Optimal. Leaf size=107 \[ \frac{(2 a-b) \sqrt{\sin (2 e+2 f x)} F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{3 f g^2 \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{2 (a+b) \sqrt{d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}} \]
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Rubi [A] time = 0.181446, antiderivative size = 114, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.189, Rules used = {3202, 457, 329, 237, 335, 275, 232} \[ \frac{2 (a+b) \sqrt{d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}-\frac{2 (2 a-b) \left (1-\csc ^2(e+f x)\right )^{3/4} (d \sin (e+f x))^{3/2} F\left (\left .\frac{1}{2} \csc ^{-1}(\sin (e+f x))\right |2\right )}{3 d^2 f g (g \cos (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3202
Rule 457
Rule 329
Rule 237
Rule 335
Rule 275
Rule 232
Rubi steps
\begin{align*} \int \frac{a+b \sin ^2(e+f x)}{(g \cos (e+f x))^{5/2} \sqrt{d \sin (e+f x)}} \, dx &=\frac{\cos ^2(e+f x)^{3/4} \operatorname{Subst}\left (\int \frac{a+b x^2}{\sqrt{d x} \left (1-x^2\right )^{7/4}} \, dx,x,\sin (e+f x)\right )}{f g (g \cos (e+f x))^{3/2}}\\ &=\frac{2 (a+b) \sqrt{d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}-\frac{\left ((-2 a+b) \cos ^2(e+f x)^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d x} \left (1-x^2\right )^{3/4}} \, dx,x,\sin (e+f x)\right )}{3 f g (g \cos (e+f x))^{3/2}}\\ &=\frac{2 (a+b) \sqrt{d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}-\frac{\left (2 (-2 a+b) \cos ^2(e+f x)^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x^4}{d^2}\right )^{3/4}} \, dx,x,\sqrt{d \sin (e+f x)}\right )}{3 d f g (g \cos (e+f x))^{3/2}}\\ &=\frac{2 (a+b) \sqrt{d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}-\frac{\left (2 (-2 a+b) \left (1-\csc ^2(e+f x)\right )^{3/4} (d \sin (e+f x))^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{d^2}{x^4}\right )^{3/4} x^3} \, dx,x,\sqrt{d \sin (e+f x)}\right )}{3 d f g (g \cos (e+f x))^{3/2}}\\ &=\frac{2 (a+b) \sqrt{d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac{\left (2 (-2 a+b) \left (1-\csc ^2(e+f x)\right )^{3/4} (d \sin (e+f x))^{3/2}\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1-d^2 x^4\right )^{3/4}} \, dx,x,\frac{1}{\sqrt{d \sin (e+f x)}}\right )}{3 d f g (g \cos (e+f x))^{3/2}}\\ &=\frac{2 (a+b) \sqrt{d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac{\left ((-2 a+b) \left (1-\csc ^2(e+f x)\right )^{3/4} (d \sin (e+f x))^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-d^2 x^2\right )^{3/4}} \, dx,x,\frac{\csc (e+f x)}{d}\right )}{3 d f g (g \cos (e+f x))^{3/2}}\\ &=\frac{2 (a+b) \sqrt{d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}-\frac{2 (2 a-b) \left (1-\csc ^2(e+f x)\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}(\csc (e+f x))\right |2\right ) (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.185173, size = 102, normalized size = 0.95 \[ \frac{2 \cos ^2(e+f x)^{3/4} \left (5 a \sin (e+f x) \, _2F_1\left (\frac{1}{4},\frac{7}{4};\frac{5}{4};\sin ^2(e+f x)\right )+b \sin ^3(e+f x) \, _2F_1\left (\frac{5}{4},\frac{7}{4};\frac{9}{4};\sin ^2(e+f x)\right )\right )}{5 f g \sqrt{d \sin (e+f x)} (g \cos (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.415, size = 327, normalized size = 3.1 \begin{align*}{\frac{\sqrt{2}\cos \left ( fx+e \right ) \sin \left ( fx+e \right ) }{3\,f \left ( -1+\cos \left ( fx+e \right ) \right ) } \left ( -2\,{\it EllipticF} \left ( \sqrt{-{\frac{-1+\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},1/2\,\sqrt{2} \right ) \cos \left ( fx+e \right ) \sin \left ( fx+e \right ) \sqrt{-{\frac{-1+\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}a+{\it EllipticF} \left ( \sqrt{-{\frac{-1+\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{\sqrt{2}}{2}} \right ) \cos \left ( fx+e \right ) \sin \left ( fx+e \right ) \sqrt{-{\frac{-1+\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}b+\sqrt{2}\cos \left ( fx+e \right ) a+\sqrt{2}\cos \left ( fx+e \right ) b-\sqrt{2}a-\sqrt{2}b \right ) \left ( g\cos \left ( fx+e \right ) \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{d\sin \left ( fx+e \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{b \sin \left (f x + e\right )^{2} + a}{\left (g \cos \left (f x + e\right )\right )^{\frac{5}{2}} \sqrt{d \sin \left (f x + e\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 \cos \left (f x + e\right )^{2} - a - b\right )} \sqrt{g \cos \left (f x + e\right )} \sqrt{d \sin \left (f x + e\right )}}{d g^{3} \cos \left (f x + e\right )^{3} \sin \left (f x + e\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{b \sin \left (f x + e\right )^{2} + a}{\left (g \cos \left (f x + e\right )\right )^{\frac{5}{2}} \sqrt{d \sin \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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