Optimal. Leaf size=319 \[ -\frac{b (a+b) (4 a+3 b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )} \text{EllipticF}\left (\sin ^{-1}(\sin (e+f x)),\frac{a}{a+b}\right )}{15 a f \left (-a \sin ^2(e+f x)+a+b\right )}+\frac{\left (8 a^2+13 a b+3 b^2\right ) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )} E\left (\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right )}{15 a f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}+\frac{a \sin (e+f x) \cos ^4(e+f x) \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}{5 f}-\frac{2 (a-3 (a+b)) \sin (e+f x) \cos ^2(e+f x) \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}{15 f} \]
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Rubi [A] time = 0.641289, antiderivative size = 395, normalized size of antiderivative = 1.24, 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} \[ \frac{\left (8 a^2+13 a b+3 b^2\right ) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right )}{15 a f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b}}+\frac{a \sin (e+f x) \cos ^4(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{5 f \sqrt{a \cos ^2(e+f x)+b}}-\frac{2 (a-3 (a+b)) \sin (e+f x) \cos ^2(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{15 f \sqrt{a \cos ^2(e+f x)+b}}-\frac{b (a+b) (4 a+3 b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)} F\left (\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right )}{15 a f \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}} \]
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 \cos ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (1-x^2\right )^2 \left (a+\frac{b}{1-x^2}\right )^{3/2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\left (\sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \sqrt{1-x^2} \left (b+a \left (1-x^2\right )\right )^{3/2} \, dx,x,\sin (e+f x)\right )}{f \sqrt{b+a \cos ^2(e+f x)}}\\ &=\frac{\left (\sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \sqrt{1-x^2} \left (a+b-a x^2\right )^{3/2} \, dx,x,\sin (e+f x)\right )}{f \sqrt{b+a \cos ^2(e+f x)}}\\ &=\frac{a \cos ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{5 f \sqrt{b+a \cos ^2(e+f x)}}-\frac{\left (\sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-x^2} \left ((a+b) (a-5 (a+b))-2 a (a-3 (a+b)) x^2\right )}{\sqrt{a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{5 f \sqrt{b+a \cos ^2(e+f x)}}\\ &=-\frac{2 (a-3 (a+b)) \cos ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{15 f \sqrt{b+a \cos ^2(e+f x)}}+\frac{a \cos ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{5 f \sqrt{b+a \cos ^2(e+f x)}}+\frac{\left (\sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{a (a+b) (8 a+9 b)-a \left (8 a^2+13 a b+3 b^2\right ) x^2}{\sqrt{1-x^2} \sqrt{a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{15 a f \sqrt{b+a \cos ^2(e+f x)}}\\ &=-\frac{2 (a-3 (a+b)) \cos ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{15 f \sqrt{b+a \cos ^2(e+f x)}}+\frac{a \cos ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{5 f \sqrt{b+a \cos ^2(e+f x)}}-\frac{\left (b (a+b) (4 a+3 b) \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^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 f \sqrt{b+a \cos ^2(e+f x)}}+\frac{\left (\left (8 a^2+13 a b+3 b^2\right ) \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^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 f \sqrt{b+a \cos ^2(e+f x)}}\\ &=-\frac{2 (a-3 (a+b)) \cos ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{15 f \sqrt{b+a \cos ^2(e+f x)}}+\frac{a \cos ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{5 f \sqrt{b+a \cos ^2(e+f x)}}+\frac{\left (\left (8 a^2+13 a b+3 b^2\right ) \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^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 f \sqrt{b+a \cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}-\frac{\left (b (a+b) (4 a+3 b) \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}}\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 f \sqrt{b+a \cos ^2(e+f x)} \sqrt{a+b-a \sin ^2(e+f x)}}\\ &=-\frac{2 (a-3 (a+b)) \cos ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{15 f \sqrt{b+a \cos ^2(e+f x)}}+\frac{a \cos ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{5 f \sqrt{b+a \cos ^2(e+f x)}}+\frac{\left (8 a^2+13 a b+3 b^2\right ) \sqrt{\cos ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right ) \sqrt{a+b \sec ^2(e+f x)} \sqrt{a+b-a \sin ^2(e+f x)}}{15 a f \sqrt{b+a \cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}-\frac{b (a+b) (4 a+3 b) \sqrt{\cos ^2(e+f x)} F\left (\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right ) \sqrt{a+b \sec ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}{15 a f \sqrt{b+a \cos ^2(e+f x)} \sqrt{a+b-a \sin ^2(e+f x)}}\\ \end{align*}
Mathematica [C] time = 9.31625, size = 350, normalized size = 1.1 \[ \frac{\cos ^3(e+f x) \csc (2 (e+f x)) \left (a+b \sec ^2(e+f x)\right )^{3/2} \left (a \left (a \sqrt{-\frac{1}{a+b}} \sin ^2(2 (e+f x)) \sqrt{a \cos (2 (e+f x))+a+2 b} (3 a \cos (2 (e+f x))+11 a+12 b)+16 i \sqrt{2} b (2 a+3 b) \sqrt{\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{-\frac{a \cos ^2(e+f x)}{b}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{1}{a+b}} \sqrt{a \cos (2 (e+f x))+a+2 b}}{\sqrt{2}}\right ),\frac{a+b}{b}\right )\right )-8 i \sqrt{2} b \left (8 a^2+13 a b+3 b^2\right ) \sqrt{\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{-\frac{a \cos ^2(e+f x)}{b}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{1}{a+b}} \sqrt{\cos (2 (e+f x)) a+a+2 b}}{\sqrt{2}}\right )|\frac{a+b}{b}\right )\right )}{30 a^2 f \sqrt{-\frac{1}{a+b}} (a \cos (2 (e+f x))+a+2 b)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.623, size = 6396, normalized size = 20.1 \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{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \cos \left (f x + e\right )^{5}\,{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 ({\left (b \cos \left (f x + e\right )^{5} \sec \left (f x + e\right )^{2} + a \cos \left (f x + e\right )^{5}\right )} \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{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \cos \left (f x + e\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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