Optimal. Leaf size=74 \[ \frac{\cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{3 a f}-\frac{(3 a+2 b) \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{3 a^2 f} \]
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Rubi [A] time = 0.0910306, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {4134, 453, 264} \[ \frac{\cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{3 a f}-\frac{(3 a+2 b) \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{3 a^2 f} \]
Antiderivative was successfully verified.
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Rule 4134
Rule 453
Rule 264
Rubi steps
\begin{align*} \int \frac{\sin ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-1+x^2}{x^4 \sqrt{a+b x^2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac{\cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{3 a f}+\frac{(3 a+2 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x^2}} \, dx,x,\sec (e+f x)\right )}{3 a f}\\ &=-\frac{(3 a+2 b) \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{3 a^2 f}+\frac{\cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{3 a f}\\ \end{align*}
Mathematica [A] time = 0.294836, size = 64, normalized size = 0.86 \[ \frac{\sec (e+f x) (a \cos (2 (e+f x))-5 a-4 b) (a \cos (2 (e+f x))+a+2 b)}{12 a^2 f \sqrt{a+b \sec ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.344, size = 69, normalized size = 0.9 \begin{align*}{\frac{ \left ( b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) \left ( a \left ( \cos \left ( fx+e \right ) \right ) ^{2}-3\,a-2\,b \right ) }{3\,f{a}^{2}\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{{\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \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 [A] time = 1.02358, size = 112, normalized size = 1.51 \begin{align*} -\frac{\frac{3 \, \sqrt{a + \frac{b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a} - \frac{{\left (a + \frac{b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac{3}{2}} \cos \left (f x + e\right )^{3} - 3 \, \sqrt{a + \frac{b}{\cos \left (f x + e\right )^{2}}} b \cos \left (f x + e\right )}{a^{2}}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.550606, size = 139, normalized size = 1.88 \begin{align*} \frac{{\left (a \cos \left (f x + e\right )^{3} -{\left (3 \, a + 2 \, b\right )} \cos \left (f x + e\right )\right )} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{3 \, a^{2} f} \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{\sin \left (f x + e\right )^{3}}{\sqrt{b \sec \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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