Optimal. Leaf size=157 \[ \frac{2 b \left (11 a^2+4 b^2\right ) \sqrt{e \sin (c+d x)}}{5 d e}+\frac{2 a \left (a^2+2 b^2\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d \sqrt{e \sin (c+d x)}}+\frac{2 b \sqrt{e \sin (c+d x)} (a+b \cos (c+d x))^2}{5 d e}+\frac{6 a b \sqrt{e \sin (c+d x)} (a+b \cos (c+d x))}{5 d e} \]
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Rubi [A] time = 0.242702, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2692, 2862, 2669, 2642, 2641} \[ \frac{2 b \left (11 a^2+4 b^2\right ) \sqrt{e \sin (c+d x)}}{5 d e}+\frac{2 a \left (a^2+2 b^2\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d \sqrt{e \sin (c+d x)}}+\frac{2 b \sqrt{e \sin (c+d x)} (a+b \cos (c+d x))^2}{5 d e}+\frac{6 a b \sqrt{e \sin (c+d x)} (a+b \cos (c+d x))}{5 d e} \]
Antiderivative was successfully verified.
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Rule 2692
Rule 2862
Rule 2669
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^3}{\sqrt{e \sin (c+d x)}} \, dx &=\frac{2 b (a+b \cos (c+d x))^2 \sqrt{e \sin (c+d x)}}{5 d e}+\frac{2}{5} \int \frac{(a+b \cos (c+d x)) \left (\frac{5 a^2}{2}+2 b^2+\frac{9}{2} a b \cos (c+d x)\right )}{\sqrt{e \sin (c+d x)}} \, dx\\ &=\frac{6 a b (a+b \cos (c+d x)) \sqrt{e \sin (c+d x)}}{5 d e}+\frac{2 b (a+b \cos (c+d x))^2 \sqrt{e \sin (c+d x)}}{5 d e}+\frac{4}{15} \int \frac{\frac{15}{4} a \left (a^2+2 b^2\right )+\frac{3}{4} b \left (11 a^2+4 b^2\right ) \cos (c+d x)}{\sqrt{e \sin (c+d x)}} \, dx\\ &=\frac{2 b \left (11 a^2+4 b^2\right ) \sqrt{e \sin (c+d x)}}{5 d e}+\frac{6 a b (a+b \cos (c+d x)) \sqrt{e \sin (c+d x)}}{5 d e}+\frac{2 b (a+b \cos (c+d x))^2 \sqrt{e \sin (c+d x)}}{5 d e}+\left (a \left (a^2+2 b^2\right )\right ) \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx\\ &=\frac{2 b \left (11 a^2+4 b^2\right ) \sqrt{e \sin (c+d x)}}{5 d e}+\frac{6 a b (a+b \cos (c+d x)) \sqrt{e \sin (c+d x)}}{5 d e}+\frac{2 b (a+b \cos (c+d x))^2 \sqrt{e \sin (c+d x)}}{5 d e}+\frac{\left (a \left (a^2+2 b^2\right ) \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{\sqrt{e \sin (c+d x)}}\\ &=\frac{2 a \left (a^2+2 b^2\right ) F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{d \sqrt{e \sin (c+d x)}}+\frac{2 b \left (11 a^2+4 b^2\right ) \sqrt{e \sin (c+d x)}}{5 d e}+\frac{6 a b (a+b \cos (c+d x)) \sqrt{e \sin (c+d x)}}{5 d e}+\frac{2 b (a+b \cos (c+d x))^2 \sqrt{e \sin (c+d x)}}{5 d e}\\ \end{align*}
Mathematica [A] time = 0.71082, size = 98, normalized size = 0.62 \[ \frac{b \sin (c+d x) \left (30 a^2+10 a b \cos (c+d x)+b^2 \cos (2 (c+d x))+9 b^2\right )-10 a \left (a^2+2 b^2\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )}{5 d \sqrt{e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.188, size = 210, normalized size = 1.3 \begin{align*} -{\frac{1}{5\,d\cos \left ( dx+c \right ) } \left ( 5\,{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) \sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{a}^{3}+10\,{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) \sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }a{b}^{2}-2\,{b}^{3}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}-10\,a{b}^{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}-30\,{a}^{2}b\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) -8\,{b}^{3}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ){\frac{1}{\sqrt{e\sin \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt{e \sin \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}\right )} \sqrt{e \sin \left (d x + c\right )}}{e \sin \left (d x + c\right )}, x\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 \frac{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt{e \sin \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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