Optimal. Leaf size=212 \[ \frac{5 (3 A+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{48 a^3 d}-\frac{(21 A+197 C) \sin (c+d x)}{24 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(19 A+163 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(A+17 C) \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.605957, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3042, 2977, 2968, 3023, 2751, 2649, 206} \[ \frac{5 (3 A+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{48 a^3 d}-\frac{(21 A+197 C) \sin (c+d x)}{24 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(19 A+163 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(A+17 C) \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3042
Rule 2977
Rule 2968
Rule 3023
Rule 2751
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^{5/2}} \, dx &=-\frac{(A+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}+\frac{\int \frac{\cos ^2(c+d x) \left (a (A-3 C)+\frac{1}{2} a (3 A+11 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac{(A+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(A+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{\int \frac{\cos (c+d x) \left (-a^2 (A+17 C)+\frac{5}{4} a^2 (3 A+19 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{8 a^4}\\ &=-\frac{(A+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(A+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{\int \frac{-a^2 (A+17 C) \cos (c+d x)+\frac{5}{4} a^2 (3 A+19 C) \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{8 a^4}\\ &=-\frac{(A+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(A+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{5 (3 A+19 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{48 a^3 d}+\frac{\int \frac{\frac{5}{8} a^3 (3 A+19 C)-\frac{1}{4} a^3 (21 A+197 C) \cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{12 a^5}\\ &=-\frac{(A+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(A+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}-\frac{(21 A+197 C) \sin (c+d x)}{24 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{5 (3 A+19 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{48 a^3 d}+\frac{(19 A+163 C) \int \frac{1}{\sqrt{a+a \cos (c+d x)}} \, dx}{32 a^2}\\ &=-\frac{(A+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(A+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}-\frac{(21 A+197 C) \sin (c+d x)}{24 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{5 (3 A+19 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{48 a^3 d}-\frac{(19 A+163 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 )}{16 a^2 d}\\ &=\frac{(19 A+163 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a+a \cos (c+d x)}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{(A+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(A+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}-\frac{(21 A+197 C) \sin (c+d x)}{24 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{5 (3 A+19 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{48 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.863963, size = 112, normalized size = 0.53 \[ \frac{6 (19 A+163 C) \cos ^3\left (\frac{1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-\tan \left (\frac{1}{2} (c+d x)\right ) ((39 A+479 C) \cos (c+d x)+27 A+80 C \cos (2 (c+d x))-8 C \cos (3 (c+d x))+379 C)}{48 a d (a (\cos (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 362, normalized size = 1.7 \begin{align*}{\frac{1}{96\,d}\sqrt{a \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 128\,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 ) ^{6}+57\,A\sqrt{2}\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 ) \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}a+489\,C\sqrt{2}\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 ) \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}a-512\,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}-39\,A\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}-87\,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}+6\,A\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a}+6\,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 ) ^{-3}{a}^{-{\frac{7}{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 = 1.71271, size = 671, normalized size = 3.17 \begin{align*} \frac{3 \, \sqrt{2}{\left ({\left (19 \, A + 163 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (19 \, A + 163 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (19 \, A + 163 \, C\right )} \cos \left (d x + c\right ) + 19 \, A + 163 \, 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 (32 \, C \cos \left (d x + c\right )^{3} - 160 \, C \cos \left (d x + c\right )^{2} -{\left (39 \, A + 503 \, C\right )} \cos \left (d x + c\right ) - 27 \, A - 299 \, C\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{192 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} 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 = 2.30582, size = 274, normalized size = 1.29 \begin{align*} \frac{\frac{{\left ({\left (3 \,{\left (\frac{2 \, \sqrt{2}{\left (A a^{5} + C a^{5}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{a^{6}} - \frac{\sqrt{2}{\left (7 \, A a^{5} + 23 \, C a^{5}\right )}}{a^{6}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - \frac{4 \, \sqrt{2}{\left (15 \, A a^{5} + 167 \, C a^{5}\right )}}{a^{6}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - \frac{3 \, \sqrt{2}{\left (11 \, A a^{5} + 155 \, C a^{5}\right )}}{a^{6}}\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}}} - \frac{3 \, \sqrt{2}{\left (19 \, A + 163 \, 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{5}{2}}}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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