Optimal. Leaf size=146 \[ -\frac {64 a^6 \sqrt {a \sin (e+f x)}}{585 b f \sqrt {b \tan (e+f x)}}-\frac {16 a^4 (a \sin (e+f x))^{5/2}}{585 b f \sqrt {b \tan (e+f x)}}-\frac {2 a^2 (a \sin (e+f x))^{9/2}}{117 b f \sqrt {b \tan (e+f x)}}+\frac {2 (a \sin (e+f x))^{13/2}}{13 b f \sqrt {b \tan (e+f x)}} \]
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Rubi [A] time = 0.21, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2596, 2598, 2589} \[ -\frac {2 a^2 (a \sin (e+f x))^{9/2}}{117 b f \sqrt {b \tan (e+f x)}}-\frac {16 a^4 (a \sin (e+f x))^{5/2}}{585 b f \sqrt {b \tan (e+f x)}}-\frac {64 a^6 \sqrt {a \sin (e+f x)}}{585 b f \sqrt {b \tan (e+f x)}}+\frac {2 (a \sin (e+f x))^{13/2}}{13 b f \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2589
Rule 2596
Rule 2598
Rubi steps
\begin {align*} \int \frac {(a \sin (e+f x))^{13/2}}{(b \tan (e+f x))^{3/2}} \, dx &=\frac {2 (a \sin (e+f x))^{13/2}}{13 b f \sqrt {b \tan (e+f x)}}+\frac {a^2 \int (a \sin (e+f x))^{9/2} \sqrt {b \tan (e+f x)} \, dx}{13 b^2}\\ &=-\frac {2 a^2 (a \sin (e+f x))^{9/2}}{117 b f \sqrt {b \tan (e+f x)}}+\frac {2 (a \sin (e+f x))^{13/2}}{13 b f \sqrt {b \tan (e+f x)}}+\frac {\left (8 a^4\right ) \int (a \sin (e+f x))^{5/2} \sqrt {b \tan (e+f x)} \, dx}{117 b^2}\\ &=-\frac {16 a^4 (a \sin (e+f x))^{5/2}}{585 b f \sqrt {b \tan (e+f x)}}-\frac {2 a^2 (a \sin (e+f x))^{9/2}}{117 b f \sqrt {b \tan (e+f x)}}+\frac {2 (a \sin (e+f x))^{13/2}}{13 b f \sqrt {b \tan (e+f x)}}+\frac {\left (32 a^6\right ) \int \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)} \, dx}{585 b^2}\\ &=-\frac {64 a^6 \sqrt {a \sin (e+f x)}}{585 b f \sqrt {b \tan (e+f x)}}-\frac {16 a^4 (a \sin (e+f x))^{5/2}}{585 b f \sqrt {b \tan (e+f x)}}-\frac {2 a^2 (a \sin (e+f x))^{9/2}}{117 b f \sqrt {b \tan (e+f x)}}+\frac {2 (a \sin (e+f x))^{13/2}}{13 b f \sqrt {b \tan (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 67, normalized size = 0.46 \[ \frac {a^6 \cos ^2(e+f x) (340 \cos (2 (e+f x))-45 \cos (4 (e+f x))-551) \sqrt {a \sin (e+f x)}}{2340 b f \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 84, normalized size = 0.58 \[ -\frac {2 \, {\left (45 \, a^{6} \cos \left (f x + e\right )^{7} - 130 \, a^{6} \cos \left (f x + e\right )^{5} + 117 \, a^{6} \cos \left (f x + e\right )^{3}\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}}}{585 \, b^{2} f \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {13}{2}}}{\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.49, size = 70, normalized size = 0.48 \[ -\frac {2 \left (45 \left (\cos ^{4}\left (f x +e \right )\right )-130 \left (\cos ^{2}\left (f x +e \right )\right )+117\right ) \left (a \sin \left (f x +e \right )\right )^{\frac {13}{2}} \cos \left (f x +e \right )}{585 f \left (\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \sin \left (f x +e \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {13}{2}}}{\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.60, size = 296, normalized size = 2.03 \[ \frac {\left (\cos \left (7\,e+7\,f\,x\right )-\sin \left (7\,e+7\,f\,x\right )\,1{}\mathrm {i}\right )\,\sqrt {\frac {b\,\left (\sin \left (2\,e+2\,f\,x\right )-\cos \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}+1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}}}\,\left (\frac {a^6\,\cos \left (3\,e+3\,f\,x\right )\,\sqrt {a\,\sin \left (e+f\,x\right )}\,\left (\cos \left (7\,e+7\,f\,x\right )+\sin \left (7\,e+7\,f\,x\right )\,1{}\mathrm {i}\right )\,217{}\mathrm {i}}{9360\,b^2\,f}-\frac {a^6\,\cos \left (5\,e+5\,f\,x\right )\,\sqrt {a\,\sin \left (e+f\,x\right )}\,\left (\cos \left (7\,e+7\,f\,x\right )+\sin \left (7\,e+7\,f\,x\right )\,1{}\mathrm {i}\right )\,41{}\mathrm {i}}{1872\,b^2\,f}+\frac {a^6\,\cos \left (7\,e+7\,f\,x\right )\,\sqrt {a\,\sin \left (e+f\,x\right )}\,\left (\cos \left (7\,e+7\,f\,x\right )+\sin \left (7\,e+7\,f\,x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{208\,b^2\,f}+\frac {a^6\,\cos \left (e+f\,x\right )\,\sqrt {a\,\sin \left (e+f\,x\right )}\,\left (\cos \left (7\,e+7\,f\,x\right )+\sin \left (7\,e+7\,f\,x\right )\,1{}\mathrm {i}\right )\,1991{}\mathrm {i}}{9360\,b^2\,f}\right )\,1{}\mathrm {i}}{2\,\sin \left (e+f\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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