Integrand size = 23, antiderivative size = 114 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {\sqrt {b} (3 a+5 b) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{2 a^{7/2} f}-\frac {(a+2 b) \cos (e+f x)}{a^3 f}+\frac {\cos ^3(e+f x)}{3 a^2 f}-\frac {b (a+b) \cos (e+f x)}{2 a^3 f \left (b+a \cos ^2(e+f x)\right )} \]
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Time = 0.13 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4218, 466, 1167, 211} \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {\sqrt {b} (3 a+5 b) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{2 a^{7/2} f}-\frac {b (a+b) \cos (e+f x)}{2 a^3 f \left (a \cos ^2(e+f x)+b\right )}-\frac {(a+2 b) \cos (e+f x)}{a^3 f}+\frac {\cos ^3(e+f x)}{3 a^2 f} \]
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Rule 211
Rule 466
Rule 1167
Rule 4218
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x^4 \left (1-x^2\right )}{\left (b+a x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{f} \\ & = -\frac {b (a+b) \cos (e+f x)}{2 a^3 f \left (b+a \cos ^2(e+f x)\right )}+\frac {\text {Subst}\left (\int \frac {b (a+b)-2 a (a+b) x^2+2 a^2 x^4}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{2 a^3 f} \\ & = -\frac {b (a+b) \cos (e+f x)}{2 a^3 f \left (b+a \cos ^2(e+f x)\right )}+\frac {\text {Subst}\left (\int \left (-2 (a+2 b)+2 a x^2+\frac {3 a b+5 b^2}{b+a x^2}\right ) \, dx,x,\cos (e+f x)\right )}{2 a^3 f} \\ & = -\frac {(a+2 b) \cos (e+f x)}{a^3 f}+\frac {\cos ^3(e+f x)}{3 a^2 f}-\frac {b (a+b) \cos (e+f x)}{2 a^3 f \left (b+a \cos ^2(e+f x)\right )}+\frac {(b (3 a+5 b)) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{2 a^3 f} \\ & = \frac {\sqrt {b} (3 a+5 b) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{2 a^{7/2} f}-\frac {(a+2 b) \cos (e+f x)}{a^3 f}+\frac {\cos ^3(e+f x)}{3 a^2 f}-\frac {b (a+b) \cos (e+f x)}{2 a^3 f \left (b+a \cos ^2(e+f x)\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.91 (sec) , antiderivative size = 403, normalized size of antiderivative = 3.54 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {\frac {3 \left (3 a^3+192 a b^2+320 b^3\right ) \arctan \left (\frac {\left (-\sqrt {a}-i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right ) \sin (e) \tan \left (\frac {f x}{2}\right )+\cos (e) \left (\sqrt {a}-\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right )}{b^{3/2}}+\frac {3 \left (3 a^3+192 a b^2+320 b^3\right ) \arctan \left (\frac {\left (-\sqrt {a}+i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right ) \sin (e) \tan \left (\frac {f x}{2}\right )+\cos (e) \left (\sqrt {a}+\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right )}{b^{3/2}}-\frac {9 a^3 \arctan \left (\frac {\sqrt {a}-\sqrt {a+b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {b}}\right )}{b^{3/2}}-\frac {9 a^3 \arctan \left (\frac {\sqrt {a}+\sqrt {a+b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {b}}\right )}{b^{3/2}}-\frac {32 \sqrt {a} \cos (e+f x) \left (9 a^2+56 a b+60 b^2+4 a (2 a+5 b) \cos (2 (e+f x))-a^2 \cos (4 (e+f x))\right )}{a+2 b+a \cos (2 (e+f x))}}{384 a^{7/2} f} \]
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Time = 2.46 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\frac {\frac {a \cos \left (f x +e \right )^{3}}{3}-\cos \left (f x +e \right ) a -2 b \cos \left (f x +e \right )}{a^{3}}+\frac {b \left (\frac {\left (-\frac {a}{2}-\frac {b}{2}\right ) \cos \left (f x +e \right )}{b +a \cos \left (f x +e \right )^{2}}+\frac {\left (3 a +5 b \right ) \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3}}}{f}\) | \(102\) |
default | \(\frac {\frac {\frac {a \cos \left (f x +e \right )^{3}}{3}-\cos \left (f x +e \right ) a -2 b \cos \left (f x +e \right )}{a^{3}}+\frac {b \left (\frac {\left (-\frac {a}{2}-\frac {b}{2}\right ) \cos \left (f x +e \right )}{b +a \cos \left (f x +e \right )^{2}}+\frac {\left (3 a +5 b \right ) \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3}}}{f}\) | \(102\) |
