Optimal. Leaf size=612 \[ -\frac{2 b g^2 \left (a^2-b^2\right ) E\left (\left .e+f x-\frac{\pi }{4}\right |2\right ) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a^4 d^5 f \sqrt{\sin (2 e+2 f x)}}+\frac{2 \sqrt{2} b^2 g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left (-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\right |-1\right )}{a^4 d^4 f \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} b^2 g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left (\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\right |-1\right )}{a^4 d^4 f \sqrt{d \sin (e+f x)}}-\frac{2 b g \left (a^2-b^2\right ) (g \cos (e+f x))^{3/2}}{a^4 d^4 f \sqrt{d \sin (e+f x)}}+\frac{2 g \left (a^2-b^2\right ) (g \cos (e+f x))^{3/2}}{3 a^3 d^3 f (d \sin (e+f x))^{3/2}}+\frac{4 b g^2 E\left (\left .e+f x-\frac{\pi }{4}\right |2\right ) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{5 a^2 d^5 f \sqrt{\sin (2 e+2 f x)}}+\frac{4 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^4 f \sqrt{d \sin (e+f x)}}+\frac{2 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^2 f (d \sin (e+f x))^{5/2}}-\frac{8 g (g \cos (e+f x))^{3/2}}{21 a d^3 f (d \sin (e+f x))^{3/2}}-\frac{2 g (g \cos (e+f x))^{3/2}}{7 a d f (d \sin (e+f x))^{7/2}} \]
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Rubi [A] time = 1.58397, antiderivative size = 612, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.27, Rules used = {2899, 2570, 2563, 2572, 2639, 2910, 2906, 2905, 490, 1218} \[ -\frac{2 b g^2 \left (a^2-b^2\right ) E\left (\left .e+f x-\frac{\pi }{4}\right |2\right ) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a^4 d^5 f \sqrt{\sin (2 e+2 f x)}}+\frac{2 \sqrt{2} b^2 g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left (-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\right |-1\right )}{a^4 d^4 f \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} b^2 g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left (\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\right |-1\right )}{a^4 d^4 f \sqrt{d \sin (e+f x)}}-\frac{2 b g \left (a^2-b^2\right ) (g \cos (e+f x))^{3/2}}{a^4 d^4 f \sqrt{d \sin (e+f x)}}+\frac{2 g \left (a^2-b^2\right ) (g \cos (e+f x))^{3/2}}{3 a^3 d^3 f (d \sin (e+f x))^{3/2}}+\frac{4 b g^2 E\left (\left .e+f x-\frac{\pi }{4}\right |2\right ) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{5 a^2 d^5 f \sqrt{\sin (2 e+2 f x)}}+\frac{4 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^4 f \sqrt{d \sin (e+f x)}}+\frac{2 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^2 f (d \sin (e+f x))^{5/2}}-\frac{8 g (g \cos (e+f x))^{3/2}}{21 a d^3 f (d \sin (e+f x))^{3/2}}-\frac{2 g (g \cos (e+f x))^{3/2}}{7 a d f (d \sin (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2899
Rule 2570
Rule 2563
Rule 2572
Rule 2639
Rule 2910
Rule 2906
Rule 2905
Rule 490
Rule 1218
Rubi steps
\begin{align*} \int \frac{(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx &=\frac{g^2 \int \frac{\sqrt{g \cos (e+f x)}}{(d \sin (e+f x))^{9/2}} \, dx}{a}-\frac{\left (\left (a^2-b^2\right ) g^2\right ) \int \frac{\sqrt{g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx}{a^2 d^2}-\frac{\left (b g^2\right ) \int \frac{\sqrt{g \cos (e+f x)}}{(d \sin (e+f x))^{7/2}} \, dx}{a^2 d}\\ &=-\frac{2 g (g \cos (e+f x))^{3/2}}{7 a d f (d \sin (e+f x))^{7/2}}+\frac{2 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^2 f (d \sin (e+f x))^{5/2}}-\frac{\left (2 b g^2\right ) \int \frac{\sqrt{g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}} \, dx}{5 a^2 d^3}+\frac{\left (b \left (a^2-b^2\right ) g^2\right ) \int \frac{\sqrt{g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx}{a^3 d^3}+\frac{\left (4 g^2\right ) \int \frac{\sqrt{g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}} \, dx}{7 a d^2}-\frac{\left (\left (a^2-b^2\right ) g^2\right ) \int \frac{\sqrt{g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}} \, dx}{a^3 d^2}\\ &=-\frac{2 g (g \cos (e+f x))^{3/2}}{7 a d f (d \sin (e+f x))^{7/2}}+\frac{2 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^2 f (d \sin (e+f x))^{5/2}}-\frac{8 g (g \cos (e+f x))^{3/2}}{21 a d^3 f (d \sin (e+f x))^{3/2}}+\frac{2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{3 a^3 d^3 f (d \sin (e+f x))^{3/2}}+\frac{4 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^4 f \sqrt{d \sin (e+f x)}}+\frac{\left (4 b g^2\right ) \int \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)} \, dx}{5 a^2 d^5}-\frac{\left (b^2 \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}{a^4 d^4}+\frac{\left (b \left (a^2-b^2\right ) g^2\right ) \int \frac{\sqrt{g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}} \, dx}{a^4 d^3}\\ &=-\frac{2 g (g \cos (e+f x))^{3/2}}{7 a d f (d \sin (e+f x))^{7/2}}+\frac{2 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^2 f (d \sin (e+f x))^{5/2}}-\frac{8 g (g \cos (e+f x))^{3/2}}{21 a d^3 f (d \sin (e+f x))^{3/2}}+\frac{2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{3 a^3 d^3 f (d \sin (e+f x))^{3/2}}+\frac{4 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^4 f \sqrt{d \sin (e+f x)}}-\frac{2 b \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{a^4 d^4 f \sqrt{d \sin (e+f x)}}-\frac{\left (2 b \left (a^2-b^2\right ) g^2\right ) \int \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)} \, dx}{a^4 d^5}-\frac{\left (b^2 \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}{a^4 d^4 \sqrt{d \sin (e+f x)}}+\frac{\left (4 b g^2 \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}\right ) \int \sqrt{\sin (2 e+2 f x)} \, dx}{5 a^2 d^5 \sqrt{\sin (2 e+2 f x)}}\\ &=-\frac{2 g (g \cos (e+f x))^{3/2}}{7 a d f (d \sin (e+f x))^{7/2}}+\frac{2 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^2 f (d \sin (e+f x))^{5/2}}-\frac{8 g (g \cos (e+f x))^{3/2}}{21 a d^3 f (d \sin (e+f x))^{3/2}}+\frac{2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{3 a^3 d^3 f (d \sin (e+f x))^{3/2}}+\frac{4 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^4 f \sqrt{d \sin (e+f x)}}-\frac{2 b \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{a^4 d^4 f \sqrt{d \sin (e+f x)}}+\frac{4 b 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)}}{5 a^2 d^5 f \sqrt{\sin (2 e+2 f x)}}+\frac{\left (4 \sqrt{2} b^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 )}{a^4 d^4 f \sqrt{d \sin (e+f x)}}-\frac{\left (2 b \left (a^2-b^2\right ) g^2 \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}\right ) \int \sqrt{\sin (2 e+2 f x)} \, dx}{a^4 d^5 \sqrt{\sin (2 e+2 f x)}}\\ &=-\frac{2 g (g \cos (e+f x))^{3/2}}{7 a d f (d \sin (e+f x))^{7/2}}+\frac{2 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^2 f (d \sin (e+f x))^{5/2}}-\frac{8 g (g \cos (e+f x))^{3/2}}{21 a d^3 f (d \sin (e+f x))^{3/2}}+\frac{2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{3 a^3 d^3 f (d \sin (e+f x))^{3/2}}+\frac{4 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^4 f \sqrt{d \sin (e+f x)}}-\frac{2 b \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{a^4 d^4 f \sqrt{d \sin (e+f x)}}+\frac{4 b 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)}}{5 a^2 d^5 f \sqrt{\sin (2 e+2 f x)}}-\frac{2 b \left (a^2-b^2\right ) 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)}}{a^4 d^5 f \sqrt{\sin (2 e+2 f x)}}+\frac{\left (2 \sqrt{2} b^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 )}{a^4 \sqrt{-a+b} d^4 f \sqrt{d \sin (e+f x)}}-\frac{\left (2 \sqrt{2} b^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 )}{a^4 \sqrt{-a+b} d^4 f \sqrt{d \sin (e+f x)}}\\ &=-\frac{2 g (g \cos (e+f x))^{3/2}}{7 a d f (d \sin (e+f x))^{7/2}}+\frac{2 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^2 f (d \sin (e+f x))^{5/2}}-\frac{8 g (g \cos (e+f x))^{3/2}}{21 a d^3 f (d \sin (e+f x))^{3/2}}+\frac{2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{3 a^3 d^3 f (d \sin (e+f x))^{3/2}}+\frac{4 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^4 f \sqrt{d \sin (e+f x)}}-\frac{2 b \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{a^4 d^4 f \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} b^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)}}{a^4 d^4 f \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} b^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)}}{a^4 d^4 f \sqrt{d \sin (e+f x)}}+\frac{4 b 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)}}{5 a^2 d^5 f \sqrt{\sin (2 e+2 f x)}}-\frac{2 b \left (a^2-b^2\right ) 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)}}{a^4 d^5 f \sqrt{\sin (2 e+2 f x)}}\\ \end{align*}
Mathematica [C] time = 24.0066, size = 1779, normalized size = 2.91 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.438, size = 10704, normalized size = 17.5 \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 )} \left (d \sin \left (f x + e\right )\right )^{\frac{9}{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{5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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