Optimal. Leaf size=525 \[ \frac{b g^2 \left (a^2-b^2\right ) \sqrt{\sin (2 e+2 f x)} F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{a^4 d^3 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} b^2 g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b-\sqrt{b^2-a^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right )\right |-1\right )}{a^4 d^{7/2} f \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} b^2 g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b+\sqrt{b^2-a^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right )\right |-1\right )}{a^4 d^{7/2} f \sqrt{g \cos (e+f x)}}+\frac{2 g \left (a^2-b^2\right ) \sqrt{g \cos (e+f x)}}{a^3 d^3 f \sqrt{d \sin (e+f x)}}-\frac{2 b g^2 \sqrt{\sin (2 e+2 f x)} F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{3 a^2 d^3 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{2 b g \sqrt{g \cos (e+f x)}}{3 a^2 d^2 f (d \sin (e+f x))^{3/2}}-\frac{8 g \sqrt{g \cos (e+f x)}}{5 a d^3 f \sqrt{d \sin (e+f x)}}-\frac{2 g \sqrt{g \cos (e+f x)}}{5 a d f (d \sin (e+f x))^{5/2}} \]
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Rubi [A] time = 1.35288, antiderivative size = 525, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.243, Rules used = {2899, 2570, 2563, 2573, 2641, 2910, 2908, 2907, 1218} \[ \frac{b g^2 \left (a^2-b^2\right ) \sqrt{\sin (2 e+2 f x)} F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{a^4 d^3 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} b^2 g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b-\sqrt{b^2-a^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right )\right |-1\right )}{a^4 d^{7/2} f \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} b^2 g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b+\sqrt{b^2-a^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right )\right |-1\right )}{a^4 d^{7/2} f \sqrt{g \cos (e+f x)}}+\frac{2 g \left (a^2-b^2\right ) \sqrt{g \cos (e+f x)}}{a^3 d^3 f \sqrt{d \sin (e+f x)}}-\frac{2 b g^2 \sqrt{\sin (2 e+2 f x)} F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{3 a^2 d^3 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{2 b g \sqrt{g \cos (e+f x)}}{3 a^2 d^2 f (d \sin (e+f x))^{3/2}}-\frac{8 g \sqrt{g \cos (e+f x)}}{5 a d^3 f \sqrt{d \sin (e+f x)}}-\frac{2 g \sqrt{g \cos (e+f x)}}{5 a d f (d \sin (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2899
Rule 2570
Rule 2563
Rule 2573
Rule 2641
Rule 2910
Rule 2908
Rule 2907
Rule 1218
Rubi steps
\begin{align*} \int \frac{(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx &=\frac{g^2 \int \frac{1}{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{7/2}} \, dx}{a}-\frac{\left (\left (a^2-b^2\right ) g^2\right ) \int \frac{1}{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx}{a^2 d^2}-\frac{\left (b g^2\right ) \int \frac{1}{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{5/2}} \, dx}{a^2 d}\\ &=-\frac{2 g \sqrt{g \cos (e+f x)}}{5 a d f (d \sin (e+f x))^{5/2}}+\frac{2 b g \sqrt{g \cos (e+f x)}}{3 a^2 d^2 f (d \sin (e+f x))^{3/2}}-\frac{\left (2 b g^2\right ) \int \frac{1}{\sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}} \, dx}{3 a^2 d^3}+\frac{\left (b \left (a^2-b^2\right ) g^2\right ) \int \frac{1}{\sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))} \, dx}{a^3 d^3}+\frac{\left (4 g^2\right ) \int \frac{1}{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{3/2}} \, dx}{5 a d^2}-\frac{\left (\left (a^2-b^2\right ) g^2\right ) \int \frac{1}{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{3/2}} \, dx}{a^3 d^2}\\ &=-\frac{2 g \sqrt{g \cos (e+f x)}}{5 a d f (d \sin (e+f x))^{5/2}}+\frac{2 b g \sqrt{g \cos (e+f x)}}{3 a^2 d^2 f (d \sin (e+f x))^{3/2}}-\frac{8 g \sqrt{g \cos (e+f x)}}{5 a d^3 f \sqrt{d \sin (e+f x)}}+\frac{2 \left (a^2-b^2\right ) g \sqrt{g \cos (e+f x)}}{a^3 d^3 f \sqrt{d \sin (e+f x)}}-\frac{\left (b^2 \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}{a^4 d^4}+\frac{\left (b \left (a^2-b^2\right ) g^2\right ) \int \frac{1}{\sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}} \, dx}{a^4 d^3}-\frac{\left (2 b g^2 \sqrt{\sin (2 e+2 f x)}\right ) \int \frac{1}{\sqrt{\sin (2 e+2 f x)}} \, dx}{3 a^2 d^3 \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}\\ &=-\frac{2 g \sqrt{g \cos (e+f x)}}{5 a d f (d \sin (e+f x))^{5/2}}+\frac{2 b g \sqrt{g \cos (e+f x)}}{3 a^2 d^2 f (d \sin (e+f x))^{3/2}}-\frac{8 g \sqrt{g \cos (e+f x)}}{5 a d^3 f \sqrt{d \sin (e+f x)}}+\frac{2 \left (a^2-b^2\right ) g \sqrt{g \cos (e+f x)}}{a^3 d^3 f \sqrt{d \sin (e+f x)}}-\frac{2 b g^2 F\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{\sin (2 e+2 f x)}}{3 a^2 d^3 f \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}-\frac{\left (b^2 \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}{a^4 d^4 \sqrt{g \cos (e+f x)}}+\frac{\left (b \left (a^2-b^2\right ) g^2 \sqrt{\sin (2 e+2 f x)}\right ) \int \frac{1}{\sqrt{\sin (2 e+2 f x)}} \, dx}{a^4 d^3 \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}\\ &=-\frac{2 g \sqrt{g \cos (e+f x)}}{5 a d f (d \sin (e+f x))^{5/2}}+\frac{2 b g \sqrt{g \cos (e+f x)}}{3 a^2 d^2 f (d \sin (e+f x))^{3/2}}-\frac{8 g \sqrt{g \cos (e+f x)}}{5 a d^3 f \sqrt{d \sin (e+f x)}}+\frac{2 \left (a^2-b^2\right ) g \sqrt{g \cos (e+f x)}}{a^3 d^3 f \sqrt{d \sin (e+f x)}}-\frac{2 b g^2 F\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{\sin (2 e+2 f x)}}{3 a^2 d^3 f \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}+\frac{b \left (a^2-b^2\right ) g^2 F\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{\sin (2 e+2 f x)}}{a^4 d^3 f \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}-\frac{\left (2 \sqrt{2} b^2 \left (a^2-b^2\right ) \left (1-\frac{b}{\sqrt{-a^2+b^2}}\right ) 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 )}{a^4 d^3 f \sqrt{g \cos (e+f x)}}-\frac{\left (2 \sqrt{2} b^2 \left (a^2-b^2\right ) \left (1+\frac{b}{\sqrt{-a^2+b^2}}\right ) 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 )}{a^4 d^3 f \sqrt{g \cos (e+f x)}}\\ &=-\frac{2 \sqrt{2} b^2 \sqrt{-a^2+b^2} 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 )}{a^4 d^{7/2} f \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} b^2 \sqrt{-a^2+b^2} 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 )}{a^4 d^{7/2} f \sqrt{g \cos (e+f x)}}-\frac{2 g \sqrt{g \cos (e+f x)}}{5 a d f (d \sin (e+f x))^{5/2}}+\frac{2 b g \sqrt{g \cos (e+f x)}}{3 a^2 d^2 f (d \sin (e+f x))^{3/2}}-\frac{8 