Integrand size = 23, antiderivative size = 154 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {5 \sqrt {b} (3 a+7 b) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{8 a^{9/2} f}-\frac {(a+3 b) \cos (e+f x)}{a^4 f}+\frac {\cos ^3(e+f x)}{3 a^3 f}+\frac {b^2 (a+b) \cos (e+f x)}{4 a^4 f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {b (9 a+13 b) \cos (e+f x)}{8 a^4 f \left (b+a \cos ^2(e+f x)\right )} \]
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Time = 0.22 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4218, 466, 1828, 1167, 211} \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {5 \sqrt {b} (3 a+7 b) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{8 a^{9/2} f}+\frac {b^2 (a+b) \cos (e+f x)}{4 a^4 f \left (a \cos ^2(e+f x)+b\right )^2}-\frac {b (9 a+13 b) \cos (e+f x)}{8 a^4 f \left (a \cos ^2(e+f x)+b\right )}-\frac {(a+3 b) \cos (e+f x)}{a^4 f}+\frac {\cos ^3(e+f x)}{3 a^3 f} \]
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Rule 211
Rule 466
Rule 1167
Rule 1828
Rule 4218
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x^6 \left (1-x^2\right )}{\left (b+a x^2\right )^3} \, dx,x,\cos (e+f x)\right )}{f} \\ & = \frac {b^2 (a+b) \cos (e+f x)}{4 a^4 f \left (b+a \cos ^2(e+f x)\right )^2}+\frac {\text {Subst}\left (\int \frac {-b^2 (a+b)+4 a b (a+b) x^2-4 a^2 (a+b) x^4+4 a^3 x^6}{\left (b+a x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{4 a^4 f} \\ & = \frac {b^2 (a+b) \cos (e+f x)}{4 a^4 f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {b (9 a+13 b) \cos (e+f x)}{8 a^4 f \left (b+a \cos ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {-b^2 (7 a+11 b)+8 a b (a+2 b) x^2-8 a^2 b x^4}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{8 a^4 b f} \\ & = \frac {b^2 (a+b) \cos (e+f x)}{4 a^4 f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {b (9 a+13 b) \cos (e+f x)}{8 a^4 f \left (b+a \cos ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \left (8 b (a+3 b)-8 a b x^2-\frac {5 \left (3 a b^2+7 b^3\right )}{b+a x^2}\right ) \, dx,x,\cos (e+f x)\right )}{8 a^4 b f} \\ & = -\frac {(a+3 b) \cos (e+f x)}{a^4 f}+\frac {\cos ^3(e+f x)}{3 a^3 f}+\frac {b^2 (a+b) \cos (e+f x)}{4 a^4 f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {b (9 a+13 b) \cos (e+f x)}{8 a^4 f \left (b+a \cos ^2(e+f x)\right )}+\frac {(5 b (3 a+7 b)) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{8 a^4 f} \\ & = \frac {5 \sqrt {b} (3 a+7 b) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{8 a^{9/2} f}-\frac {(a+3 b) \cos (e+f x)}{a^4 f}+\frac {\cos ^3(e+f x)}{3 a^3 f}+\frac {b^2 (a+b) \cos (e+f x)}{4 a^4 f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {b (9 a+13 b) \cos (e+f x)}{8 a^4 f \left (b+a \cos ^2(e+f x)\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.74 (sec) , antiderivative size = 1153, normalized size of antiderivative = 7.49 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {(a+2 b+a \cos (2 (e+f x)))^3 \sec ^6(e+f x) \left (3 \left (9 a^4+1920 a b^3+4480 b^4\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 )-\frac {-27 a^{11/2} \sqrt {b} \cos (e+f x)+162 a^{9/2} b^{3/2} \cos (e+f x)+10816 a^{7/2} b^{5/2} \cos (e+f x)+51552 a^{5/2} b^{7/2} \cos (e+f x)+87424 a^{3/2} b^{9/2} \cos (e+f x)+53760 \sqrt {a} b^{11/2} \cos (e+f x)-27 a^{11/2} \sqrt {b} \cos (e+f x) \cos (2 (e+f x))+47936 a^{5/2} b^{7/2} \cos (e+f x) \cos (2 (e+f x))+44800 a^{3/2} b^{9/2} \cos (e+f x) \cos (2 (e+f x))+27 a^{9/2} \sqrt {b} \cos (e+f x) (a+2 b+a \cos (2 (e+f x)))-216 a^{7/2} b^{3/2} \cos (e+f x) (a+2 b+a \cos (2 (e+f x)))-3600 a^{5/2} b^{5/2} \cos (e+f x) (a+2 b+a \cos (2 (e+f x)))-5184 a^{3/2} b^{7/2} \cos (e+f x) (a+2 b+a \cos (2 (e+f x)))-27 a^4 \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 ) (a+2 