Integrand size = 23, antiderivative size = 238 \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {3 \left (a^2+12 a b+16 b^2\right ) x}{8 a^5}-\frac {3 \sqrt {b} \left (5 a^2+20 a b+16 b^2\right ) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^5 \sqrt {a+b} f}-\frac {(5 a+8 b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {b (7 a+12 b) \tan (e+f x)}{8 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {3 b (a+2 b) \tan (e+f x)}{2 a^4 f \left (a+b+b \tan ^2(e+f x)\right )} \]
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Time = 0.39 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4217, 481, 541, 536, 209, 211} \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=-\frac {3 b (a+2 b) \tan (e+f x)}{2 a^4 f \left (a+b \tan ^2(e+f x)+b\right )}-\frac {b (7 a+12 b) \tan (e+f x)}{8 a^3 f \left (a+b \tan ^2(e+f x)+b\right )^2}-\frac {(5 a+8 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)+b\right )^2}-\frac {3 \sqrt {b} \left (5 a^2+20 a b+16 b^2\right ) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^5 f \sqrt {a+b}}+\frac {3 x \left (a^2+12 a b+16 b^2\right )}{8 a^5}+\frac {\sin (e+f x) \cos ^3(e+f x)}{4 a f \left (a+b \tan ^2(e+f x)+b\right )^2} \]
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Rule 209
Rule 211
Rule 481
Rule 536
Rule 541
Rule 4217
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^3 \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\cos ^3(e+f x) \sin (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {a+b+(-4 a-7 b) x^2}{\left (1+x^2\right )^2 \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{4 a f} \\ & = -\frac {(5 a+8 b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\text {Subst}\left (\int \frac {(a+b) (3 a+8 b)-5 b (5 a+8 b) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{8 a^2 f} \\ & = -\frac {(5 a+8 b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {b (7 a+12 b) \tan (e+f x)}{8 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\text {Subst}\left (\int \frac {12 (a+b)^2 (a+4 b)-12 b (a+b) (7 a+12 b) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{32 a^3 (a+b) f} \\ & = -\frac {(5 a+8 b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {b (7 a+12 b) \tan (e+f x)}{8 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {3 b (a+2 b) \tan (e+f x)}{2 a^4 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\text {Subst}\left (\int \frac {24 (a+b)^2 \left (a^2+8 a b+8 b^2\right )-96 b (a+b)^2 (a+2 b) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{64 a^4 (a+b)^2 f} \\ & = -\frac {(5 a+8 b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {b (7 a+12 b) \tan (e+f x)}{8 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {3 b (a+2 b) \tan (e+f x)}{2 a^4 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\left (3 \left (a^2+12 a b+16 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{8 a^5 f}-\frac {\left (3 b \left (5 a^2+20 a b+16 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{8 a^5 f} \\ & = \frac {3 \left (a^2+12 a b+16 b^2\right ) x}{8 a^5}-\frac {3 \sqrt {b} \left (5 a^2+20 a b+16 b^2\right ) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^5 \sqrt {a+b} f}-\frac {(5 a+8 b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {b (7 a+12 b) \tan (e+f x)}{8 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {3 b (a+2 b) \tan (e+f x)}{2 a^4 f \left (a+b+b \tan ^2(e+f x)\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 21.95 (sec) , antiderivative size = 2469, normalized size of antiderivative = 10.37 \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Result too large to show} \]
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Time = 6.37 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (-\frac {3}{2} a b -\frac {5}{8} a^{2}\right ) \tan \left (f x +e \right )^{3}+\left (-\frac {3}{8} a^{2}-\frac {3}{2} a b \right ) \tan \left (f x +e \right )}{\left (1+\tan \left (f x +e \right )^{2}\right )^{2}}+\frac {3 \left (a^{2}+12 a b +16 b^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{8}}{a^{5}}-\frac {b \left (\frac {\left (\frac {7}{8} a^{2} b +\frac {3}{2} a \,b^{2}\right ) \tan \left (f x +e \right )^{3}+\frac {3 a \left (3 a^{2}+7 a b +4 b^{2}\right ) \tan \left (f x +e \right )}{8}}{\left (a +b +b \tan \left (f x +e \right )^{2}\right )^{2}}+\frac {3 \left (5 a^{2}+20 a b +16 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{8 \sqrt {\left (a +b \right ) b}}\right )}{a^{5}}}{f}\) | \(193\) |
