Optimal. Leaf size=101 \[ \frac{b \sin (e+f x) \cos (e+f x)}{a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac{b}{a}\right .\right )}{a f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}} \]
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Rubi [A] time = 0.0573837, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3184, 21, 3178, 3177} \[ \frac{b \sin (e+f x) \cos (e+f x)}{a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac{b}{a}\right .\right )}{a f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 3184
Rule 21
Rule 3178
Rule 3177
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx &=\frac{b \cos (e+f x) \sin (e+f x)}{a (a+b) f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\int \frac{-a-b \sin ^2(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx}{a (a+b)}\\ &=\frac{b \cos (e+f x) \sin (e+f x)}{a (a+b) f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\int \sqrt{a+b \sin ^2(e+f x)} \, dx}{a (a+b)}\\ &=\frac{b \cos (e+f x) \sin (e+f x)}{a (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}{a (a+b) \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}\\ &=\frac{b \cos (e+f x) \sin (e+f x)}{a (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)}}{a (a+b) f \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}\\ \end{align*}
Mathematica [A] time = 0.147582, size = 90, normalized size = 0.89 \[ \frac{2 a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left (e+f x\left |-\frac{b}{a}\right .\right )+\sqrt{2} b \sin (2 (e+f x))}{2 a f (a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.217, size = 103, normalized size = 1. \begin{align*}{\frac{1}{a \left ( a+b \right ) \cos \left ( fx+e \right ) f} \left ( \sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{-{\frac{b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{a}}+{\frac{a+b}{a}}}a{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) +\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}b \right ){\frac{1}{\sqrt{a+b \left ( \sin \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}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}}}\,{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^{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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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