(2 A+C) \sec (c+d x) \tan (c+d x)}{2 a d \sqrt{a+a \cos (c+d x)}}+\frac{\int \frac{\left (-a^2 (7 A+2 C)+3 a^2 (2 A+C) \cos (c+d x)\right ) \sec ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{4 a^3}\\ &=-\frac{(7 A+2 C) \tan (c+d x)}{4 a d \sqrt{a+a \cos (c+d x)}}-\frac{(A+C) \sec (c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(2 A+C) \sec (c+d x) \tan (c+d x)}{2 a d \sqrt{a+a \cos (c+d x)}}+\frac{\int \frac{\left (\frac{1}{2} a^3 (19 A+8 C)-\frac{1}{2} a^3 (7 A+2 C) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{4 a^4}\\ &=-\frac{(7 A+2 C) \tan (c+d x)}{4 a d \sqrt{a+a \cos (c+d x)}}-\frac{(A+C) \sec (c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(2 A+C) \sec (c+d x) \tan (c+d x)}{2 a d \sqrt{a+a \cos (c+d x)}}-\frac{(13 A+5 C) \int \frac{1}{\sqrt{a+a \cos (c+d x)}} \, dx}{4 a}+\frac{(19 A+8 C) \int \sqrt{a+a \cos (c+d x)} \sec (c+d x) \, dx}{8 a^2}\\ &=-\frac{(7 A+2 C) \tan (c+d x)}{4 a d \sqrt{a+a \cos (c+d x)}}-\frac{(A+C) \sec (c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(2 A+C) \sec (c+d x) \tan (c+d x)}{2 a d \sqrt{a+a \cos (c+d x)}}+\frac{(13 A+5 C) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{2 a d}-\frac{(19 A+8 C) \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{4 a d}\\ &=\frac{(19 A+8 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{4 a^{3/2} d}-\frac{(13 A+5 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a+a \cos (c+d x)}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(7 A+2 C) \tan (c+d x)}{4 a d \sqrt{a+a \cos (c+d x)}}-\frac{(A+C) \sec (c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(2 A+C) \sec (c+d x) \tan (c+d x)}{2 a d \sqrt{a+a \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 3.04186, size = 211, normalized size = 0.97 \[ -\frac{\sec \left (\frac{1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (A+C \cos ^2(c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right ) ((7 A+2 C) \cos (2 (c+d x))+6 A \cos (c+d x)+3 A+2 C)+(13 A+5 C) \left (\cos \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{3}{2} (c+d x)\right )\right )^2 \tanh ^{-1}\left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-\frac{(19 A+8 C) \left (\cos \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{3}{2} (c+d x)\right )\right )^2 \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{2}}\right )}{4 a d \sqrt{a (\cos (c+d x)+1)} (2 A+C \cos (2 (c+d x))+C)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.124, size = 1540, normalized size = 7.1 \begin{align*} \text{result too large to display} \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 = 2.29836, size = 927, normalized size = 4.27 \begin{align*} \frac{2 \, \sqrt{2}{\left ({\left (13 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (13 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (13 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt{a} \log \left (-\frac{a \cos \left (d x + c\right )^{2} + 2 \, \sqrt{2} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{a} \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 ) +{\left ({\left (19 \, A + 8 \, C\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (19 \, A + 8 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (19 \, A + 8 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt{a} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{a}{\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) - 4 \,{\left ({\left (7 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, A \cos \left (d x + c\right ) - 2 \, A\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{16 \,{\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} + a^{2} d \cos \left (d x + c\right )^{2}\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 = 3.53705, size = 612, normalized size = 2.82 \begin{align*} \frac{\frac{\sqrt{2}{\left (13 \, A \sqrt{a} + 5 \, C \sqrt{a}\right )} \log \left ({\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2}\right )}{a^{2}} + \frac{{\left (19 \, A \sqrt{a} + 8 \, C \sqrt{a}\right )} \log \left ({\left |{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2} - a{\left (2 \, \sqrt{2} + 3\right )} \right |}\right )}{a^{2}} - \frac{{\left (19 \, A \sqrt{a} + 8 \, C \sqrt{a}\right )} \log \left ({\left |{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2} + a{\left (2 \, \sqrt{2} - 3\right )} \right |}\right )}{a^{2}} - \frac{2 \, \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}{\left (\sqrt{2} A a + \sqrt{2} C a\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{3}} - \frac{4 \, \sqrt{2}{\left (29 \,{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{6} A \sqrt{a} - 133 \,{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{4} A a^{\frac{3}{2}} + 55 \,{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2} A a^{\frac{5}{2}} - 7 \, A a^{\frac{7}{2}}\right )}}{{\left ({\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{4} - 6 \,{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2} a + a^{2}\right )}^{2} a}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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