Optimal. Leaf size=345 \[ -\frac{b \left (4 a^2-3 a b+8 b^2\right ) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \text{EllipticF}\left (\sin ^{-1}(\sin (e+f x)),\frac{a}{a+b}\right )}{15 a^3 f \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}+\frac{\left (8 a^2-7 a b+8 b^2\right ) \left (-a \sin ^2(e+f x)+a+b\right ) E\left (\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right )}{15 a^3 f \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}+\frac{4 (a-b) \sin (e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{15 a^2 f \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}+\frac{\sin (e+f x) \cos ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{5 a f \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}} \]
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Rubi [A] time = 0.573637, antiderivative size = 395, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4148, 6722, 1974, 416, 528, 524, 426, 424, 421, 419} \[ -\frac{b \left (4 a^2-3 a b+8 b^2\right ) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left (\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right )}{15 a^3 f \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}+\frac{\left (8 a^2-7 a b+8 b^2\right ) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left (\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right )}{15 a^3 f \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}+\frac{4 (a-b) \sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}{15 a^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\sin (e+f x) \cos ^2(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}{5 a f \sqrt{a+b \sec ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 4148
Rule 6722
Rule 1974
Rule 416
Rule 528
Rule 524
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{\cos ^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}{\sqrt{a+\frac{b}{1-x^2}}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\sqrt{b+a \cos ^2(e+f x)} \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^{5/2}}{\sqrt{b+a \left (1-x^2\right )}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}}\\ &=\frac{\sqrt{b+a \cos ^2(e+f x)} \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^{5/2}}{\sqrt{a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}}\\ &=\frac{\cos ^2(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{5 a f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\sqrt{b+a \cos ^2(e+f x)} \operatorname{Subst}\left (\int \frac{\sqrt{1-x^2} \left (-4 a+b+4 (a-b) x^2\right )}{\sqrt{a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{5 a f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}}\\ &=\frac{4 (a-b) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{15 a^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\cos ^2(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{5 a f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\sqrt{b+a \cos ^2(e+f x)} \operatorname{Subst}\left (\int \frac{8 a^2-3 a b+4 b^2+\left (-8 a^2+7 a b-8 b^2\right ) x^2}{\sqrt{1-x^2} \sqrt{a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{15 a^2 f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}}\\ &=\frac{4 (a-b) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{15 a^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\cos ^2(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{5 a f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\left (\left (-8 a^2+7 a b-8 b^2\right ) \sqrt{b+a \cos ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b-a x^2}}{\sqrt{1-x^2}} \, dx,x,\sin (e+f x)\right )}{15 a^3 f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}}-\frac{\left (\left (-(a+b) \left (-8 a^2+7 a b-8 b^2\right )-a \left (8 a^2-3 a b+4 b^2\right )\right ) \sqrt{b+a \cos ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{15 a^3 f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}}\\ &=\frac{4 (a-b) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{15 a^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\cos ^2(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{5 a f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\left (\left (-8 a^2+7 a b-8 b^2\right ) \sqrt{b+a \cos ^2(e+f x)} \sqrt{a+b-a \sin ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{a x^2}{a+b}}}{\sqrt{1-x^2}} \, dx,x,\sin (e+f x)\right )}{15 a^3 f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}-\frac{\left (\left (-(a+b) \left (-8 a^2+7 a b-8 b^2\right )-a \left (8 a^2-3 a b+4 b^2\right )\right ) \sqrt{b+a \cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1-\frac{a x^2}{a+b}}} \, dx,x,\sin (e+f x)\right )}{15 a^3 f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)} \sqrt{a+b-a \sin ^2(e+f x)}}\\ &=\frac{4 (a-b) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{15 a^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\cos ^2(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{5 a f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\left (8 a^2-7 a b+8 b^2\right ) \sqrt{b+a \cos ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right ) \sqrt{a+b-a \sin ^2(e+f x)}}{15 a^3 f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}-\frac{b \left (4 a^2-3 a b+8 b^2\right ) \sqrt{b+a \cos ^2(e+f x)} F\left (\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right ) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}{15 a^3 f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)} \sqrt{a+b-a \sin ^2(e+f x)}}\\ \end{align*}
Mathematica [F] time = 12.874, size = 0, normalized size = 0. \[ \int \frac{\cos ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx \]
Verification is Not applicable to the result.
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Maple [C] time = 0.744, size = 6382, normalized size = 18.5 \begin{align*} \text{output too large to display} \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{\cos \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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (f x + e\right )^{5}}{\sqrt{b \sec \left (f x + e\right )^{2} + a}}, 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{\cos \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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