Integrand size = 25, antiderivative size = 274 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {a \cos (e+f x) \sin ^3(e+f x)}{b (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(4 a+b) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 b^2 (a+b) f}-\frac {\left (8 a^2+3 a b-2 b^2\right ) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 b^3 (a+b) f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {a (8 a-b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 b^3 f \sqrt {a+b \sin ^2(e+f x)}} \]
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Time = 0.21 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3267, 481, 596, 538, 437, 435, 432, 430} \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\left (8 a^2+3 a b-2 b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{3 b^3 f (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}+\frac {a (8 a-b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{3 b^3 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(4 a+b) \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 b^2 f (a+b)}+\frac {a \sin ^3(e+f x) \cos (e+f x)}{b f (a+b) \sqrt {a+b \sin ^2(e+f x)}} \]
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Rule 430
Rule 432
Rule 435
Rule 437
Rule 481
Rule 538
Rule 596
Rule 3267
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt {1-x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {a \cos (e+f x) \sin ^3(e+f x)}{b (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {x^2 \left (3 a+(-4 a-b) x^2\right )}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{b (a+b) f} \\ & = \frac {a \cos (e+f x) \sin ^3(e+f x)}{b (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(4 a+b) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 b^2 (a+b) f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-a (4 a+b)+\left (8 a^2+3 a b-2 b^2\right ) x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 b^2 (a+b) f} \\ & = \frac {a \cos (e+f x) \sin ^3(e+f x)}{b (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(4 a+b) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 b^2 (a+b) f}+\frac {\left (a (8 a-b) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 b^3 f}-\frac {\left (\left (8 a^2+3 a b-2 b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 b^3 (a+b) f} \\ & = \frac {a \cos (e+f x) \sin ^3(e+f x)}{b (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(4 a+b) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 b^2 (a+b) f}-\frac {\left (\left (8 a^2+3 a b-2 b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {b x^2}{a}}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 b^3 (a+b) f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\left (a (8 a-b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {b x^2}{a}}} \, dx,x,\sin (e+f x)\right )}{3 b^3 f \sqrt {a+b \sin ^2(e+f x)}} \\ & = \frac {a \cos (e+f x) \sin ^3(e+f x)}{b (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(4 a+b) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 b^2 (a+b) f}-\frac {\left (8 a^2+3 a b-2 b^2\right ) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 b^3 (a+b) f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {a (8 a-b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 b^3 f \sqrt {a+b \sin ^2(e+f x)}} \\ \end{align*}
Time = 1.08 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.72 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {-2 \sqrt {2} a \left (8 a^2+3 a b-2 b^2\right ) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )+2 \sqrt {2} a \left (8 a^2+7 a b-b^2\right ) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )+b \left (-8 a^2-3 a b-b^2+b (a+b) \cos (2 (e+f x))\right ) \sin (2 (e+f x))}{6 \sqrt {2} b^3 (a+b) f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]
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Time = 3.34 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.48
method | result | size |
default | \(\frac {a \,b^{2} \left (\sin ^{5}\left (f x +e \right )\right )+b^{3} \left (\sin ^{5}\left (f x +e \right )\right )+8 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+7 a^{2} \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -a \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}-8 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}-3 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b +2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}+4 a^{2} b \left (\sin ^{3}\left (f x +e \right )\right )-b^{3} \left (\sin ^{3}\left (f x +e \right )\right )-4 a^{2} b \sin \left (f x +e \right )-a \,b^{2} \sin \left (f x +e \right )}{3 b^{3} \left (a +b \right ) \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(405\) |
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\[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{6}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{6}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{6}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^6}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]
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