Optimal. Leaf size=169 \[ \frac{(3 A+7 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{6 a^2 d}+\frac{(3 A+11 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(3 A+13 C) \sin (c+d x)}{3 a d \sqrt{a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.341496, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3042, 2968, 3023, 2751, 2649, 206} \[ \frac{(3 A+7 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{6 a^2 d}+\frac{(3 A+11 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(3 A+13 C) \sin (c+d x)}{3 a d \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3042
Rule 2968
Rule 3023
Rule 2751
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx &=-\frac{(A+C) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{\int \frac{\cos (c+d x) \left (-2 a C+\frac{1}{2} a (3 A+7 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{(A+C) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{\int \frac{-2 a C \cos (c+d x)+\frac{1}{2} a (3 A+7 C) \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{(A+C) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(3 A+7 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{6 a^2 d}+\frac{\int \frac{\frac{1}{4} a^2 (3 A+7 C)-\frac{1}{2} a^2 (3 A+13 C) \cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{3 a^3}\\ &=-\frac{(A+C) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{(3 A+13 C) \sin (c+d x)}{3 a d \sqrt{a+a \cos (c+d x)}}+\frac{(3 A+7 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{6 a^2 d}+\frac{(3 A+11 C) \int \frac{1}{\sqrt{a+a \cos (c+d x)}} \, dx}{4 a}\\ &=-\frac{(A+C) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{(3 A+13 C) \sin (c+d x)}{3 a d \sqrt{a+a \cos (c+d x)}}+\frac{(3 A+7 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{6 a^2 d}-\frac{(3 A+11 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{(3 A+11 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{(A+C) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{(3 A+13 C) \sin (c+d x)}{3 a d \sqrt{a+a \cos (c+d x)}}+\frac{(3 A+7 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{6 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.605505, size = 94, normalized size = 0.56 \[ \frac{3 (3 A+11 C) \cos \left (\frac{1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-\tan \left (\frac{1}{2} (c+d x)\right ) (3 A+12 C \cos (c+d x)-2 C \cos (2 (c+d x))+17 C)}{6 a d \sqrt{a (\cos (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 292, normalized size = 1.7 \begin{align*}{\frac{1}{12\,d}\sqrt{a \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 16\,C\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+9\,A\ln \left ( 2\,{\frac{2\,\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+2\,a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) \sqrt{2} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+33\,C\ln \left ( 2\,{\frac{2\,\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+2\,a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) \sqrt{2} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-40\,C\sqrt{a}\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-3\,A\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a}-3\,C\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \right ) \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{a}^{-{\frac{5}{2}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{a \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{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 = 1.68336, size = 533, normalized size = 3.15 \begin{align*} \frac{3 \, \sqrt{2}{\left ({\left (3 \, A + 11 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (3 \, A + 11 \, C\right )} \cos \left (d x + c\right ) + 3 \, A + 11 \, C\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 ) + 4 \,{\left (4 \, C \cos \left (d x + c\right )^{2} - 12 \, C \cos \left (d x + c\right ) - 3 \, A - 19 \, C\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \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 )}} \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 [A] time = 1.9682, size = 225, normalized size = 1.33 \begin{align*} -\frac{\frac{3 \,{\left (3 \, \sqrt{2} A + 11 \, \sqrt{2} C\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 |}\right )}{a^{\frac{3}{2}}} + \frac{{\left ({\left (\frac{3 \,{\left (\sqrt{2} A a + \sqrt{2} C a\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{a} + \frac{2 \,{\left (3 \, \sqrt{2} A a + 23 \, \sqrt{2} C a\right )}}{a}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{3 \,{\left (\sqrt{2} A a + 9 \, \sqrt{2} C a\right )}}{a}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}^{\frac{3}{2}}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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