default | \(\frac {\frac {\frac {\left (-\frac {3}{2} a b -\frac {5}{8} a^{2}\right ) \tan \left (f x +e \right )^{3}+\left (-\frac {3}{8} a^{2}-\frac {3}{2} a b \right ) \tan \left (f x +e \right )}{\left (1+\tan \left (f x +e \right )^{2}\right )^{2}}+\frac {3 \left (a^{2}+12 a b +16 b^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{8}}{a^{5}}-\frac {b \left (\frac {\left (\frac {7}{8} a^{2} b +\frac {3}{2} a \,b^{2}\right ) \tan \left (f x +e \right )^{3}+\frac {3 a \left (3 a^{2}+7 a b +4 b^{2}\right ) \tan \left (f x +e \right )}{8}}{\left (a +b +b \tan \left (f x +e \right )^{2}\right )^{2}}+\frac {3 \left (5 a^{2}+20 a b +16 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{8 \sqrt {\left (a +b \right ) b}}\right )}{a^{5}}}{f}\) | \(193\) |
risch | \(\frac {3 x}{8 a^{3}}+\frac {9 x b}{2 a^{4}}+\frac {6 x \,b^{2}}{a^{5}}-\frac {i b \left (9 a^{3} {\mathrm e}^{6 i \left (f x +e \right )}+36 a^{2} b \,{\mathrm e}^{6 i \left (f x +e \right )}+32 a \,b^{2} {\mathrm e}^{6 i \left (f x +e \right )}+27 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}+114 a^{2} b \,{\mathrm e}^{4 i \left (f x +e \right )}+184 a \,b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+112 b^{3} {\mathrm e}^{4 i \left (f x +e \right )}+27 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}+92 a^{2} b \,{\mathrm e}^{2 i \left (f x +e \right )}+80 a \,b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+9 a^{3}+14 a^{2} b \right )}{4 a^{5} 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 {3 i {\mathrm e}^{-2 i \left (f x +e \right )} b}{8 a^{4} f}-\frac {i {\mathrm e}^{4 i \left (f x +e \right )}}{64 a^{3} f}+\frac {i {\mathrm e}^{2 i \left (f x +e \right )}}{8 a^{3} f}-\frac {i {\mathrm e}^{-2 i \left (f x +e \right )}}{8 a^{3} f}+\frac {3 i {\mathrm e}^{2 i \left (f x +e \right )} b}{8 a^{4} f}+\frac {i {\mathrm e}^{-4 i \left (f x +e \right )}}{64 a^{3} f}+\frac {15 \sqrt {-\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-\left (a +b \right ) b}+a +2 b}{a}\right )}{16 \left (a +b \right ) f \,a^{3}}+\frac {15 \sqrt {-\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-\left (a +b \right ) b}+a +2 b}{a}\right ) b}{4 \left (a +b \right ) f \,a^{4}}+\frac {3 \sqrt {-\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-\left (a +b \right ) b}+a +2 b}{a}\right ) b^{2}}{\left (a +b \right ) f \,a^{5}}-\frac {15 \sqrt {-\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-\left (a +b \right ) b}-a -2 b}{a}\right )}{16 \left (a +b \right ) f \,a^{3}}-\frac {15 \sqrt {-\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-\left (a +b \right ) b}-a -2 b}{a}\right ) b}{4 \left (a +b \right ) f \,a^{4}}-\frac {3 \sqrt {-\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-\left (a +b \right ) b}-a -2 b}{a}\right ) b^{2}}{\left (a +b \right ) f \,a^{5}}\) | \(670\) |
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Time = 0.34 (sec) , antiderivative size = 803, normalized size of antiderivative = 3.37 \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\left [\frac {12 \, {\left (a^{4} + 12 \, a^{3} b + 16 \, a^{2} b^{2}\right )} f x \cos \left (f x + e\right )^{4} + 24 \, {\left (a^{3} b + 12 \, a^{2} b^{2} + 16 \, a b^{3}\right )} f x \cos \left (f x + e\right )^{2} + 12 \, {\left (a^{2} b^{2} + 12 \, a b^{3} + 16 \, b^{4}\right )} f x + 3 \, {\left ({\left (5 \, a^{4} + 20 \, a^{3} b + 16 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 5 \, a^{2} b^{2} + 20 \, a b^{3} + 16 \, b^{4} + 2 \, {\left (5 \, a^{3} b + 20 \, a^{2} b^{2} + 16 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-\frac {b}{a + b}} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (a b + b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {b}{a + b}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right ) + 4 \, {\left (2 \, a^{4} \cos \left (f x + e\right )^{7} - {\left (5 \, a^{4} + 