Integrand size = 25, antiderivative size = 285 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {a \cos (e+f x) \sin ^3(e+f x)}{3 b (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 a (2 a+3 b) \cos (e+f x) \sin (e+f x)}{3 b^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left (8 a^2+13 a b+3 b^2\right ) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 b^3 (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {a (8 a+9 b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 b^3 (a+b) f \sqrt {a+b \sin ^2(e+f x)}} \]
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Time = 0.23 (sec) , antiderivative size = 285, 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, 592, 538, 437, 435, 432, 430} \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {\left (8 a^2+13 a b+3 b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{3 b^3 f (a+b)^2 \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (8 a+9 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{3 b^3 f (a+b) \sqrt {a+b \sin ^2(e+f x)}}+\frac {2 a (2 a+3 b) \sin (e+f x) \cos (e+f x)}{3 b^2 f (a+b)^2 \sqrt {a+b \sin ^2(e+f x)}}+\frac {a \sin ^3(e+f x) \cos (e+f x)}{3 b f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
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Rule 430
Rule 432
Rule 435
Rule 437
Rule 481
Rule 538
Rule 592
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 )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {a \cos (e+f x) \sin ^3(e+f x)}{3 b (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\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-3 b) x^2\right )}{\sqrt {1-x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 b (a+b) f} \\ & = \frac {a \cos (e+f x) \sin ^3(e+f x)}{3 b (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 a (2 a+3 b) \cos (e+f x) \sin (e+f x)}{3 b^2 (a+b)^2 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 {2 a (2 a+3 b)+\left (-8 a^2-13 a b-3 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)^2 f} \\ & = \frac {a \cos (e+f x) \sin ^3(e+f x)}{3 b (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 a (2 a+3 b) \cos (e+f x) \sin (e+f x)}{3 b^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (a (8 a+9 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 (a+b) f}-\frac {\left (\left (-8 a^2-13 a b-3 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)^2 f} \\ & = \frac {a \cos (e+f x) \sin ^3(e+f x)}{3 b (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 a (2 a+3 b) \cos (e+f x) \sin (e+f x)}{3 b^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (\left (-8 a^2-13 a b-3 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)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {\left (a (8 a+9 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 (a+b) f \sqrt {a+b \sin ^2(e+f x)}} \\ & = \frac {a \cos (e+f x) \sin ^3(e+f x)}{3 b (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 a (2 a+3 b) \cos (e+f x) \sin (e+f x)}{3 b^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left (8 a^2+13 a b+3 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)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {a (8 a+9 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 (a+b) f \sqrt {a+b \sin ^2(e+f x)}} \\ \end{align*}
Time = 1.66 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.67 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {a \left (-2 a \left (8 a^2+13 a b+3 b^2\right ) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} E\left (e+f x\left |-\frac {b}{a}\right .\right )+2 a \left (8 a^2+17 a b+9 b^2\right ) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )+\sqrt {2} b \left (-8 a^2-17 a b-7 b^2+b (5 a+7 b) \cos (2 (e+f x))\right ) \sin (2 (e+f x))\right )}{6 b^3 (a+b)^2 f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(697\) vs. \(2(263)=526\).
Time = 3.98 (sec) , antiderivative size = 698, normalized size of antiderivative = 2.45
method | result | size |
default | \(-\frac {\left (\left (5 a \,b^{2}+7 b^{3}\right ) \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-4 a^{2} b -11 a \,b^{2}-7 b^{3}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-\sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, b \left (8 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}+17 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b +9 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}-8 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}-13 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b -3 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+8 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+25 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b +26 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}+9 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{3}-8 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}-21 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b -16 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}-3 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{3}\right ) a}{3 {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}} \left (a +b \right )^{2} b^{3} \cos \left (f x +e \right ) f}\) | \(698\) |
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\[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{6}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{6}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{6}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^6}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
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