b+a \cos (2 (e+f x)))^2-5760 a b^3 \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 ) (a+2 b+a \cos (2 (e+f x)))^2-13440 b^4 \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 ) (a+2 b+a \cos (2 (e+f x)))^2+27 a^4 \arctan \left (\frac {\sqrt {a}-\sqrt {a+b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {b}}\right ) (a+2 b+a \cos (2 (e+f x)))^2+27 a^4 \arctan \left (\frac {\sqrt {a}+\sqrt {a+b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {b}}\right ) (a+2 b+a \cos (2 (e+f x)))^2-2304 a^{3/2} b^{5/2} \cos (e) \cos (f x) (a+2 b+a \cos (2 (e+f x)))^2+1920 a^{7/2} b^{5/2} \cos (e+f x) \cos (4 (e+f x))+3584 a^{5/2} b^{7/2} \cos (e+f x) \cos (4 (e+f x))-128 a^{7/2} b^{5/2} \cos (e+f x) \cos (6 (e+f x))+2304 a^{3/2} b^{5/2} (a+2 b+a \cos (2 (e+f x)))^2 \sin (e) \sin (f x)+54 a^{9/2} b^{3/2} \csc (e+f x) \sin (4 (e+f x))+3108 a^{7/2} b^{5/2} \csc (e+f x) \sin (4 (e+f x))}{(a+2 b+a \cos (2 (e+f x)))^2}\right )}{24576 a^{9/2} b^{5/2} f \left (a+b \sec ^2(e+f x)\right )^3} \]
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Time = 5.74 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {\frac {\frac {a \cos \left (f x +e \right )^{3}}{3}-\cos \left (f x +e \right ) a -3 b \cos \left (f x +e \right )}{a^{4}}+\frac {b \left (\frac {\left (-\frac {9}{8} a^{2}-\frac {13}{8} a b \right ) \cos \left (f x +e \right )^{3}-\frac {b \left (7 a +11 b \right ) \cos \left (f x +e \right )}{8}}{\left (b +a \cos \left (f x +e \right )^{2}\right )^{2}}+\frac {5 \left (3 a +7 b \right ) \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}}{f}\) | \(125\) |
default | \(\frac {\frac {\frac {a \cos \left (f x +e \right )^{3}}{3}-\cos \left (f x +e \right ) a -3 b \cos \left (f x +e \right )}{a^{4}}+\frac {b \left (\frac {\left (-\frac {9}{8} a^{2}-\frac {13}{8} a b \right ) \cos \left (f x +e \right )^{3}-\frac {b \left (7 a +11 b \right ) \cos \left (f x +e \right )}{8}}{\left (b +a \cos \left (f x +e \right )^{2}\right )^{2}}+\frac {5 \left (3 a +7 b \right ) \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}}{f}\) | \(125\) |
risch | \(\frac {{\mathrm e}^{3 i \left (f x +e \right )}}{24 a^{3} f}-\frac {3 \,{\mathrm e}^{i \left (f x +e \right )}}{8 a^{3} f}-\frac {3 \,{\mathrm e}^{i \left (f x +e \right )} b}{2 a^{4} f}-\frac {3 \,{\mathrm e}^{-i \left (f x +e \right )}}{8 a^{3} f}-\frac {3 \,{\mathrm e}^{-i \left (f x +e \right )} b}{2 a^{4} f}+\frac {{\mathrm e}^{-3 i \left (f x +e \right )}}{24 a^{3} f}-\frac {b \left (9 a^{2} {\mathrm e}^{7 i \left (f x +e \right )}+13 a b \,{\mathrm e}^{7 i \left (f x +e \right )}+27 a^{2} {\mathrm e}^{5 i \left (f x +e \right )}+67 a b \,{\mathrm e}^{5 i \left (f x +e \right )}+44 b^{2} {\mathrm e}^{5 i \left (f x +e \right )}+27 a^{2} {\mathrm e}^{3 i \left (f x +e \right )}+67 a b \,{\mathrm e}^{3 i \left (f x +e \right )}+44 b^{2} {\mathrm e}^{3 i \left (f x +e \right )}+9 a^{2} {\mathrm e}^{i \left (f x +e \right )}+13 a b \,{\mathrm e}^{i \left (f x +e \right )}\right )}{4 a^{4} f \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )^{2}}+\frac {15 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right )}{16 a^{4} f}+\frac {35 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right ) b}{16 a^{5} f}-\frac {15 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right )}{16 a^{4} f}-\frac {35 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right ) b}{16 a^{5} f}\) | \(477\) |