risch | \(\frac {{\mathrm e}^{3 i \left (f x +e \right )}}{24 a^{2} f}-\frac {3 \,{\mathrm e}^{i \left (f x +e \right )}}{8 f \,a^{2}}-\frac {{\mathrm e}^{i \left (f x +e \right )} b}{f \,a^{3}}-\frac {3 \,{\mathrm e}^{-i \left (f x +e \right )}}{8 f \,a^{2}}-\frac {{\mathrm e}^{-i \left (f x +e \right )} b}{f \,a^{3}}+\frac {{\mathrm e}^{-3 i \left (f x +e \right )}}{24 a^{2} f}-\frac {\left (a +b \right ) b \left ({\mathrm e}^{3 i \left (f x +e \right )}+{\mathrm e}^{i \left (f x +e \right )}\right )}{a^{3} f \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )}+\frac {3 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right )}{4 a^{3} f}+\frac {5 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right ) b}{4 a^{4} f}-\frac {3 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right )}{4 a^{3} f}-\frac {5 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right ) b}{4 a^{4} f}\) | \(362\) |
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Time = 0.28 (sec) , antiderivative size = 297, normalized size of antiderivative = 2.61 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\left [\frac {4 \, a^{2} \cos \left (f x + e\right )^{5} - 4 \, {\left (3 \, a^{2} + 5 \, a b\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left ({\left (3 \, a^{2} + 5 \, a b\right )} \cos \left (f x + e\right )^{2} + 3 \, a b + 5 \, b^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (-\frac {a \cos \left (f x + e\right )^{2} + 2 \, a \sqrt {-\frac {b}{a}} \cos \left (f x + e\right ) - b}{a \cos \left (f x + e\right )^{2} + b}\right ) - 6 \, {\left (3 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )}{12 \, {\left (a^{4} f \cos \left (f x + e\right )^{2} + a^{3} b f\right )}}, \frac {2 \, a^{2} \cos \left (f x + e\right )^{5} - 2 \, {\left (3 \, a^{2} + 5 \, a b\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left ({\left (3 \, a^{2} + 5 \, a b\right )} \cos \left (f x + e\right )^{2} + 3 \, a b + 5 \, b^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}} \cos \left (f x + e\right )}{b}\right ) - 3 \, {\left (3 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )}{6 \, {\left (a^{4} f \cos \left (f x + e\right )^{2} + a^{3} b f\right )}}\right ] \]
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Timed out. \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\text {Timed out} \]
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Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.91 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=-\frac {\frac {3 \, {\left (a b + b^{2}\right )} \cos \left (f x + e\right )}{a^{4} \cos \left (f x + e\right )^{2} + a^{3} b} - \frac {3 \, {\left (3 \, a b + 5 \, b^{2}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {2 \, {\left (a \cos \left (f x + e\right )^{3} - 3 \, {\left (a + 2 \, b\right )} \cos \left (f x + e\right )\right )}}{a^{3}}}{6 \, f} \]
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Time = 0.34 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.19 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {{\left (3 \, a b + 5 \, b^{2}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3} f} - \frac {\frac {a b \cos \left (f x + e\right )}{f} + \frac {b^{2} \cos \left (f x + e\right )}{f}}{2 \, {\left (a \cos \left (f x + e\right )^{2} + b\right )} a^{3}} + \frac {a^{4} f^{11} \cos \left (f x + e\right )^{3} - 3 \, a^{4} f^{11} \cos \left (f x + e\right ) - 6 \, a^{3} b f^{11} \cos \left (f x + e\right )}{3 \, a^{6} f^{12}} \]
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Time = 0.14 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.14 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {{\cos \left (e+f\,x\right )}^3}{3\,a^2\,f}-\frac {\cos \left (e+f\,x\right )\,\left (\frac {2\,b}{a^3}+\frac {1}{a^2}\right )}{f}-\frac {\cos \left (e+f\,x\right )\,\left (\frac {b^2}{2}+\frac {a\,b}{2}\right )}{f\,\left (a^4\,{\cos \left (e+f\,x\right )}^2+b\,a^3\right )}+\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,\cos \left (e+f\,x\right )\,\left (3\,a+5\,b\right )}{5\,b^2+3\,a\,b}\right )\,\left (3\,a+5\,b\right )}{2\,a^{7/2}\,f} \]
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