Optimal. Leaf size=453 \[ \frac{a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt [4]{b^2-a^2}}\right )}{\sqrt{b} f g^{3/2} \left (b^2-a^2\right )^{5/4}}-\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt [4]{b^2-a^2}}\right )}{\sqrt{b} f g^{3/2} \left (b^2-a^2\right )^{5/4}}-\frac{2 a E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{f g^2 \left (a^2-b^2\right ) \sqrt{\cos (e+f x)}}-\frac{2 b}{f g \left (a^2-b^2\right ) \sqrt{g \cos (e+f x)}}+\frac{2 a \sin (e+f x)}{f g \left (a^2-b^2\right ) \sqrt{g \cos (e+f x)}}-\frac{a^3 \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{b f g \left (a^2-b^2\right ) \left (b-\sqrt{b^2-a^2}\right ) \sqrt{g \cos (e+f x)}}-\frac{a^3 \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{b f g \left (a^2-b^2\right ) \left (\sqrt{b^2-a^2}+b\right ) \sqrt{g \cos (e+f x)}} \]
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Rubi [A] time = 0.880679, antiderivative size = 453, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 13, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.394, Rules used = {2902, 2636, 2640, 2639, 2565, 30, 2701, 2807, 2805, 329, 298, 205, 208} \[ \frac{a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt [4]{b^2-a^2}}\right )}{\sqrt{b} f g^{3/2} \left (b^2-a^2\right )^{5/4}}-\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt [4]{b^2-a^2}}\right )}{\sqrt{b} f g^{3/2} \left (b^2-a^2\right )^{5/4}}-\frac{2 a E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{f g^2 \left (a^2-b^2\right ) \sqrt{\cos (e+f x)}}-\frac{2 b}{f g \left (a^2-b^2\right ) \sqrt{g \cos (e+f x)}}+\frac{2 a \sin (e+f x)}{f g \left (a^2-b^2\right ) \sqrt{g \cos (e+f x)}}-\frac{a^3 \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{b f g \left (a^2-b^2\right ) \left (b-\sqrt{b^2-a^2}\right ) \sqrt{g \cos (e+f x)}}-\frac{a^3 \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{b f g \left (a^2-b^2\right ) \left (\sqrt{b^2-a^2}+b\right ) \sqrt{g \cos (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2902
Rule 2636
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{\sin ^2(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx &=\frac{a \int \frac{1}{(g \cos (e+f x))^{3/2}} \, dx}{a^2-b^2}-\frac{b \int \frac{\sin (e+f x)}{(g \cos (e+f x))^{3/2}} \, dx}{a^2-b^2}-\frac{a^2 \int \frac{\sqrt{g \cos (e+f x)}}{a+b \sin (e+f x)} \, dx}{\left (a^2-b^2\right ) g^2}\\ &=\frac{2 a \sin (e+f x)}{\left (a^2-b^2\right ) f g \sqrt{g \cos (e+f x)}}-\frac{a \int \sqrt{g \cos (e+f x)} \, dx}{\left (a^2-b^2\right ) g^2}+\frac{a^3 \int \frac{1}{\sqrt{g \cos (e+f x)} \left (\sqrt{-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 b \left (a^2-b^2\right ) g}-\frac{a^3 \int \frac{1}{\sqrt{g \cos (e+f x)} \left (\sqrt{-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 b \left (a^2-b^2\right ) g}+\frac{b \operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,g \cos (e+f x)\right )}{\left (a^2-b^2\right ) f g}-\frac{\left (a^2 b\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 )}{\left (a^2-b^2\right ) f g}\\ &=-\frac{2 b}{\left (a^2-b^2\right ) f g \sqrt{g \cos (e+f x)}}+\frac{2 a \sin (e+f x)}{\left (a^2-b^2\right ) f g \sqrt{g \cos (e+f x)}}-\frac{\left (2 a^2 b\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 )}{\left (a^2-b^2\right ) f g}+\frac{\left (a^3 \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 \left (a^2-b^2\right ) g \sqrt{g \cos (e+f x)}}-\frac{\left (a^3 \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 \left (a^2-b^2\right ) g \sqrt{g \cos (e+f x)}}-\frac{\left (a \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{\left (a^2-b^2\right ) g^2 \sqrt{\cos (e+f x)}}\\ &=-\frac{2 b}{\left (a^2-b^2\right ) f g \sqrt{g \cos (e+f x)}}-\frac{2 a \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) f g^2 \sqrt{\cos (e+f x)}}-\frac{a^3 \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 \left (a^2-b^2\right ) \left (b-\sqrt{-a^2+b^2}\right ) f g \sqrt{g \cos (e+f x)}}-\frac{a^3 \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 \left (a^2-b^2\right ) \left (b+\sqrt{-a^2+b^2}\right ) f g \sqrt{g \cos (e+f x)}}+\frac{2 a \sin (e+f x)}{\left (a^2-b^2\right ) f g \sqrt{g \cos (e+f x)}}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a^2+b^2} g-b x^2} \, dx,x,\sqrt{g \cos (e+f x)}\right )}{\left (a^2-b^2\right ) f g}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a^2+b^2} g+b x^2} \, dx,x,\sqrt{g \cos (e+f x)}\right )}{\left (a^2-b^2\right ) f g}\\ &=\frac{a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt{g}}\right )}{\sqrt{b} \left (-a^2+b^2\right )^{5/4} f g^{3/2}}-\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt{g}}\right )}{\sqrt{b} \left (-a^2+b^2\right )^{5/4} f g^{3/2}}-\frac{2 b}{\left (a^2-b^2\right ) f g \sqrt{g \cos (e+f x)}}-\frac{2 a \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) f g^2 \sqrt{\cos (e+f x)}}-\frac{a^3 \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 \left (a^2-b^2\right ) \left (b-\sqrt{-a^2+b^2}\right ) f g \sqrt{g \cos (e+f x)}}-\frac{a^3 \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 \left (a^2-b^2\right ) \left (b+\sqrt{-a^2+b^2}\right ) f g \sqrt{g \cos (e+f x)}}+\frac{2 a \sin (e+f x)}{\left (a^2-b^2\right ) f g \sqrt{g \cos (e+f x)}}\\ \end{align*}
Mathematica [C] time = 16.6881, size = 785, normalized size = 1.73 \[ \frac{2 \cos (e+f x) (a \sin (e+f x)-b)}{f \left (a^2-b^2\right ) (g \cos (e+f x))^{3/2}}-\frac{a \cos ^{\frac{3}{2}}(e+f x) \left (-\frac{\sin ^2(e+f x) \left (a+b \sqrt{1-\cos ^2(e+f x)}\right ) \left (8 b^{5/2} \cos ^{\frac{3}{2}}(e+f x) F_1\left (\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right )+3 \sqrt{2} a \left (a^2-b^2\right )^{3/4} \left (-\log \left (-\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right )+\log \left (\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right )+2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}+1\right )\right )\right )}{12 \sqrt{b} \left (b^2-a^2\right ) \left (1-\cos ^2(e+f x)\right ) (a+b \sin (e+f x))}-\frac{4 a \sin (e+f x) \left (a+b \sqrt{1-\cos ^2(e+f x)}\right ) \left (\frac{a \cos ^{\frac{3}{2}}(e+f x) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right )}{3 \left (a^2-b^2\right )}+\frac{\left (\frac{1}{8}+\frac{i}{8}\right ) \left (-\log \left (-(1+i) \sqrt{b} \sqrt [4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}+i b \cos (e+f x)\right )+\log \left ((1+i) \sqrt{b} \sqrt [4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}+i b \cos (e+f x)\right )+2 \tan ^{-1}\left (1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (1+\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt [4]{b^2-a^2}}\right )\right )}{\sqrt{b} \sqrt [4]{b^2-a^2}}\right )}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}\right )}{f (a-b) (a+b) (g \cos (e+f x))^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 7.4, size = 1105, normalized size = 2.4 \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{\sin \left (f x + e\right )^{2}}{\left (g \cos \left (f x + e\right )\right )^{\frac{3}{2}}{\left (b \sin \left (f x + e\right ) + a\right )}}\,{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{\sin \left (f x + e\right )^{2}}{\left (g \cos \left (f x + e\right )\right )^{\frac{3}{2}}{\left (b \sin \left (f x + e\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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