g \sqrt{g \cos (e+f x)}}{5 a d^3 f \sqrt{d \sin (e+f x)}}+\frac{2 \left (a^2-b^2\right ) g \sqrt{g \cos (e+f x)}}{a^3 d^3 f \sqrt{d \sin (e+f x)}}-\frac{2 b g^2 F\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{\sin (2 e+2 f x)}}{3 a^2 d^3 f \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}+\frac{b \left (a^2-b^2\right ) g^2 F\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{\sin (2 e+2 f x)}}{a^4 d^3 f \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 20.498, size = 1162, normalized size = 2.21 \[ \frac{b (g \cos (e+f x))^{3/2} \left (-\frac{2 \left (a^2-3 b^2\right ) \left (a+b \sqrt{1-\cos ^2(e+f x)}\right ) \sqrt{\sin (e+f x)} \left (\frac{5 a \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 ) \sqrt{\cos (e+f x)}}{\left (1-\cos ^2(e+f x)\right )^{3/4} \left (\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 ) \cos ^2(e+f x)+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 ) \left (a^2+b^2 \left (\cos ^2(e+f x)-1\right )\right )}-\frac{\left (\frac{1}{8}-\frac{i}{8}\right ) b \left (2 \tan ^{-1}\left (1-\frac{(1+i) \sqrt{a} \sqrt{\cos (e+f x)}}{\sqrt [4]{b^2-a^2} \sqrt [4]{\cos ^2(e+f x)-1}}\right )-2 \tan ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\cos (e+f x)}}{\sqrt [4]{b^2-a^2} \sqrt [4]{\cos ^2(e+f x)-1}}+1\right )+\log \left (\frac{i a \cos (e+f x)}{\sqrt{\cos ^2(e+f x)-1}}-\frac{(1+i) \sqrt{a} \sqrt [4]{b^2-a^2} \sqrt{\cos (e+f x)}}{\sqrt [4]{\cos ^2(e+f x)-1}}+\sqrt{b^2-a^2}\right )-\log \left (\frac{i a \cos (e+f x)}{\sqrt{\cos ^2(e+f x)-1}}+\frac{(1+i) \sqrt{a} \sqrt [4]{b^2-a^2} \sqrt{\cos (e+f x)}}{\sqrt [4]{\cos ^2(e+f x)-1}}+\sqrt{b^2-a^2}\right )\right )}{\sqrt{a} \left (b^2-a^2\right )^{3/4}}\right )}{\sqrt [4]{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}-\frac{4 a b \sqrt{\sin (e+f x)} \left (\frac{\sqrt{a} \left (-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right )+\log \left (-a+\sqrt{2} \sqrt [4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right )-\log \left (a+\sqrt{2} \sqrt [4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right )\right )}{4 \sqrt{2} \left (a^2-b^2\right )^{3/4}}-\frac{b F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\tan ^2(e+f x),\left (\frac{b^2}{a^2}-1\right ) \tan ^2(e+f x)\right ) \tan ^{\frac{5}{2}}(e+f x)}{5 a^2}\right ) \left (\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right )}{\cos ^{\frac{5}{2}}(e+f x) (a+b \sin (e+f x)) \sqrt{\tan (e+f x)} \left (\tan ^2(e+f x)+1\right )^{3/2}}\right ) \sin ^{\frac{7}{2}}(e+f x)}{3 a^3 f \cos ^{\frac{3}{2}}(e+f x) (d \sin (e+f x))^{7/2}}+\frac{(g \cos (e+f x))^{3/2} \left (-\frac{2 \csc ^3(e+f x)}{5 a}+\frac{2 b \csc ^2(e+f x)}{3 a^2}+\frac{2 \left (a^2-5 b^2\right ) \csc (e+f x)}{5 a^3}\right ) \tan (e+f x) \sin ^3(e+f x)}{f (d \sin (e+f x))^{7/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.312, size = 5828, normalized size = 11.1 \begin{align*} \text{output 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}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac{7}{2}}}\,{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}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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