Optimal. Leaf size=123 \[ -\frac{\left (15 a^2+20 a b+8 b^2\right ) \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{15 a^3 f}+\frac{2 (5 a+2 b) \cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{15 a^2 f}-\frac{\cos ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{5 a f} \]
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Rubi [A] time = 0.138716, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {4134, 462, 453, 264} \[ -\frac{\left (15 a^2+20 a b+8 b^2\right ) \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{15 a^3 f}+\frac{2 (5 a+2 b) \cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{15 a^2 f}-\frac{\cos ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{5 a f} \]
Antiderivative was successfully verified.
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Rule 4134
Rule 462
Rule 453
Rule 264
Rubi steps
\begin{align*} \int \frac{\sin ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right )^2}{x^6 \sqrt{a+b x^2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac{\cos ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{5 a f}+\frac{\operatorname{Subst}\left (\int \frac{-2 (5 a+2 b)+5 a x^2}{x^4 \sqrt{a+b x^2}} \, dx,x,\sec (e+f x)\right )}{5 a f}\\ &=\frac{2 (5 a+2 b) \cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{15 a^2 f}-\frac{\cos ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{5 a f}+\frac{\left (15 a^2+20 a b+8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x^2}} \, dx,x,\sec (e+f x)\right )}{15 a^2 f}\\ &=-\frac{\left (15 a^2+20 a b+8 b^2\right ) \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{15 a^3 f}+\frac{2 (5 a+2 b) \cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{15 a^2 f}-\frac{\cos ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{5 a f}\\ \end{align*}
Mathematica [A] time = 0.980078, size = 93, normalized size = 0.76 \[ -\frac{\sec (e+f x) (a \cos (2 (e+f x))+a+2 b) \left (3 a^2 \cos (4 (e+f x))+89 a^2-4 a (7 a+4 b) \cos (2 (e+f x))+144 a b+64 b^2\right )}{240 a^3 f \sqrt{a+b \sec ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.427, size = 105, normalized size = 0.9 \begin{align*} -{\frac{ \left ( b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) \left ( 3\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}{a}^{2}-10\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}{a}^{2}-4\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}ab+15\,{a}^{2}+20\,ab+8\,{b}^{2} \right ) }{15\,f{a}^{3}\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 = 1.21052, size = 219, normalized size = 1.78 \begin{align*} -\frac{\frac{15 \, \sqrt{a + \frac{b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a} - \frac{10 \,{\left ({\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 )\right )}}{a^{2}} + \frac{3 \,{\left (a + \frac{b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac{5}{2}} \cos \left (f x + e\right )^{5} - 10 \,{\left (a + \frac{b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac{3}{2}} b \cos \left (f x + e\right )^{3} + 15 \, \sqrt{a + \frac{b}{\cos \left (f x + e\right )^{2}}} b^{2} \cos \left (f x + e\right )}{a^{3}}}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.573482, size = 213, normalized size = 1.73 \begin{align*} -\frac{{\left (3 \, a^{2} \cos \left (f x + e\right )^{5} - 2 \,{\left (5 \, a^{2} + 2 \, a b\right )} \cos \left (f x + e\right )^{3} +{\left (15 \, a^{2} + 20 \, a b + 8 \, b^{2}\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}}}}{15 \, a^{3} 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 )^{5}}{\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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