Integrand size = 25, antiderivative size = 168 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=-\frac {(a+b) \cos (e+f x)}{(a-b)^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}+\frac {\cos ^3(e+f x)}{3 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {4 b (a+b) \sec (e+f x)}{3 (a-b)^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {8 b (a+b) \sec (e+f x)}{3 (a-b)^4 f \sqrt {a-b+b \sec ^2(e+f x)}} \]
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Time = 0.19 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3745, 464, 277, 198, 197} \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=-\frac {8 b (a+b) \sec (e+f x)}{3 f (a-b)^4 \sqrt {a+b \sec ^2(e+f x)-b}}-\frac {4 b (a+b) \sec (e+f x)}{3 f (a-b)^3 \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}+\frac {\cos ^3(e+f x)}{3 f (a-b) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}-\frac {(a+b) \cos (e+f x)}{f (a-b)^2 \left (a+b \sec ^2(e+f x)-b\right )^{3/2}} \]
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Rule 197
Rule 198
Rule 277
Rule 464
Rule 3745
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {-1+x^2}{x^4 \left (a-b+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{f} \\ & = \frac {\cos ^3(e+f x)}{3 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}+\frac {(a+b) \text {Subst}\left (\int \frac {1}{x^2 \left (a-b+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{(a-b) f} \\ & = -\frac {(a+b) \cos (e+f x)}{(a-b)^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}+\frac {\cos ^3(e+f x)}{3 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(4 b (a+b)) \text {Subst}\left (\int \frac {1}{\left (a-b+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{(a-b)^2 f} \\ & = -\frac {(a+b) \cos (e+f x)}{(a-b)^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}+\frac {\cos ^3(e+f x)}{3 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {4 b (a+b) \sec (e+f x)}{3 (a-b)^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(8 b (a+b)) \text {Subst}\left (\int \frac {1}{\left (a-b+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{3 (a-b)^3 f} \\ & = -\frac {(a+b) \cos (e+f x)}{(a-b)^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}+\frac {\cos ^3(e+f x)}{3 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {4 b (a+b) \sec (e+f x)}{3 (a-b)^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {8 b (a+b) \sec (e+f x)}{3 (a-b)^4 f \sqrt {a-b+b \sec ^2(e+f x)}} \\ \end{align*}
Time = 6.91 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.22 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\cos (e+f x) \left (26 a^3+186 a^2 b+190 a b^2+110 b^3+3 \left (11 a^3+63 a^2 b-31 a b^2-43 b^3\right ) \cos (2 (e+f x))+6 (a-b)^2 (a+3 b) \cos (4 (e+f x))-a^3 \cos (6 (e+f x))+3 a^2 b \cos (6 (e+f x))-3 a b^2 \cos (6 (e+f x))+b^3 \cos (6 (e+f x))\right ) \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)}}{24 \sqrt {2} (a-b)^4 f (a+b+(a-b) \cos (2 (e+f x)))^2} \]
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Time = 1.33 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.47
method | result | size |
default | \(-\frac {a^{5} \left (\sin \left (f x +e \right )^{2} \cos \left (f x +e \right )^{4} b^{3}+a^{3} \cos \left (f x +e \right )^{6}-3 a^{2} b \cos \left (f x +e \right )^{6}+3 a \,b^{2} \cos \left (f x +e \right )^{6}+4 \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )^{2} b^{3}-3 a^{3} \cos \left (f x +e \right )^{4}+3 a^{2} b \cos \left (f x +e \right )^{4}+3 a \,b^{2} \cos \left (f x +e \right )^{4}-8 \sin \left (f x +e \right )^{2} b^{3}-12 a^{2} b \cos \left (f x +e \right )^{2}-8 a \,b^{2}\right ) \left (a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}\right ) \left (a -b \right ) \sec \left (f x +e \right )^{5}}{3 f \left (\sqrt {-b \left (a -b \right )}+a -b \right )^{5} \left (\sqrt {-b \left (a -b \right )}-a +b \right )^{5} \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}\) | \(247\) |
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Time = 0.45 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.61 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\frac {{\left ({\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{7} - 3 \, {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{5} - 12 \, {\left (a^{2} b - b^{3}\right )} \cos \left (f x + e\right )^{3} - 8 \, {\left (a b^{2} + b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{3 \, {\left ({\left (a^{6} - 6 \, a^{5} b + 15 \, a^{4} b^{2} - 20 \, a^{3} b^{3} + 15 \, a^{2} b^{4} - 6 \, a b^{5} + b^{6}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{5} b - 5 \, a^{4} b^{2} + 10 \, a^{3} b^{3} - 10 \, a^{2} b^{4} + 5 \, a b^{5} - b^{6}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b^{2} - 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}\right )} f\right )}} \]
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Timed out. \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.83 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\frac {3 \, \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {{\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{3} - 9 \, \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} b \cos \left (f x + e\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {9 \, {\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )} b^{2} \cos \left (f x + e\right )^{2} - b^{3}}{{\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} {\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{3}} + \frac {6 \, {\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )} b \cos \left (f x + e\right )^{2} - b^{2}}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} {\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{3}}}{3 \, f} \]
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\[ \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{3}}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^3}{{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
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