Integrand size = 16, antiderivative size = 154 \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=-\frac {b \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}+\frac {2 (2 a+b) E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{3 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {a (a+b) \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 f \sqrt {a+b \sin ^2(e+f x)}} \]
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Time = 0.12 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3259, 3251, 3257, 3256, 3262, 3261} \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=-\frac {b \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}-\frac {a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )}{3 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {2 (2 a+b) \sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{3 f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}} \]
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Rule 3251
Rule 3256
Rule 3257
Rule 3259
Rule 3261
Rule 3262
Rubi steps \begin{align*} \text {integral}& = -\frac {b \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}+\frac {1}{3} \int \frac {a (3 a+b)+2 b (2 a+b) \sin ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx \\ & = -\frac {b \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}-\frac {1}{3} (a (a+b)) \int \frac {1}{\sqrt {a+b \sin ^2(e+f x)}} \, dx+\frac {1}{3} (2 (2 a+b)) \int \sqrt {a+b \sin ^2(e+f x)} \, dx \\ & = -\frac {b \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}+\frac {\left (2 (2 a+b) \sqrt {a+b \sin ^2(e+f x)}\right ) \int \sqrt {1+\frac {b \sin ^2(e+f x)}{a}} \, dx}{3 \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {\left (a (a+b) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}\right ) \int \frac {1}{\sqrt {1+\frac {b \sin ^2(e+f x)}{a}}} \, dx}{3 \sqrt {a+b \sin ^2(e+f x)}} \\ & = -\frac {b \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}+\frac {2 (2 a+b) E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{3 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {a (a+b) \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 f \sqrt {a+b \sin ^2(e+f x)}} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.01 \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {4 \sqrt {2} a (2 a+b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )-2 \sqrt {2} a (a+b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )+b (-2 a-b+b \cos (2 (e+f x))) \sin (2 (e+f x))}{6 \sqrt {2} f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]
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Time = 2.39 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.73
method | result | size |
default | \(\frac {-\frac {\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}}{3}-\frac {a \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b}{3}+\frac {4 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}}{3}+\frac {2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b}{3}+\frac {b^{2} \left (\sin ^{5}\left (f x +e \right )\right )}{3}+\frac {a b \left (\sin ^{3}\left (f x +e \right )\right )}{3}-\frac {b^{2} \left (\sin ^{3}\left (f x +e \right )\right )}{3}-\frac {a b \sin \left (f x +e \right )}{3}}{\cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(266\) |
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\[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int \left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int {\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]
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