Optimal. Leaf size=241 \[ -\frac{b (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )} \text{EllipticF}\left (\sin ^{-1}(\sin (e+f x)),\frac{a}{a+b}\right )}{3 f \left (-a \sin ^2(e+f x)+a+b\right )}+\frac{a \sin (e+f x) \cos ^2(e+f x) \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}{3 f}+\frac{2 (a+2 b) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )} E\left (\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right )}{3 f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}} \]
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Rubi [A] time = 0.421035, antiderivative size = 294, normalized size of antiderivative = 1.22, number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {4148, 6722, 1974, 416, 524, 426, 424, 421, 419} \[ \frac{a \sin (e+f x) \cos ^2(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{3 f \sqrt{a \cos ^2(e+f x)+b}}-\frac{b (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)} F\left (\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right )}{3 f \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}+\frac{2 (a+2 b) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right )}{3 f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b}} \]
Antiderivative was successfully verified.
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Rule 4148
Rule 6722
Rule 1974
Rule 416
Rule 524
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \cos ^3(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 ) \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 \frac{\left (b+a \left (1-x^2\right )\right )^{3/2}}{\sqrt{1-x^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 \frac{\left (a+b-a x^2\right )^{3/2}}{\sqrt{1-x^2}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{b+a \cos ^2(e+f x)}}\\ &=\frac{a \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)}}{3 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+b) (a-3 (a+b))+2 a (a+2 b) x^2}{\sqrt{1-x^2} \sqrt{a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{3 f \sqrt{b+a \cos ^2(e+f x)}}\\ &=\frac{a \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)}}{3 f \sqrt{b+a \cos ^2(e+f x)}}-\frac{\left (b (a+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 )}{3 f \sqrt{b+a \cos ^2(e+f x)}}+\frac{\left (2 (a+2 b) \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 )}{3 f \sqrt{b+a \cos ^2(e+f x)}}\\ &=\frac{a \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)}}{3 f \sqrt{b+a \cos ^2(e+f x)}}+\frac{\left (2 (a+2 b) \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 )}{3 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) \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 )}{3 f \sqrt{b+a \cos ^2(e+f x)} \sqrt{a+b-a \sin ^2(e+f x)}}\\ &=\frac{a \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)}}{3 f \sqrt{b+a \cos ^2(e+f x)}}+\frac{2 (a+2 b) \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)}}{3 f \sqrt{b+a \cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}-\frac{b (a+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}}}{3 f \sqrt{b+a \cos ^2(e+f x)} \sqrt{a+b-a \sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.96235, size = 179, normalized size = 0.74 \[ \frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left (-2 \sqrt{2} b (a+b) \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}} \text{EllipticF}\left (e+f x,\frac{a}{a+b}\right )+4 \sqrt{2} \left (a^2+3 a b+2 b^2\right ) \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}} E\left (e+f x\left |\frac{a}{a+b}\right .\right )+a \sin (2 (e+f x)) (a \cos (2 (e+f x))+a+2 b)\right )}{6 f (a \cos (2 (e+f x))+a+2 b)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.439, size = 5069, normalized size = 21. \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 )^{3}\,{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 )^{3} \sec \left (f x + e\right )^{2} + a \cos \left (f x + e\right )^{3}\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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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