Optimal. Leaf size=528 \[ \frac{a^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt [4]{b^2-a^2}}\right )}{\sqrt{b} f g^{5/2} \left (b^2-a^2\right )^{7/4}}+\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt [4]{b^2-a^2}}\right )}{\sqrt{b} f g^{5/2} \left (b^2-a^2\right )^{7/4}}-\frac{2 a^2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{b f g^2 \left (a^2-b^2\right ) \sqrt{g \cos (e+f x)}}+\frac{4 b \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{3 f g^2 \left (a^2-b^2\right ) \sqrt{g \cos (e+f x)}}+\frac{a^4 \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{b f g^2 \left (a^2-b^2\right ) \left (a^2-b \left (b-\sqrt{b^2-a^2}\right )\right ) \sqrt{g \cos (e+f x)}}+\frac{a^4 \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{b f g^2 \left (a^2-b^2\right ) \left (a^2-b \left (\sqrt{b^2-a^2}+b\right )\right ) \sqrt{g \cos (e+f x)}}+\frac{2 a}{3 f g \left (a^2-b^2\right ) (g \cos (e+f x))^{3/2}}-\frac{2 b \sin (e+f x)}{3 f g \left (a^2-b^2\right ) (g \cos (e+f x))^{3/2}} \]
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Rubi [A] time = 1.08649, antiderivative size = 528, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 14, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.424, Rules used = {2902, 2565, 30, 2566, 2642, 2641, 2867, 2702, 2807, 2805, 329, 212, 208, 205} \[ \frac{a^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt [4]{b^2-a^2}}\right )}{\sqrt{b} f g^{5/2} \left (b^2-a^2\right )^{7/4}}+\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt [4]{b^2-a^2}}\right )}{\sqrt{b} f g^{5/2} \left (b^2-a^2\right )^{7/4}}-\frac{2 a^2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{b f g^2 \left (a^2-b^2\right ) \sqrt{g \cos (e+f x)}}+\frac{4 b \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{3 f g^2 \left (a^2-b^2\right ) \sqrt{g \cos (e+f x)}}+\frac{a^4 \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{b f g^2 \left (a^2-b^2\right ) \left (a^2-b \left (b-\sqrt{b^2-a^2}\right )\right ) \sqrt{g \cos (e+f x)}}+\frac{a^4 \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{b f g^2 \left (a^2-b^2\right ) \left (a^2-b \left (\sqrt{b^2-a^2}+b\right )\right ) \sqrt{g \cos (e+f x)}}+\frac{2 a}{3 f g \left (a^2-b^2\right ) (g \cos (e+f x))^{3/2}}-\frac{2 b \sin (e+f x)}{3 f g \left (a^2-b^2\right ) (g \cos (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2902
Rule 2565
Rule 30
Rule 2566
Rule 2642
Rule 2641
Rule 2867
Rule 2702
Rule 2807
Rule 2805
Rule 329
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{\sin ^3(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx &=\frac{a \int \frac{\sin (e+f x)}{(g \cos (e+f x))^{5/2}} \, dx}{a^2-b^2}-\frac{b \int \frac{\sin ^2(e+f x)}{(g \cos (e+f x))^{5/2}} \, dx}{a^2-b^2}-\frac{a^2 \int \frac{\sin (e+f x)}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx}{\left (a^2-b^2\right ) g^2}\\ &=-\frac{2 b \sin (e+f x)}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}}-\frac{a^2 \int \frac{1}{\sqrt{g \cos (e+f x)}} \, dx}{b \left (a^2-b^2\right ) g^2}+\frac{a^3 \int \frac{1}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx}{b \left (a^2-b^2\right ) g^2}+\frac{(2 b) \int \frac{1}{\sqrt{g \cos (e+f x)}} \, dx}{3 \left (a^2-b^2\right ) g^2}-\frac{a \operatorname{Subst}\left (\int \frac{1}{x^{5/2}} \, dx,x,g \cos (e+f x)\right )}{\left (a^2-b^2\right ) f g}\\ &=\frac{2 a}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}}-\frac{2 b \sin (e+f x)}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}}+\frac{a^4 \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 )^{3/2} g^2}+\frac{a^4 \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 )^{3/2} g^2}+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (\left (a^2-b^2\right ) g^2+b^2 x^2\right )} \, dx,x,g \cos (e+f x)\right )}{\left (a^2-b^2\right ) f g}-\frac{\left (a^2 \sqrt{\cos (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)}} \, dx}{b \left (a^2-b^2\right ) g^2 \sqrt{g \cos (e+f x)}}+\frac{\left (2 b \sqrt{\cos (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)}} \, dx}{3 \left (a^2-b^2\right ) g^2 \sqrt{g \cos (e+f x)}}\\ &=\frac{2 a}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}}-\frac{2 a^2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{b \left (a^2-b^2\right ) f g^2 \sqrt{g \cos (e+f x)}}+\frac{4 b \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{3 \left (a^2-b^2\right ) f g^2 \sqrt{g \cos (e+f x)}}-\frac{2 b \sin (e+f x)}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}}+\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\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^4 \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 )^{3/2} g^2 \sqrt{g \cos (e+f x)}}+\frac{\left (a^4 \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 )^{3/2} g^2 \sqrt{g \cos (e+f x)}}\\ &=\frac{2 a}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}}-\frac{2 a^2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{b \left (a^2-b^2\right ) f g^2 \sqrt{g \cos (e+f x)}}+\frac{4 b \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{3 \left (a^2-b^2\right ) f g^2 \sqrt{g \cos (e+f x)}}-\frac{a^4 \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 )^{3/2} \left (b-\sqrt{-a^2+b^2}\right ) f g^2 \sqrt{g \cos (e+f x)}}+\frac{a^4 \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 )^{3/2} \left (b+\sqrt{-a^2+b^2}\right ) f g^2 \sqrt{g \cos (e+f x)}}-\frac{2 b \sin (e+f x)}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}}+\frac{a^3 \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 )^{3/2} f g^2}+\frac{a^3 \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 )^{3/2} f g^2}\\ &=\frac{a^3 \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 )^{7/4} f g^{5/2}}+\frac{a^3 \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 )^{7/4} f g^{5/2}}+\frac{2 a}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}}-\frac{2 a^2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{b \left (a^2-b^2\right ) f g^2 \sqrt{g \cos (e+f x)}}+\frac{4 b \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{3 \left (a^2-b^2\right ) f g^2 \sqrt{g \cos (e+f x)}}-\frac{a^4 \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 )^{3/2} \left (b-\sqrt{-a^2+b^2}\right ) f g^2 \sqrt{g \cos (e+f x)}}+\frac{a^4 \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 )^{3/2} \left (b+\sqrt{-a^2+b^2}\right ) f g^2 \sqrt{g \cos (e+f x)}}-\frac{2 b \sin (e+f x)}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}}\\ \end{align*}
Mathematica [C] time = 24.43, size = 1193, normalized size = 2.26 \[ \frac{2 \cos (e+f x) (a-b \sin (e+f x))}{3 \left (a^2-b^2\right ) f (g \cos (e+f x))^{5/2}}-\frac{\cos ^{\frac{5}{2}}(e+f x) \left (\frac{4 a b \left (a+b \sqrt{1-\cos ^2(e+f x)}\right ) \left (\frac{5 a \left (a^2-b^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},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)}}{\sqrt{1-\cos ^2(e+f x)} \left (5 \left (a^2-b^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right )-2 \left (2 F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right ) b^2+\left (b^2-a^2\right ) F_1\left (\frac{5}{4};\frac{3}{2},1;\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)\right ) \left (a^2+b^2 \left (\cos ^2(e+f x)-1\right )\right )}-\frac{\left (\frac{1}{8}-\frac{i}{8}\right ) \sqrt{b} \left (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 (\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt [4]{b^2-a^2}}+1\right )+\log \left (i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt [4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right )-\log \left (i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt [4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right )\right )}{\left (b^2-a^2\right )^{3/4}}\right ) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}-\frac{2 \left (3 a^2-2 b^2\right ) \left (a+b \sqrt{1-\cos ^2(e+f x)}\right ) \left (\frac{5 b \left (a^2-b^2\right ) \sqrt{\cos (e+f x)} \sqrt{1-\cos ^2(e+f x)} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right )}{\left (2 \left (2 F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right ) b^2+\left (a^2-b^2\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\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{1}{2},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{a \left (-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 )-\log \left (b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right )+\log \left (b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right )\right )}{4 \sqrt{2} \sqrt{b} \left (a^2-b^2\right )^{3/4}}\right ) \sin ^2(e+f x)}{\left (1-\cos ^2(e+f x)\right ) (a+b \sin (e+f x))}\right )}{3 (a-b) (a+b) f (g \cos (e+f x))^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 12.217, size = 2331, normalized size = 4.4 \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 )^{3}}{\left (g \cos \left (f x + e\right )\right )^{\frac{5}{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 )^{3}}{\left (g \cos \left (f x + e\right )\right )^{\frac{5}{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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