Optimal. Leaf size=124 \[ \frac{2 \left (a^2-2 b^2\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{3 d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 a b \sqrt{e \sin (c+d x)}}{3 d e^3}-\frac{2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{3 d e (e \sin (c+d x))^{3/2}} \]
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Rubi [A] time = 0.139355, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2691, 2669, 2642, 2641} \[ \frac{2 \left (a^2-2 b^2\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{3 d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 a b \sqrt{e \sin (c+d x)}}{3 d e^3}-\frac{2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{3 d e (e \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2691
Rule 2669
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{5/2}} \, dx &=-\frac{2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 \int \frac{-\frac{a^2}{2}+b^2+\frac{1}{2} a b \cos (c+d x)}{\sqrt{e \sin (c+d x)}} \, dx}{3 e^2}\\ &=-\frac{2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 a b \sqrt{e \sin (c+d x)}}{3 d e^3}+\frac{\left (a^2-2 b^2\right ) \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx}{3 e^2}\\ &=-\frac{2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 a b \sqrt{e \sin (c+d x)}}{3 d e^3}+\frac{\left (\left (a^2-2 b^2\right ) \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{3 e^2 \sqrt{e \sin (c+d x)}}\\ &=-\frac{2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{3 d e (e \sin (c+d x))^{3/2}}+\frac{2 \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)}}{3 d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 a b \sqrt{e \sin (c+d x)}}{3 d e^3}\\ \end{align*}
Mathematica [A] time = 0.26479, size = 76, normalized size = 0.61 \[ -\frac{2 \left (\left (a^2+b^2\right ) \cos (c+d x)+\left (a^2-2 b^2\right ) \sin ^{\frac{3}{2}}(c+d x) F\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )+2 a b\right )}{3 d e (e \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.883, size = 190, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ( -{\frac{4\,ab}{3\,e} \left ( e\sin \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{3\,{e}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) } \left ( \left ( 2\,{a}^{2}+2\,{b}^{2} \right ) \sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) } \left ( \sin \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },{\frac{\sqrt{2}}{2}} \right ){a}^{2}-2\,{b}^{2}\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) } \left ( \sin \left ( dx+c \right ) \right ) ^{5/2}{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) \right ){\frac{1}{\sqrt{e\sin \left ( dx+c \right ) }}}} \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 )}^{2}}{\left (e \sin \left (d x + c\right )\right )^{\frac{5}{2}}}\,{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^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}\right )} \sqrt{e \sin \left (d x + c\right )}}{{\left (e^{3} \cos \left (d x + c\right )^{2} - e^{3}\right )} \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 )}^{2}}{\left (e \sin \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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