Integrand size = 25, antiderivative size = 88 \[ \int \frac {\sin ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=-\frac {(3 a-b) \cos (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{3 (a-b)^2 f}+\frac {\cos ^3(e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{3 (a-b) f} \]
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Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3745, 464, 270} \[ \int \frac {\sin ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\frac {\cos ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{3 f (a-b)}-\frac {(3 a-b) \cos (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{3 f (a-b)^2} \]
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Rule 270
Rule 464
Rule 3745
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {-1+x^2}{x^4 \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{f} \\ & = \frac {\cos ^3(e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{3 (a-b) f}+\frac {(3 a-b) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{3 (a-b) f} \\ & = -\frac {(3 a-b) \cos (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{3 (a-b)^2 f}+\frac {\cos ^3(e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{3 (a-b) f} \\ \end{align*}
Time = 2.33 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.84 \[ \int \frac {\sin ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\frac {\cos (e+f x) (-5 a+b+(a-b) \cos (2 (e+f x))) \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)}}{6 \sqrt {2} (a-b)^2 f} \]
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Time = 0.84 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.89
method | result | size |
default | \(\frac {\left (a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}\right ) \left (b \sin \left (f x +e \right )^{2}+a \cos \left (f x +e \right )^{2}-3 a \right ) \sec \left (f x +e \right )}{3 f \left (a -b \right )^{2} \sqrt {a +b \tan \left (f x +e \right )^{2}}}\) | \(78\) |
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Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.85 \[ \int \frac {\sin ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\frac {{\left ({\left (a - b\right )} \cos \left (f x + e\right )^{3} - {\left (3 \, a - b\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{3 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} f} \]
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Timed out. \[ \int \frac {\sin ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.20 \[ \int \frac {\sin ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=-\frac {\frac {3 \, \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a - b} - \frac {{\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{3} - 3 \, \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} b \cos \left (f x + e\right )}{a^{2} - 2 \, a b + b^{2}}}{3 \, f} \]
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\[ \int \frac {\sin ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{3}}{\sqrt {b \tan \left (f x + e\right )^{2} + a}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^3}{\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}} \,d x \]
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