Optimal. Leaf size=611 \[ -\frac{2 \sqrt{2} \sqrt{d} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b-\sqrt{b^2-a^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right )\right |-1\right )}{b^2 f \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} \sqrt{d} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b+\sqrt{b^2-a^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right )\right |-1\right )}{b^2 f \sqrt{g \cos (e+f x)}}-\frac{a \sqrt{d} g^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right )}{\sqrt{2} b^2 f}+\frac{a \sqrt{d} g^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}+1\right )}{\sqrt{2} b^2 f}+\frac{a \sqrt{d} g^{3/2} \log \left (-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right )}{2 \sqrt{2} b^2 f}-\frac{a \sqrt{d} g^{3/2} \log \left (\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right )}{2 \sqrt{2} b^2 f}-\frac{d g^2 \sqrt{\sin (2 e+2 f x)} F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{2 b f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{g \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{b f} \]
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Rubi [A] time = 1.0299, antiderivative size = 611, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 15, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.405, Rules used = {2901, 2838, 2574, 297, 1162, 617, 204, 1165, 628, 2568, 2573, 2641, 2908, 2907, 1218} \[ -\frac{2 \sqrt{2} \sqrt{d} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b-\sqrt{b^2-a^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right )\right |-1\right )}{b^2 f \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} \sqrt{d} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b+\sqrt{b^2-a^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right )\right |-1\right )}{b^2 f \sqrt{g \cos (e+f x)}}-\frac{a \sqrt{d} g^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right )}{\sqrt{2} b^2 f}+\frac{a \sqrt{d} g^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}+1\right )}{\sqrt{2} b^2 f}+\frac{a \sqrt{d} g^{3/2} \log \left (-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right )}{2 \sqrt{2} b^2 f}-\frac{a \sqrt{d} g^{3/2} \log \left (\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right )}{2 \sqrt{2} b^2 f}-\frac{d g^2 \sqrt{\sin (2 e+2 f x)} F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{2 b f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{g \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{b f} \]
Antiderivative was successfully verified.
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Rule 2901
Rule 2838
Rule 2574
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 2568
Rule 2573
Rule 2641
Rule 2908
Rule 2907
Rule 1218
Rubi steps
\begin{align*} \int \frac{(g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx &=\frac{g^2 \int \frac{\sqrt{d \sin (e+f x)} (a-b \sin (e+f x))}{\sqrt{g \cos (e+f x)}} \, dx}{b^2}-\frac{\left (\left (a^2-b^2\right ) g^2\right ) \int \frac{\sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx}{b^2}\\ &=\frac{\left (a g^2\right ) \int \frac{\sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}} \, dx}{b^2}-\frac{g^2 \int \frac{(d \sin (e+f x))^{3/2}}{\sqrt{g \cos (e+f x)}} \, dx}{b d}-\frac{\left (\left (a^2-b^2\right ) g^2 \sqrt{\cos (e+f x)}\right ) \int \frac{\sqrt{d \sin (e+f x)}}{\sqrt{\cos (e+f x)} (a+b \sin (e+f x))} \, dx}{b^2 \sqrt{g \cos (e+f x)}}\\ &=\frac{g \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}{b f}-\frac{\left (d g^2\right ) \int \frac{1}{\sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}} \, dx}{2 b}+\frac{\left (2 a d g^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{d^2+g^2 x^4} \, dx,x,\frac{\sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}\right )}{b^2 f}-\frac{\left (2 \sqrt{2} \left (a^2-b^2\right ) \left (1-\frac{b}{\sqrt{-a^2+b^2}}\right ) d g^2 \sqrt{\cos (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\left (b-\sqrt{-a^2+b^2}\right ) d+a x^2\right ) \sqrt{1-\frac{x^4}{d^2}}} \, dx,x,\frac{\sqrt{d \sin (e+f x)}}{\sqrt{1+\cos (e+f x)}}\right )}{b^2 f \sqrt{g \cos (e+f x)}}-\frac{\left (2 \sqrt{2} \left (a^2-b^2\right ) \left (1+\frac{b}{\sqrt{-a^2+b^2}}\right ) d g^2 \sqrt{\cos (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\left (b+\sqrt{-a^2+b^2}\right ) d+a x^2\right ) \sqrt{1-\frac{x^4}{d^2}}} \, dx,x,\frac{\sqrt{d \sin (e+f x)}}{\sqrt{1+\cos (e+f x)}}\right )}{b^2 f \sqrt{g \cos (e+f x)}}\\ &=-\frac{2 \sqrt{2} \sqrt{-a^2+b^2} \sqrt{d} g^2 \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b-\sqrt{-a^2+b^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{1+\cos (e+f x)}}\right )\right |-1\right )}{b^2 f \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} \sqrt{-a^2+b^2} \sqrt{d} g^2 \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b+\sqrt{-a^2+b^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{1+\cos (e+f x)}}\right )\right |-1\right )}{b^2 f \sqrt{g \cos (e+f x)}}+\frac{g \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}{b f}-\frac{\left (a d g^2\right ) \operatorname{Subst}\left (\int \frac{d-g x^2}{d^2+g^2 x^4} \, dx,x,\frac{\sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}\right )}{b^2 f}+\frac{\left (a d g^2\right ) \operatorname{Subst}\left (\int \frac{d+g x^2}{d^2+g^2 x^4} \, dx,x,\frac{\sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}\right )}{b^2 f}-\frac{\left (d g^2 \sqrt{\sin (2 e+2 f x)}\right ) \int \frac{1}{\sqrt{\sin (2 e+2 f x)}} \, dx}{2 b \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}\\ &=-\frac{2 \sqrt{2} \sqrt{-a^2+b^2} \sqrt{d} g^2 \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b-\sqrt{-a^2+b^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{1+\cos (e+f x)}}\right )\right |-1\right )}{b^2 f \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} \sqrt{-a^2+b^2} \sqrt{d} g^2 \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b+\sqrt{-a^2+b^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{1+\cos (e+f x)}}\right )\right |-1\right )}{b^2 f \sqrt{g \cos (e+f x)}}+\frac{g \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}{b f}-\frac{d g^2 F\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{\sin (2 e+2 f x)}}{2 b f \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}+\frac{(a d g) \operatorname{Subst}\left (\int \frac{1}{\frac{d}{g}-\frac{\sqrt{2} \sqrt{d} x}{\sqrt{g}}+x^2} \, dx,x,\frac{\sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}\right )}{2 b^2 f}+\frac{(a d g) \operatorname{Subst}\left (\int \frac{1}{\frac{d}{g}+\frac{\sqrt{2} \sqrt{d} x}{\sqrt{g}}+x^2} \, dx,x,\frac{\sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}\right )}{2 b^2 f}+\frac{\left (a \sqrt{d} g^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{d}}{\sqrt{g}}+2 x}{-\frac{d}{g}-\frac{\sqrt{2} \sqrt{d} x}{\sqrt{g}}-x^2} \, dx,x,\frac{\sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}\right )}{2 \sqrt{2} b^2 f}+\frac{\left (a \sqrt{d} g^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{d}}{\sqrt{g}}-2 x}{-\frac{d}{g}+\frac{\sqrt{2} \sqrt{d} x}{\sqrt{g}}-x^2} \, dx,x,\frac{\sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}\right )}{2 \sqrt{2} b^2 f}\\ &=-\frac{2 \sqrt{2} \sqrt{-a^2+b^2} \sqrt{d} g^2 \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b-\sqrt{-a^2+b^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{1+\cos (e+f x)}}\right )\right |-1\right )}{b^2 f \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} \sqrt{-a^2+b^2} \sqrt{d} g^2 \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b+\sqrt{-a^2+b^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{1+\cos (e+f x)}}\right )\right |-1\right )}{b^2 f \sqrt{g \cos (e+f x)}}+\frac{a \sqrt{d} g^{3/2} \log \left (\sqrt{d}-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)\right )}{2 \sqrt{2} b^2 f}-\frac{a \sqrt{d} g^{3/2} \log \left (\sqrt{d}+\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)\right )}{2 \sqrt{2} b^2 f}+\frac{g \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}{b f}-\frac{d g^2 F\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{\sin (2 e+2 f x)}}{2 b f \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}+\frac{\left (a \sqrt{d} g^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right )}{\sqrt{2} b^2 f}-\frac{\left (a \sqrt{d} g^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right )}{\sqrt{2} b^2 f}\\ &=-\frac{a \sqrt{d} g^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right )}{\sqrt{2} b^2 f}+\frac{a \sqrt{d} g^{3/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right )}{\sqrt{2} b^2 f}-\frac{2 \sqrt{2} \sqrt{-a^2+b^2} \sqrt{d} g^2 \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b-\sqrt{-a^2+b^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{1+\cos (e+f x)}}\right )\right |-1\right )}{b^2 f \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} \sqrt{-a^2+b^2} \sqrt{d} g^2 \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b+\sqrt{-a^2+b^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{1+\cos (e+f x)}}\right )\right |-1\right )}{b^2 f \sqrt{g \cos (e+f x)}}+\frac{a \sqrt{d} g^{3/2} \log \left (\sqrt{d}-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)\right )}{2 \sqrt{2} b^2 f}-\frac{a \sqrt{d} g^{3/2} \log \left (\sqrt{d}+\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)\right )}{2 \sqrt{2} b^2 f}+\frac{g \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}{b f}-\frac{d g^2 F\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{\sin (2 e+2 f x)}}{2 b f \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 21.5511, size = 604, normalized size = 0.99 \[ -\frac{(d \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2} \left (a+b \sqrt{\sin ^2(e+f x)}\right ) \left (\frac{2 a F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right )}{a^2-b^2}+\frac{\left (2 a^2-b^2\right ) F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right )}{b^3-a^2 b}+\frac{5 \left (\sin ^2(e+f x) \left (a^2-b^2 \sin ^2(e+f x)\right ) \left (3 \left (a^2-b^2\right ) F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right )-4 b^2 F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right )\right )-5 \left (a^2-b^2\right ) \left (a^2+b^2 \cos ^2(e+f x)-2 b^2\right ) F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right )\right )}{b \sin ^2(e+f x)^{3/4} \left (a^2-b^2 \sin ^2(e+f x)\right ) \left (\cos ^2(e+f x) \left (4 b^2 F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right )+3 \left (b^2-a^2\right ) F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right )\right )-5 \left (a^2-b^2\right ) F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right )\right )}\right )}{5 d f g \sin ^2(e+f x)^{3/4} (a+b \sin (e+f x))} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.264, size = 1926, normalized size = 3.2 \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{\left (g \cos \left (f x + e\right )\right )^{\frac{3}{2}} \sqrt{d \sin \left (f x + e\right )}}{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{\left (g \cos \left (f x + e\right )\right )^{\frac{3}{2}} \sqrt{d \sin \left (f x + e\right )}}{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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