Optimal. Leaf size=111 \[ \frac {E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{b f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {a F\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{b f \sqrt {a+b \sin ^2(e+f x)}} \]
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Rubi [A]
time = 0.09, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3251, 3257,
3256, 3262, 3261} \begin {gather*} \frac {\sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3251
Rule 3256
Rule 3257
Rule 3261
Rule 3262
Rubi steps
\begin {align*} \int \frac {\sin ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx &=\frac {\int \sqrt {a+b \sin ^2(e+f x)} \, dx}{b}-\frac {a \int \frac {1}{\sqrt {a+b \sin ^2(e+f x)}} \, dx}{b}\\ &=\frac {\sqrt {a+b \sin ^2(e+f x)} \int \sqrt {1+\frac {b \sin ^2(e+f x)}{a}} \, dx}{b \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {\left (a \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}{b \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 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {a 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 = 0.17, size = 78, normalized size = 0.70 \begin {gather*} \frac {\sqrt {2 a+b-b \cos (2 (e+f x))} \left (E\left (e+f x\left |-\frac {b}{a}\right .\right )-F\left (e+f x\left |-\frac {b}{a}\right .\right )\right )}{b f \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.92, size = 93, normalized size = 0.84
method | result | size |
default | \(-\frac {a \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \left (\EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right )-\EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right )\right )}{b \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(93\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.10, size = 48, normalized size = 0.43 \begin {gather*} {\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 \cos \left (f x + e\right )^{2} - a - b}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin ^{2}{\left (e + f x \right )}}{\sqrt {a + b \sin ^{2}{\left (e + f x \right )}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\sin \left (e+f\,x\right )}^2}{\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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