Optimal. Leaf size=369 \[ \frac{a^2 \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt [4]{b^2-a^2}}\right )}{b^{5/2} f \sqrt [4]{b^2-a^2}}-\frac{a^2 \sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt [4]{b^2-a^2}}\right )}{b^{5/2} f \sqrt [4]{b^2-a^2}}+\frac{a^3 g \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{b^3 f \left (b-\sqrt{b^2-a^2}\right ) \sqrt{g \cos (e+f x)}}+\frac{a^3 g \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{b^3 f \left (\sqrt{b^2-a^2}+b\right ) \sqrt{g \cos (e+f x)}}-\frac{2 a E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{b^2 f \sqrt{\cos (e+f x)}}-\frac{2 (g \cos (e+f x))^{3/2}}{3 b f g} \]
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Rubi [A] time = 0.872078, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2898, 2640, 2639, 2565, 30, 2701, 2807, 2805, 329, 298, 205, 208} \[ \frac{a^2 \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt [4]{b^2-a^2}}\right )}{b^{5/2} f \sqrt [4]{b^2-a^2}}-\frac{a^2 \sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt [4]{b^2-a^2}}\right )}{b^{5/2} f \sqrt [4]{b^2-a^2}}+\frac{a^3 g \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{b^3 f \left (b-\sqrt{b^2-a^2}\right ) \sqrt{g \cos (e+f x)}}+\frac{a^3 g \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{b^3 f \left (\sqrt{b^2-a^2}+b\right ) \sqrt{g \cos (e+f x)}}-\frac{2 a E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{b^2 f \sqrt{\cos (e+f x)}}-\frac{2 (g \cos (e+f x))^{3/2}}{3 b f g} \]
Antiderivative was successfully verified.
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Rule 2898
Rule 2640
Rule 2639
Rule 2565
Rule 30
Rule 2701
Rule 2807
Rule 2805
Rule 329
Rule 298
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{g \cos (e+f x)} \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx &=\int \left (-\frac{a \sqrt{g \cos (e+f x)}}{b^2}+\frac{\sqrt{g \cos (e+f x)} \sin (e+f x)}{b}+\frac{a^2 \sqrt{g \cos (e+f x)}}{b^2 (a+b \sin (e+f x))}\right ) \, dx\\ &=-\frac{a \int \sqrt{g \cos (e+f x)} \, dx}{b^2}+\frac{a^2 \int \frac{\sqrt{g \cos (e+f x)}}{a+b \sin (e+f x)} \, dx}{b^2}+\frac{\int \sqrt{g \cos (e+f x)} \sin (e+f x) \, dx}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \sqrt{x} \, dx,x,g \cos (e+f x)\right )}{b f g}-\frac{\left (a^3 g\right ) \int \frac{1}{\sqrt{g \cos (e+f x)} \left (\sqrt{-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 b^3}+\frac{\left (a^3 g\right ) \int \frac{1}{\sqrt{g \cos (e+f x)} \left (\sqrt{-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 b^3}+\frac{\left (a^2 g\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\left (a^2-b^2\right ) g^2+b^2 x^2} \, dx,x,g \cos (e+f x)\right )}{b f}-\frac{\left (a \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{b^2 \sqrt{\cos (e+f x)}}\\ &=-\frac{2 (g \cos (e+f x))^{3/2}}{3 b f g}-\frac{2 a \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{b^2 f \sqrt{\cos (e+f x)}}+\frac{\left (2 a^2 g\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (a^2-b^2\right ) g^2+b^2 x^4} \, dx,x,\sqrt{g \cos (e+f x)}\right )}{b f}-\frac{\left (a^3 g \sqrt{\cos (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)} \left (\sqrt{-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 b^3 \sqrt{g \cos (e+f x)}}+\frac{\left (a^3 g \sqrt{\cos (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)} \left (\sqrt{-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 b^3 \sqrt{g \cos (e+f x)}}\\ &=-\frac{2 (g \cos (e+f x))^{3/2}}{3 b f g}-\frac{2 a \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{b^2 f \sqrt{\cos (e+f x)}}+\frac{a^3 g \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b-\sqrt{-a^2+b^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{b^3 \left (b-\sqrt{-a^2+b^2}\right ) f \sqrt{g \cos (e+f x)}}+\frac{a^3 g \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b+\sqrt{-a^2+b^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{b^3 \left (b+\sqrt{-a^2+b^2}\right ) f \sqrt{g \cos (e+f x)}}-\frac{\left (a^2 g\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a^2+b^2} g-b x^2} \, dx,x,\sqrt{g \cos (e+f x)}\right )}{b^2 f}+\frac{\left (a^2 g\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a^2+b^2} g+b x^2} \, dx,x,\sqrt{g \cos (e+f x)}\right )}{b^2 f}\\ &=\frac{a^2 \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt{g}}\right )}{b^{5/2} \sqrt [4]{-a^2+b^2} f}-\frac{a^2 \sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt{g}}\right )}{b^{5/2} \sqrt [4]{-a^2+b^2} f}-\frac{2 (g \cos (e+f x))^{3/2}}{3 b f g}-\frac{2 a \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{b^2 f \sqrt{\cos (e+f x)}}+\frac{a^3 g \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b-\sqrt{-a^2+b^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{b^3 \left (b-\sqrt{-a^2+b^2}\right ) f \sqrt{g \cos (e+f x)}}+\frac{a^3 g \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b+\sqrt{-a^2+b^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{b^3 \left (b+\sqrt{-a^2+b^2}\right ) f \sqrt{g \cos (e+f x)}}\\ \end{align*}
Mathematica [C] time = 17.6111, size = 372, normalized size = 1.01 \[ \frac{\sqrt{g \cos (e+f x)} \left (-\frac{a \left (a+b \sqrt{\sin ^2(e+f x)}\right ) \left (8 b^{5/2} \cos ^{\frac{3}{2}}(e+f x) F_1\left (\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right )+3 \sqrt{2} a \left (a^2-b^2\right )^{3/4} \left (-\log \left (-\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right )+\log \left (\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right )+2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}+1\right )\right )\right )}{\left (a^2-b^2\right ) (a+b \sin (e+f x))}-8 b^{3/2} \cos ^{\frac{3}{2}}(e+f x)\right )}{12 b^{5/2} f \sqrt{\cos (e+f x)}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 5.386, size = 924, normalized size = 2.5 \begin{align*} \text{result too large to display} \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{\sqrt{g \cos \left (f x + e\right )} \sin \left (f x + e\right )^{2}}{b \sin \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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{\sqrt{g \cos \left (f x + e\right )} \sin \left (f x + e\right )^{2}}{b \sin \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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