Optimal. Leaf size=572 \[ \frac{a g^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right )}{\sqrt{2} b^2 \sqrt{d} f}-\frac{a g^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right )}{\sqrt{2} b^2 \sqrt{d} f}+\frac{2 \sqrt{2} g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left (-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\right |-1\right )}{b^2 f \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left (\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\right |-1\right )}{b^2 f \sqrt{d \sin (e+f x)}}-\frac{a g^{5/2} \log \left (-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right )}{2 \sqrt{2} b^2 \sqrt{d} f}+\frac{a g^{5/2} \log \left (\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right )}{2 \sqrt{2} b^2 \sqrt{d} f}-\frac{g^2 E\left (\left .e+f x-\frac{\pi }{4}\right |2\right ) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{b d f \sqrt{\sin (2 e+2 f x)}} \]
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Rubi [A] time = 1.0231, antiderivative size = 572, 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, 2575, 297, 1162, 617, 204, 1165, 628, 2572, 2639, 2906, 2905, 490, 1218} \[ \frac{a g^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right )}{\sqrt{2} b^2 \sqrt{d} f}-\frac{a g^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right )}{\sqrt{2} b^2 \sqrt{d} f}+\frac{2 \sqrt{2} g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left (-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\right |-1\right )}{b^2 f \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left (\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\right |-1\right )}{b^2 f \sqrt{d \sin (e+f x)}}-\frac{a g^{5/2} \log \left (-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right )}{2 \sqrt{2} b^2 \sqrt{d} f}+\frac{a g^{5/2} \log \left (\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right )}{2 \sqrt{2} b^2 \sqrt{d} f}-\frac{g^2 E\left (\left .e+f x-\frac{\pi }{4}\right |2\right ) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{b d f \sqrt{\sin (2 e+2 f x)}} \]
Antiderivative was successfully verified.
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Rule 2901
Rule 2838
Rule 2575
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 2572
Rule 2639
Rule 2906
Rule 2905
Rule 490
Rule 1218
Rubi steps
\begin{align*} \int \frac{(g \cos (e+f x))^{5/2}}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))} \, dx &=\frac{g^2 \int \frac{\sqrt{g \cos (e+f x)} (a-b \sin (e+f x))}{\sqrt{d \sin (e+f x)}} \, dx}{b^2}-\frac{\left (\left (a^2-b^2\right ) g^2\right ) \int \frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))} \, dx}{b^2}\\ &=\frac{\left (a g^2\right ) \int \frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}} \, dx}{b^2}-\frac{g^2 \int \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)} \, dx}{b d}-\frac{\left (\left (a^2-b^2\right ) g^2 \sqrt{\sin (e+f x)}\right ) \int \frac{\sqrt{g \cos (e+f x)}}{\sqrt{\sin (e+f x)} (a+b \sin (e+f x))} \, dx}{b^2 \sqrt{d \sin (e+f x)}}\\ &=-\frac{\left (2 a d g^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{g^2+d^2 x^4} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{b^2 f}+\frac{\left (4 \sqrt{2} \left (a^2-b^2\right ) g^3 \sqrt{\sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left ((a+b) g^2+(a-b) x^4\right ) \sqrt{1-\frac{x^4}{g^2}}} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{1+\sin (e+f x)}}\right )}{b^2 f \sqrt{d \sin (e+f x)}}-\frac{\left (g^2 \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}\right ) \int \sqrt{\sin (2 e+2 f x)} \, dx}{b d \sqrt{\sin (2 e+2 f x)}}\\ &=-\frac{g^2 \sqrt{g \cos (e+f x)} E\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{d \sin (e+f x)}}{b d f \sqrt{\sin (2 e+2 f x)}}+\frac{\left (a g^3\right ) \operatorname{Subst}\left (\int \frac{g-d x^2}{g^2+d^2 x^4} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{b^2 f}-\frac{\left (a g^3\right ) \operatorname{Subst}\left (\int \frac{g+d x^2}{g^2+d^2 x^4} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{b^2 f}+\frac{\left (2 \sqrt{2} \left (a^2-b^2\right ) g^3 \sqrt{\sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{a+b} g-\sqrt{-a+b} x^2\right ) \sqrt{1-\frac{x^4}{g^2}}} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{1+\sin (e+f x)}}\right )}{b^2 \sqrt{-a+b} f \sqrt{d \sin (e+f x)}}-\frac{\left (2 \sqrt{2} \left (a^2-b^2\right ) g^3 \sqrt{\sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{a+b} g+\sqrt{-a+b} x^2\right ) \sqrt{1-\frac{x^4}{g^2}}} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{1+\sin (e+f x)}}\right )}{b^2 \sqrt{-a+b} f \sqrt{d \sin (e+f x)}}\\ &=\frac{2 \sqrt{2} \sqrt{-a+b} \sqrt{a+b} g^{5/2} \Pi \left (-\frac{\sqrt{-a+b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt{\sin (e+f x)}}{b^2 f \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} \sqrt{-a+b} \sqrt{a+b} g^{5/2} \Pi \left (\frac{\sqrt{-a+b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt{\sin (e+f x)}}{b^2 f \sqrt{d \sin (e+f x)}}-\frac{g^2 \sqrt{g \cos (e+f x)} E\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{d \sin (e+f x)}}{b d f \sqrt{\sin (2 e+2 f x)}}-\frac{\left (a g^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{g}}{\sqrt{d}}+2 x}{-\frac{g}{d}-\frac{\sqrt{2} \sqrt{g} x}{\sqrt{d}}-x^2} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 \sqrt{2} b^2 \sqrt{d} f}-\frac{\left (a g^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{g}}{\sqrt{d}}-2 x}{-\frac{g}{d}+\frac{\sqrt{2} \sqrt{g} x}{\sqrt{d}}-x^2} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 \sqrt{2} b^2 \sqrt{d} f}-\frac{\left (a g^3\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{g}{d}-\frac{\sqrt{2} \sqrt{g} x}{\sqrt{d}}+x^2} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 b^2 d f}-\frac{\left (a g^3\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{g}{d}+\frac{\sqrt{2} \sqrt{g} x}{\sqrt{d}}+x^2} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 b^2 d f}\\ &=-\frac{a g^{5/2} \log \left (\sqrt{g}+\sqrt{g} \cot (e+f x)-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 \sqrt{2} b^2 \sqrt{d} f}+\frac{a g^{5/2} \log \left (\sqrt{g}+\sqrt{g} \cot (e+f x)+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 \sqrt{2} b^2 \sqrt{d} f}+\frac{2 \sqrt{2} \sqrt{-a+b} \sqrt{a+b} g^{5/2} \Pi \left (-\frac{\sqrt{-a+b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt{\sin (e+f x)}}{b^2 f \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} \sqrt{-a+b} \sqrt{a+b} g^{5/2} \Pi \left (\frac{\sqrt{-a+b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt{\sin (e+f x)}}{b^2 f \sqrt{d \sin (e+f x)}}-\frac{g^2 \sqrt{g \cos (e+f x)} E\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{d \sin (e+f x)}}{b d f \sqrt{\sin (2 e+2 f x)}}-\frac{\left (a g^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right )}{\sqrt{2} b^2 \sqrt{d} f}+\frac{\left (a g^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right )}{\sqrt{2} b^2 \sqrt{d} f}\\ &=\frac{a g^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right )}{\sqrt{2} b^2 \sqrt{d} f}-\frac{a g^{5/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right )}{\sqrt{2} b^2 \sqrt{d} f}-\frac{a g^{5/2} \log \left (\sqrt{g}+\sqrt{g} \cot (e+f x)-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 \sqrt{2} b^2 \sqrt{d} f}+\frac{a g^{5/2} \log \left (\sqrt{g}+\sqrt{g} \cot (e+f x)+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 \sqrt{2} b^2 \sqrt{d} f}+\frac{2 \sqrt{2} \sqrt{-a+b} \sqrt{a+b} g^{5/2} \Pi \left (-\frac{\sqrt{-a+b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt{\sin (e+f x)}}{b^2 f \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} \sqrt{-a+b} \sqrt{a+b} g^{5/2} \Pi \left (\frac{\sqrt{-a+b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt{\sin (e+f x)}}{b^2 f \sqrt{d \sin (e+f x)}}-\frac{g^2 \sqrt{g \cos (e+f x)} E\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{d \sin (e+f x)}}{b d f \sqrt{\sin (2 e+2 f x)}}\\ \end{align*}
Mathematica [C] time = 25.765, size = 1402, normalized size = 2.45 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.303, size = 5148, normalized size = 9. \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{5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \sqrt{d \sin \left (f x + e\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{\left (g \cos \left (f x + e\right )\right )^{\frac{5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \sqrt{d \sin \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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