Optimal. Leaf size=174 \[ \frac{8 a^2 (63 A+47 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (63 A+22 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{315 d}+\frac{2 a (63 A+47 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{21 a d} \]
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Rubi [A] time = 0.364194, antiderivative size = 174, 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 = {3046, 2968, 3023, 2751, 2647, 2646} \[ \frac{8 a^2 (63 A+47 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (63 A+22 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{315 d}+\frac{2 a (63 A+47 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{21 a d} \]
Antiderivative was successfully verified.
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Rule 3046
Rule 2968
Rule 3023
Rule 2751
Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac{2 \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (\frac{1}{2} a (9 A+4 C)+\frac{3}{2} a C \cos (c+d x)\right ) \, dx}{9 a}\\ &=\frac{2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac{2 \int (a+a \cos (c+d x))^{3/2} \left (\frac{1}{2} a (9 A+4 C) \cos (c+d x)+\frac{3}{2} a C \cos ^2(c+d x)\right ) \, dx}{9 a}\\ &=\frac{2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{21 a d}+\frac{4 \int (a+a \cos (c+d x))^{3/2} \left (\frac{15 a^2 C}{4}+\frac{1}{4} a^2 (63 A+22 C) \cos (c+d x)\right ) \, dx}{63 a^2}\\ &=\frac{2 (63 A+22 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{315 d}+\frac{2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{21 a d}+\frac{1}{105} (63 A+47 C) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac{2 a (63 A+47 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac{2 (63 A+22 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{315 d}+\frac{2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{21 a d}+\frac{1}{315} (4 a (63 A+47 C)) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{8 a^2 (63 A+47 C) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (63 A+47 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac{2 (63 A+22 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{315 d}+\frac{2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{21 a d}\\ \end{align*}
Mathematica [A] time = 0.540507, size = 93, normalized size = 0.53 \[ \frac{a \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} (2 (756 A+799 C) \cos (c+d x)+4 (63 A+137 C) \cos (2 (c+d x))+3276 A+170 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))+2689 C)}{1260 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 118, normalized size = 0.7 \begin{align*}{\frac{4\,{a}^{2}\sqrt{2}}{315\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 280\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}-900\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+ \left ( 126\,A+1134\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -315\,A-735\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+315\,A+315\,C \right ){\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 [A] time = 1.93453, size = 186, normalized size = 1.07 \begin{align*} \frac{252 \,{\left (\sqrt{2} a \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 5 \, \sqrt{2} a \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 20 \, \sqrt{2} a \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} A \sqrt{a} +{\left (35 \, \sqrt{2} a \sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ) + 135 \, \sqrt{2} a \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 378 \, \sqrt{2} a \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 1050 \, \sqrt{2} a \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3780 \, \sqrt{2} a \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} C \sqrt{a}}{2520 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34964, size = 275, normalized size = 1.58 \begin{align*} \frac{2 \,{\left (35 \, C a \cos \left (d x + c\right )^{4} + 85 \, C a \cos \left (d x + c\right )^{3} + 3 \,{\left (21 \, A + 34 \, C\right )} a \cos \left (d x + c\right )^{2} +{\left (189 \, A + 136 \, C\right )} a \cos \left (d x + c\right ) + 2 \,{\left (189 \, A + 136 \, C\right )} a\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \,{\left (d \cos \left (d x + c\right ) + 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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