Integrand size = 25, antiderivative size = 325 \[ \int \sin ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=-\frac {\left (a^2+11 a b+8 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}-\frac {2 (4 a+3 b) \cos (e+f x) \sin ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos (e+f x) \sin ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {a (a+b) \left (2 a^2-5 a b-8 b^2\right ) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{35 b^2 f \sqrt {a+b \sin ^2(e+f x)}} \]
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Time = 0.30 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3267, 488, 596, 538, 437, 435, 432, 430} \[ \int \sin ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {a (a+b) \left (2 a^2-5 a b-8 b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{35 b^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{35 b^2 f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {\left (a^2+11 a b+8 b^2\right ) \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}-\frac {b \sin ^5(e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {2 (4 a+3 b) \sin ^3(e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f} \]
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Rule 430
Rule 432
Rule 435
Rule 437
Rule 488
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^4 \left (a+b x^2\right )^{3/2}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = -\frac {b \cos (e+f x) \sin ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {x^4 \left (-a (7 a+5 b)-2 b (4 a+3 b) x^2\right )}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{7 f} \\ & = -\frac {2 (4 a+3 b) \cos (e+f x) \sin ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos (e+f x) \sin ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {x^2 \left (-6 a b (4 a+3 b)-3 b \left (a^2+11 a b+8 b^2\right ) x^2\right )}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{35 b f} \\ & = -\frac {\left (a^2+11 a b+8 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}-\frac {2 (4 a+3 b) \cos (e+f x) \sin ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos (e+f x) \sin ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-3 a b \left (a^2+11 a b+8 b^2\right )+6 b (a+2 b) \left (a^2-4 a b-4 b^2\right ) x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{105 b^2 f} \\ & = -\frac {\left (a^2+11 a b+8 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}-\frac {2 (4 a+3 b) \cos (e+f x) \sin ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos (e+f x) \sin ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}+\frac {\left (a (a+b) \left (2 a^2-5 a b-8 b^2\right ) \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 )}{35 b^2 f}-\frac {\left (2 (a+2 b) \left (a^2-4 a b-4 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 )}{35 b^2 f} \\ & = -\frac {\left (a^2+11 a b+8 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}-\frac {2 (4 a+3 b) \cos (e+f x) \sin ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos (e+f x) \sin ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {\left (2 (a+2 b) \left (a^2-4 a b-4 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 )}{35 b^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\left (a (a+b) \left (2 a^2-5 a b-8 b^2\right ) \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 )}{35 b^2 f \sqrt {a+b \sin ^2(e+f x)}} \\ & = -\frac {\left (a^2+11 a b+8 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}-\frac {2 (4 a+3 b) \cos (e+f x) \sin ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos (e+f x) \sin ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {2 (a+2 b) \left (a^2-4 a b-4 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)}}{35 b^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {a (a+b) \left (2 a^2-5 a b-8 b^2\right ) \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}}}{35 b^2 f \sqrt {a+b \sin ^2(e+f x)}} \\ \end{align*}
Time = 1.86 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.77 \[ \int \sin ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {-128 a \left (a^3-2 a^2 b-12 a b^2-8 b^3\right ) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )+64 a \left (2 a^3-3 a^2 b-13 a b^2-8 b^3\right ) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )+\sqrt {2} b \left (-32 a^3-496 a^2 b-684 a b^2-250 b^3+b \left (144 a^2+480 a b+299 b^2\right ) \cos (2 (e+f x))-2 b^2 (26 a+27 b) \cos (4 (e+f x))+5 b^3 \cos (6 (e+f x))\right ) \sin (2 (e+f x))}{2240 b^2 f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(601\) vs. \(2(299)=598\).
Time = 4.04 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.85
method | result | size |
default | \(\frac {5 b^{4} \left (\sin ^{9}\left (f x +e \right )\right )+13 a \,b^{3} \left (\sin ^{7}\left (f x +e \right )\right )+b^{4} \left (\sin ^{7}\left (f x +e \right )\right )+9 a^{2} b^{2} \left (\sin ^{5}\left (f x +e \right )\right )+4 a \,b^{3} \left (\sin ^{5}\left (f x +e \right )\right )+2 b^{4} \left (\sin ^{5}\left (f x +e \right )\right )+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 ) a^{4}-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}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3} b -13 \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^{2} 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}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{3}-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^{4}+4 \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} b +24 \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}+16 \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^{3}+a^{3} b \left (\sin ^{3}\left (f x +e \right )\right )+2 a^{2} b^{2} \left (\sin ^{3}\left (f x +e \right )\right )-9 a \,b^{3} \left (\sin ^{3}\left (f x +e \right )\right )-8 b^{4} \left (\sin ^{3}\left (f x +e \right )\right )-a^{3} b \sin \left (f x +e \right )-11 a^{2} b^{2} \sin \left (f x +e \right )-8 a \,b^{3} \sin \left (f x +e \right )}{35 b^{2} \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(602\) |
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\[ \int \sin ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sin \left (f x + e\right )^{4} \,d x } \]
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Timed out. \[ \int \sin ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
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\[ \int \sin ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sin \left (f x + e\right )^{4} \,d x } \]
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\[ \int \sin ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sin \left (f x + e\right )^{4} \,d x } \]
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Timed out. \[ \int \sin ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int {\sin \left (e+f\,x\right )}^4\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]
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