Optimal. Leaf size=138 \[ \frac{8 a^2 (21 B+19 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (7 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 a (21 B+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d} \]
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Rubi [A] time = 0.188278, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3023, 2751, 2647, 2646} \[ \frac{8 a^2 (21 B+19 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (7 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 a (21 B+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d} \]
Antiderivative was successfully verified.
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Rule 3023
Rule 2751
Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 a d}+\frac{2 \int (a+a \cos (c+d x))^{3/2} \left (\frac{5 a C}{2}+\frac{1}{2} a (7 B-2 C) \cos (c+d x)\right ) \, dx}{7 a}\\ &=\frac{2 (7 B-2 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 a d}+\frac{1}{35} (21 B+19 C) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac{2 a (21 B+19 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 (7 B-2 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 a d}+\frac{1}{105} (4 a (21 B+19 C)) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{8 a^2 (21 B+19 C) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (21 B+19 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 (7 B-2 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 a d}\\ \end{align*}
Mathematica [A] time = 0.393073, size = 81, normalized size = 0.59 \[ \frac{a \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} ((252 B+253 C) \cos (c+d x)+6 (7 B+13 C) \cos (2 (c+d x))+546 B+15 C \cos (3 (c+d x))+494 C)}{210 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 104, normalized size = 0.8 \begin{align*}{\frac{4\,{a}^{2}\sqrt{2}}{105\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( -60\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+ \left ( 42\,B+168\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -105\,B-175\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+105\,B+105\,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 = 1.86838, size = 166, normalized size = 1.2 \begin{align*} \frac{42 \,{\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 )} B \sqrt{a} +{\left (15 \, \sqrt{2} a \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 63 \, \sqrt{2} a \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 175 \, \sqrt{2} a \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 735 \, \sqrt{2} a \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} C \sqrt{a}}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57489, size = 236, normalized size = 1.71 \begin{align*} \frac{2 \,{\left (15 \, C a \cos \left (d x + c\right )^{3} + 3 \,{\left (7 \, B + 13 \, C\right )} a \cos \left (d x + c\right )^{2} +{\left (63 \, B + 52 \, C\right )} a \cos \left (d x + c\right ) + 2 \,{\left (63 \, B + 52 \, C\right )} a\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{105 \,{\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(-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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