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Time = 0.30 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.85 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\left [\frac {16 \, a^{3} \cos \left (f x + e\right )^{7} - 16 \, {\left (3 \, a^{3} + 7 \, a^{2} b\right )} \cos \left (f x + e\right )^{5} - 50 \, {\left (3 \, a^{2} b + 7 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + 15 \, {\left ({\left (3 \, a^{3} + 7 \, a^{2} b\right )} \cos \left (f x + e\right )^{4} + 3 \, a b^{2} + 7 \, b^{3} + 2 \, {\left (3 \, a^{2} b + 7 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (-\frac {a \cos \left (f x + e\right )^{2} + 2 \, a \sqrt {-\frac {b}{a}} \cos \left (f x + e\right ) - b}{a \cos \left (f x + e\right )^{2} + b}\right ) - 30 \, {\left (3 \, a b^{2} + 7 \, b^{3}\right )} \cos \left (f x + e\right )}{48 \, {\left (a^{6} f \cos \left (f x + e\right )^{4} + 2 \, a^{5} b f \cos \left (f x + e\right )^{2} + a^{4} b^{2} f\right )}}, \frac {8 \, a^{3} \cos \left (f x + e\right )^{7} - 8 \, {\left (3 \, a^{3} + 7 \, a^{2} b\right )} \cos \left (f x + e\right )^{5} - 25 \, {\left (3 \, a^{2} b + 7 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + 15 \, {\left ({\left (3 \, a^{3} + 7 \, a^{2} b\right )} \cos \left (f x + e\right )^{4} + 3 \, a b^{2} + 7 \, b^{3} + 2 \, {\left (3 \, a^{2} b + 7 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}} \cos \left (f x + e\right )}{b}\right ) - 15 \, {\left (3 \, a b^{2} + 7 \, b^{3}\right )} \cos \left (f x + e\right )}{24 \, {\left (a^{6} f \cos \left (f x + e\right )^{4} + 2 \, a^{5} b f \cos \left (f x + e\right )^{2} + a^{4} b^{2} f\right )}}\right ] \]
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Timed out. \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Timed out} \]
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Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.97 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=-\frac {\frac {3 \, {\left ({\left (9 \, a^{2} b + 13 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (7 \, a b^{2} + 11 \, b^{3}\right )} \cos \left (f x + e\right )\right )}}{a^{6} \cos \left (f x + e\right )^{4} + 2 \, a^{5} b \cos \left (f x + e\right )^{2} + a^{4} b^{2}} - \frac {15 \, {\left (3 \, a b + 7 \, b^{2}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} - \frac {8 \, {\left (a \cos \left (f x + e\right )^{3} - 3 \, {\left (a + 3 \, b\right )} \cos \left (f x + e\right )\right )}}{a^{4}}}{24 \, f} \]
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Time = 0.42 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.13 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {5 \, {\left (3 \, a b + 7 \, b^{2}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4} f} - \frac {\frac {9 \, a^{2} b \cos \left (f x + e\right )^{3}}{f} + \frac {13 \, a b^{2} \cos \left (f x + e\right )^{3}}{f} + \frac {7 \, a b^{2} \cos \left (f x + e\right )}{f} + \frac {11 \, b^{3} \cos \left (f x + e\right )}{f}}{8 \, {\left (a \cos \left (f x + e\right )^{2} + b\right )}^{2} a^{4}} + \frac {a^{6} f^{17} \cos \left (f x + e\right )^{3} - 3 \, a^{6} f^{17} \cos \left (f x + e\right ) - 9 \, a^{5} b f^{17} \cos \left (f x + e\right )}{3 \, a^{9} f^{18}} \]
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Time = 0.18 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.12 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {{\cos \left (e+f\,x\right )}^3}{3\,a^3\,f}-\frac {\left (\frac {9\,a^2\,b}{8}+\frac {13\,a\,b^2}{8}\right )\,{\cos \left (e+f\,x\right )}^3+\left (\frac {11\,b^3}{8}+\frac {7\,a\,b^2}{8}\right )\,\cos \left (e+f\,x\right )}{f\,\left (a^6\,{\cos \left (e+f\,x\right )}^4+2\,a^5\,b\,{\cos \left (e+f\,x\right )}^2+a^4\,b^2\right )}-\frac {\cos \left (e+f\,x\right )\,\left (\frac {3\,b}{a^4}+\frac {1}{a^3}\right )}{f}+\frac {5\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,\cos \left (e+f\,x\right )\,\left (3\,a+7\,b\right )}{7\,b^2+3\,a\,b}\right )\,\left (3\,a+7\,b\right )}{8\,a^{9/2}\,f} \]
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