8 \, a^{3} b\right )} \cos \left (f x + e\right )^{5} - {\left (19 \, a^{3} b + 36 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{3} - 12 \, {\left (a^{2} b^{2} + 2 \, a b^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{32 \, {\left (a^{7} f \cos \left (f x + e\right )^{4} + 2 \, a^{6} b f \cos \left (f x + e\right )^{2} + a^{5} b^{2} f\right )}}, \frac {6 \, {\left (a^{4} + 12 \, a^{3} b + 16 \, a^{2} b^{2}\right )} f x \cos \left (f x + e\right )^{4} + 12 \, {\left (a^{3} b + 12 \, a^{2} b^{2} + 16 \, a b^{3}\right )} f x \cos \left (f x + e\right )^{2} + 6 \, {\left (a^{2} b^{2} + 12 \, a b^{3} + 16 \, b^{4}\right )} f x + 3 \, {\left ({\left (5 \, a^{4} + 20 \, a^{3} b + 16 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 5 \, a^{2} b^{2} + 20 \, a b^{3} + 16 \, b^{4} + 2 \, {\left (5 \, a^{3} b + 20 \, a^{2} b^{2} + 16 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{a + b}} \arctan \left (\frac {{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {b}{a + b}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) + 2 \, {\left (2 \, a^{4} \cos \left (f x + e\right )^{7} - {\left (5 \, a^{4} + 8 \, a^{3} b\right )} \cos \left (f x + e\right )^{5} - {\left (19 \, a^{3} b + 36 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{3} - 12 \, {\left (a^{2} b^{2} + 2 \, a b^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{16 \, {\left (a^{7} f \cos \left (f x + e\right )^{4} + 2 \, a^{6} b f \cos \left (f x + e\right )^{2} + a^{5} b^{2} f\right )}}\right ] \]
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Timed out. \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.26 \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=-\frac {\frac {12 \, {\left (a b^{2} + 2 \, b^{3}\right )} \tan \left (f x + e\right )^{7} + {\left (19 \, a^{2} b + 72 \, a b^{2} + 72 \, b^{3}\right )} \tan \left (f x + e\right )^{5} + {\left (5 \, a^{3} + 46 \, a^{2} b + 108 \, a b^{2} + 72 \, b^{3}\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (a^{3} + 9 \, a^{2} b + 16 \, a b^{2} + 8 \, b^{3}\right )} \tan \left (f x + e\right )}{a^{4} b^{2} \tan \left (f x + e\right )^{8} + 2 \, {\left (a^{5} b + 2 \, a^{4} b^{2}\right )} \tan \left (f x + e\right )^{6} + a^{6} + 2 \, a^{5} b + a^{4} b^{2} + {\left (a^{6} + 6 \, a^{5} b + 6 \, a^{4} b^{2}\right )} \tan \left (f x + e\right )^{4} + 2 \, {\left (a^{6} + 3 \, a^{5} b + 2 \, a^{4} b^{2}\right )} \tan \left (f x + e\right )^{2}} - \frac {3 \, {\left (a^{2} + 12 \, a b + 16 \, b^{2}\right )} {\left (f x + e\right )}}{a^{5}} + \frac {3 \, {\left (5 \, a^{2} b + 20 \, a b^{2} + 16 \, b^{3}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {{\left (a + b\right )} b} a^{5}}}{8 \, f} \]
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Time = 0.44 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.29 \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {\frac {3 \, {\left (a^{2} + 12 \, a b + 16 \, b^{2}\right )} {\left (f x + e\right )}}{a^{5}} - \frac {3 \, {\left (5 \, a^{2} b + 20 \, a b^{2} + 16 \, b^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )\right )}}{\sqrt {a b + b^{2}} a^{5}} - \frac {12 \, a b^{2} \tan \left (f x + e\right )^{7} + 24 \, b^{3} \tan \left (f x + e\right )^{7} + 19 \, a^{2} b \tan \left (f x + e\right )^{5} + 72 \, a b^{2} \tan \left (f x + e\right )^{5} + 72 \, b^{3} \tan \left (f x + e\right )^{5} + 5 \, a^{3} \tan \left (f x + e\right )^{3} + 46 \, a^{2} b \tan \left (f x + e\right )^{3} + 108 \, a b^{2} \tan \left (f x + e\right )^{3} + 72 \, b^{3} \tan \left (f x + e\right )^{3} + 3 \, a^{3} \tan \left (f x + e\right ) + 27 \, a^{2} b \tan \left (f x + e\right ) + 48 \, a b^{2} \tan \left (f x + e\right ) + 24 \, b^{3} \tan \left (f x + e\right )}{{\left (b \tan \left (f x + e\right )^{4} + a \tan \left (f x + e\right )^{2} + 2 \, b \tan \left (f x + e\right )^{2} + a + b\right )}^{2} a^{4}}}{8 \, f} \]
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Time = 21.11 (sec) , antiderivative size = 1317, normalized size of antiderivative = 5.53 \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \]
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