Optimal. Leaf size=51 \[ \frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (e+f x\left |-\frac{b}{a}\right .\right )}{f \sqrt{a+b \sin ^2(e+f x)}} \]
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Rubi [A] time = 0.0328417, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3183, 3182} \[ \frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (e+f x\left |-\frac{b}{a}\right .\right )}{f \sqrt{a+b \sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3183
Rule 3182
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b \sin ^2(e+f x)}} \, dx &=\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}{\sqrt{a+b \sin ^2(e+f x)}}\\ &=\frac{F\left (e+f x\left |-\frac{b}{a}\right .\right ) \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}{f \sqrt{a+b \sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.0846094, size = 60, normalized size = 1.18 \[ \frac{\sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left (e+f x\left |-\frac{b}{a}\right .\right )}{f \sqrt{2 a-b \cos (2 (e+f x))+b}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.266, size = 60, normalized size = 1.2 \begin{align*}{\frac{1}{f}\sqrt{-{\frac{b \left ( \cos \left ( fx+e \right ) \right ) ^{2}-a-b}{a}}}{\it InverseJacobiAM} \left ( fx+e,{i\sqrt{b}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sin \left (f x + e\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-b \cos \left (f x + e\right )^{2} + a + b}}{b \cos \left (f x + e\right )^{2} - a - b}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b \sin ^{2}{\left (e + f x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sin \left (f x + e\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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