Integrand size = 25, antiderivative size = 175 \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\frac {3 (a+b) (a+5 b) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{8 a^{7/2} f}-\frac {5 (a+b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 a f \sqrt {a+b+b \tan ^2(e+f x)}}-\frac {b (13 a+15 b) \tan (e+f x)}{8 a^3 f \sqrt {a+b+b \tan ^2(e+f x)}} \]
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Time = 0.25 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4217, 481, 541, 12, 385, 209} \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\frac {3 (a+b) (a+5 b) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)+b}}\right )}{8 a^{7/2} f}-\frac {b (13 a+15 b) \tan (e+f x)}{8 a^3 f \sqrt {a+b \tan ^2(e+f x)+b}}-\frac {5 (a+b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \sqrt {a+b \tan ^2(e+f x)+b}}+\frac {\sin (e+f x) \cos ^3(e+f x)}{4 a f \sqrt {a+b \tan ^2(e+f x)+b}} \]
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Rule 12
Rule 209
Rule 385
Rule 481
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 )^{3/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\cos ^3(e+f x) \sin (e+f x)}{4 a f \sqrt {a+b+b \tan ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {a+b-4 (a+b) x^2}{\left (1+x^2\right )^2 \left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{4 a f} \\ & = -\frac {5 (a+b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 a f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {(a+b) (3 a+5 b)-10 b (a+b) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{8 a^2 f} \\ & = -\frac {5 (a+b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 a f \sqrt {a+b+b \tan ^2(e+f x)}}-\frac {b (13 a+15 b) \tan (e+f x)}{8 a^3 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {3 (a+b)^2 (a+5 b)}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{8 a^3 (a+b) f} \\ & = -\frac {5 (a+b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 a f \sqrt {a+b+b \tan ^2(e+f x)}}-\frac {b (13 a+15 b) \tan (e+f x)}{8 a^3 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {(3 (a+b) (a+5 b)) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{8 a^3 f} \\ & = -\frac {5 (a+b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 a f \sqrt {a+b+b \tan ^2(e+f x)}}-\frac {b (13 a+15 b) \tan (e+f x)}{8 a^3 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {(3 (a+b) (a+5 b)) \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{8 a^3 f} \\ & = \frac {3 (a+b) (a+5 b) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{8 a^{7/2} f}-\frac {5 (a+b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 a f \sqrt {a+b+b \tan ^2(e+f x)}}-\frac {b (13 a+15 b) \tan (e+f x)}{8 a^3 f \sqrt {a+b+b \tan ^2(e+f x)}} \\ \end{align*}
Time = 3.33 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.31 \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\frac {(a+2 b+a \cos (2 (e+f x))) \sec ^3(e+f x) \left (24 \left (a^2+6 a b+5 b^2\right ) \arcsin \left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right ) (a+2 b+a \cos (2 (e+f x)))-2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\frac {a+2 b+a \cos (2 (e+f x))}{a+b}} \left (7 a^2+62 a b+60 b^2+2 a (3 a+5 b) \cos (2 (e+f x))-a^2 \cos (4 (e+f x))\right ) \sin (e+f x)\right )}{256 a^{7/2} \sqrt {a+b} f \left (a+b \sec ^2(e+f x)\right )^{3/2} \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(832\) vs. \(2(155)=310\).
Time = 7.29 (sec) , antiderivative size = 833, normalized size of antiderivative = 4.76
method | result | size |
default | \(\frac {\left (b +a \cos \left (f x +e \right )^{2}\right ) \left (2 \cos \left (f x +e \right )^{4} \sin \left (f x +e \right ) \sqrt {-a}\, a^{2}-5 \cos \left (f x +e \right )^{2} \sin \left (f x +e \right ) \sqrt {-a}\, a^{2}-5 \sqrt {-a}\, a b \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )+3 \cos \left (f x +e \right ) \ln \left (4 \sqrt {-a}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \cos \left (f x +e \right )+4 \sqrt {-a}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}-4 \sin \left (f x +e \right ) a \right ) \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, a^{2}+18 \ln \left (4 \sqrt {-a}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \cos \left (f x +e \right )+4 \sqrt {-a}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}-4 \sin \left (f x +e \right ) a \right ) \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, a b \cos \left (f x +e \right )+15 \ln \left (4 \sqrt {-a}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \cos \left (f x +e \right )+4 \sqrt {-a}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}-4 \sin \left (f x +e \right ) a \right ) \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, b^{2} \cos \left (f x +e \right )-13 \sqrt {-a}\, a b \sin \left (f x +e \right )-15 \sqrt {-a}\, b^{2} \sin \left (f x +e \right )+3 \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \ln \left (4 \sqrt {-a}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \cos \left (f x +e \right )+4 \sqrt {-a}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}-4 \sin \left (f x +e \right ) a \right ) a^{2}+18 \ln \left (4 \sqrt {-a}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \cos \left (f x +e \right )+4 \sqrt {-a}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}-4 \sin \left (f x +e \right ) a \right ) \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, a b +15 \ln \left (4 \sqrt {-a}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \cos \left (f x +e \right )+4 \sqrt {-a}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}-4 \sin \left (f x +e \right ) a \right ) \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, b^{2}\right ) \sec \left (f x +e \right )^{3}}{8 f \,a^{3} \sqrt {-a}\, \left (a +b \sec \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}\) | \(833\) |
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Time = 1.83 (sec) , antiderivative size = 703, normalized size of antiderivative = 4.02 \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (a^{2} b + 6 \, a b^{2} + 5 \, b^{3} + {\left (a^{3} + 6 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a} \log \left (128 \, a^{4} \cos \left (f x + e\right )^{8} - 256 \, {\left (a^{4} - a^{3} b\right )} \cos \left (f x + e\right )^{6} + 32 \, {\left (5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left (a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right )} \cos \left (f x + e\right )^{2} + 8 \, {\left (16 \, a^{3} \cos \left (f x + e\right )^{7} - 24 \, {\left (a^{3} - a^{2} b\right )} \cos \left (f x + e\right )^{5} + 2 \, {\left (5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right )\right ) - 8 \, {\left (2 \, a^{3} \cos \left (f x + e\right )^{5} - 5 \, {\left (a^{3} + a^{2} b\right )} \cos \left (f x + e\right )^{3} - {\left (13 \, a^{2} b + 15 \, a b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right )}{64 \, {\left (a^{5} f \cos \left (f x + e\right )^{2} + a^{4} b f\right )}}, -\frac {3 \, {\left (a^{2} b + 6 \, a b^{2} + 5 \, b^{3} + {\left (a^{3} + 6 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {a} \arctan \left (\frac {{\left (8 \, a^{2} \cos \left (f x + e\right )^{5} - 8 \, {\left (a^{2} - a b\right )} \cos \left (f x + e\right )^{3} + {\left (a^{2} - 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{4 \, {\left (2 \, a^{3} \cos \left (f x + e\right )^{4} - a^{2} b + a b^{2} - {\left (a^{3} - 3 \, a^{2} b\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}\right ) - 4 \, {\left (2 \, a^{3} \cos \left (f x + e\right )^{5} - 5 \, {\left (a^{3} + a^{2} b\right )} \cos \left (f x + e\right )^{3} - {\left (13 \, a^{2} b + 15 \, a b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right )}{32 \, {\left (a^{5} f \cos \left (f x + e\right )^{2} + a^{4} b f\right )}}\right ] \]
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\[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\sin ^{4}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{4}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{4}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^4}{{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2}} \,d x \]
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