Optimal. Leaf size=183 \[ -\frac{13 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{10 a^2 d}-\frac{15 \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{9 \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{31 \sin (c+d x)}{5 a d \sqrt{a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.399834, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {2765, 2983, 2968, 3023, 2751, 2649, 206} \[ -\frac{13 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{10 a^2 d}-\frac{15 \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{9 \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{31 \sin (c+d x)}{5 a d \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2765
Rule 2983
Rule 2968
Rule 3023
Rule 2751
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{\int \frac{\cos ^2(c+d x) \left (3 a-\frac{9}{2} a \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{9 \cos ^2(c+d x) \sin (c+d x)}{10 a d \sqrt{a+a \cos (c+d x)}}-\frac{\int \frac{\cos (c+d x) \left (-9 a^2+\frac{39}{4} a^2 \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{5 a^3}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{9 \cos ^2(c+d x) \sin (c+d x)}{10 a d \sqrt{a+a \cos (c+d x)}}-\frac{\int \frac{-9 a^2 \cos (c+d x)+\frac{39}{4} a^2 \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{5 a^3}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{9 \cos ^2(c+d x) \sin (c+d x)}{10 a d \sqrt{a+a \cos (c+d x)}}-\frac{13 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{10 a^2 d}-\frac{2 \int \frac{\frac{39 a^3}{8}-\frac{93}{4} a^3 \cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{15 a^4}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{31 \sin (c+d x)}{5 a d \sqrt{a+a \cos (c+d x)}}+\frac{9 \cos ^2(c+d x) \sin (c+d x)}{10 a d \sqrt{a+a \cos (c+d x)}}-\frac{13 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{10 a^2 d}-\frac{15 \int \frac{1}{\sqrt{a+a \cos (c+d x)}} \, dx}{4 a}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{31 \sin (c+d x)}{5 a d \sqrt{a+a \cos (c+d x)}}+\frac{9 \cos ^2(c+d x) \sin (c+d x)}{10 a d \sqrt{a+a \cos (c+d x)}}-\frac{13 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{10 a^2 d}+\frac{15 \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{15 \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{\cos ^3(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{31 \sin (c+d x)}{5 a d \sqrt{a+a \cos (c+d x)}}+\frac{9 \cos ^2(c+d x) \sin (c+d x)}{10 a d \sqrt{a+a \cos (c+d x)}}-\frac{13 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{10 a^2 d}\\ \end{align*}
Mathematica [A] time = 1.41112, size = 226, normalized size = 1.23 \[ \frac{\cos ^3\left (\frac{1}{2} (c+d x)\right ) \left (200 \sin \left (\frac{c}{2}\right ) \cos \left (\frac{d x}{2}\right )-20 \sin \left (\frac{3 c}{2}\right ) \cos \left (\frac{3 d x}{2}\right )+4 \sin \left (\frac{5 c}{2}\right ) \cos \left (\frac{5 d x}{2}\right )+200 \cos \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right )-20 \cos \left (\frac{3 c}{2}\right ) \sin \left (\frac{3 d x}{2}\right )+4 \cos \left (\frac{5 c}{2}\right ) \sin \left (\frac{5 d x}{2}\right )+\frac{5}{\left (\cos \left (\frac{1}{4} (c+d x)\right )-\sin \left (\frac{1}{4} (c+d x)\right )\right )^2}-\frac{5}{\left (\sin \left (\frac{1}{4} (c+d x)\right )+\cos \left (\frac{1}{4} (c+d x)\right )\right )^2}+150 \log \left (\cos \left (\frac{1}{4} (c+d x)\right )-\sin \left (\frac{1}{4} (c+d x)\right )\right )-150 \log \left (\sin \left (\frac{1}{4} (c+d x)\right )+\cos \left (\frac{1}{4} (c+d x)\right )\right )\right )}{10 d (a (\cos (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.523, size = 265, normalized size = 1.5 \begin{align*}{\frac{\sqrt{2}}{20\,d}\sqrt{a \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( -32\,\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+32\,\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+75\,\ln \left ( 4\,{\frac{\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-80\,\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-75\,\ln \left ( 4\,{\frac{\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) a+85\,\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \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{ \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a}}}} \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 = 1.64834, size = 497, normalized size = 2.72 \begin{align*} \frac{75 \, \sqrt{2}{\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\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 \, \cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )^{2} + 36 \, \cos \left (d x + c\right ) + 49\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{40 \,{\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 = 2.07941, size = 185, normalized size = 1.01 \begin{align*} \frac{\frac{75 \, \sqrt{2} \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 ({\left (5 \, \sqrt{2} a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 127 \, \sqrt{2} a\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 175 \, \sqrt{2} a\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 85 \, \sqrt{2} 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{5}{2}}}}{20 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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