Integrand size = 25, antiderivative size = 68 \[ \int \frac {(a \sin (e+f x))^{7/2}}{\sqrt {b \tan (e+f x)}} \, dx=-\frac {8 a^2 b (a \sin (e+f x))^{3/2}}{21 f (b \tan (e+f x))^{3/2}}-\frac {2 b (a \sin (e+f x))^{7/2}}{7 f (b \tan (e+f x))^{3/2}} \]
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Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2678, 2669} \[ \int \frac {(a \sin (e+f x))^{7/2}}{\sqrt {b \tan (e+f x)}} \, dx=-\frac {8 a^2 b (a \sin (e+f x))^{3/2}}{21 f (b \tan (e+f x))^{3/2}}-\frac {2 b (a \sin (e+f x))^{7/2}}{7 f (b \tan (e+f x))^{3/2}} \]
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Rule 2669
Rule 2678
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b (a \sin (e+f x))^{7/2}}{7 f (b \tan (e+f x))^{3/2}}+\frac {1}{7} \left (4 a^2\right ) \int \frac {(a \sin (e+f x))^{3/2}}{\sqrt {b \tan (e+f x)}} \, dx \\ & = -\frac {8 a^2 b (a \sin (e+f x))^{3/2}}{21 f (b \tan (e+f x))^{3/2}}-\frac {2 b (a \sin (e+f x))^{7/2}}{7 f (b \tan (e+f x))^{3/2}} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \frac {(a \sin (e+f x))^{7/2}}{\sqrt {b \tan (e+f x)}} \, dx=\frac {a^3 \cos (e+f x) (-11+3 \cos (2 (e+f x))) \sqrt {a \sin (e+f x)}}{21 f \sqrt {b \tan (e+f x)}} \]
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Time = 0.85 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {2 a^{3} \sqrt {\sin \left (f x +e \right ) a}\, \left (3 \left (\cos ^{3}\left (f x +e \right )\right )-7 \cos \left (f x +e \right )\right )}{21 f \sqrt {b \tan \left (f x +e \right )}}\) | \(48\) |
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Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.04 \[ \int \frac {(a \sin (e+f x))^{7/2}}{\sqrt {b \tan (e+f x)}} \, dx=\frac {2 \, {\left (3 \, a^{3} \cos \left (f x + e\right )^{4} - 7 \, a^{3} \cos \left (f x + e\right )^{2}\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}}}{21 \, b f \sin \left (f x + e\right )} \]
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Timed out. \[ \int \frac {(a \sin (e+f x))^{7/2}}{\sqrt {b \tan (e+f x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a \sin (e+f x))^{7/2}}{\sqrt {b \tan (e+f x)}} \, dx=\int { \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}}}{\sqrt {b \tan \left (f x + e\right )}} \,d x } \]
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\[ \int \frac {(a \sin (e+f x))^{7/2}}{\sqrt {b \tan (e+f x)}} \, dx=\int { \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}}}{\sqrt {b \tan \left (f x + e\right )}} \,d x } \]
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Time = 5.39 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.29 \[ \int \frac {(a \sin (e+f x))^{7/2}}{\sqrt {b \tan (e+f x)}} \, dx=-\frac {a^3\,\sqrt {a\,\sin \left (e+f\,x\right )}\,\sqrt {-\frac {b\,\sin \left (2\,e+2\,f\,x\right )}{2\,{\sin \left (e+f\,x\right )}^2-2}}\,\left (22\,\sin \left (e+f\,x\right )+19\,\sin \left (3\,e+3\,f\,x\right )-3\,\sin \left (5\,e+5\,f\,x\right )\right )}{168\,b\,f\,{\sin \left (e+f\,x\right )}^2} \]
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