Optimal. Leaf size=153 \[ -\frac {\sin (e+f x) \cos (e+f x)}{f (a+b) \sqrt {a+b \sin ^2(e+f x)}}+\frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1} F\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}} \]
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Rubi [A] time = 0.19, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3173, 3172, 3178, 3177, 3183, 3182} \[ -\frac {\sin (e+f x) \cos (e+f x)}{f (a+b) \sqrt {a+b \sin ^2(e+f x)}}+\frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1} F\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 3172
Rule 3173
Rule 3177
Rule 3178
Rule 3182
Rule 3183
Rubi steps
\begin {align*} \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx &=-\frac {\cos (e+f x) \sin (e+f x)}{(a+b) f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\int \frac {a-a \sin ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx}{a (a+b)}\\ &=-\frac {\cos (e+f x) \sin (e+f x)}{(a+b) f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\int \frac {1}{\sqrt {a+b \sin ^2(e+f x)}} \, dx}{b}-\frac {\int \sqrt {a+b \sin ^2(e+f x)} \, dx}{b (a+b)}\\ &=-\frac {\cos (e+f x) \sin (e+f x)}{(a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\sqrt {a+b \sin ^2(e+f x)} \int \sqrt {1+\frac {b \sin ^2(e+f x)}{a}} \, dx}{b (a+b) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\sqrt {1+\frac {b \sin ^2(e+f x)}{a}} \int \frac {1}{\sqrt {1+\frac {b \sin ^2(e+f x)}{a}}} \, dx}{b \sqrt {a+b \sin ^2(e+f x)}}\\ &=-\frac {\cos (e+f x) \sin (e+f x)}{(a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{b (a+b) f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {F\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{b f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 138, normalized size = 0.90 \[ \frac {\sqrt {2} (a+b) \sqrt {\frac {2 a-b \cos (2 (e+f x))+b}{a}} F\left (e+f x\left |-\frac {b}{a}\right .\right )-\sqrt {2} a \sqrt {\frac {2 a-b \cos (2 (e+f x))+b}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )-b \sin (2 (e+f x))}{\sqrt {2} b f (a+b) \sqrt {2 a-b \cos (2 (e+f x))+b}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (\cos \left (f x + e\right )^{2} - 1\right )}}{b^{2} \cos \left (f x + e\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + 2 \, a b + b^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.52, size = 191, normalized size = 1.25 \[ \frac {a \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right )+\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -a \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right )+b \left (\sin ^{3}\left (f x +e \right )\right )-b \sin \left (f x +e \right )}{b \left (a +b \right ) \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (e+f\,x\right )}^2}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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