Optimal. Leaf size=559 \[ -\frac{b \left (-9 a^2 b+4 a^3+120 a b^2+128 b^3\right ) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \text{EllipticF}\left (\sin ^{-1}(\sin (e+f x)),\frac{a}{a+b}\right )}{15 a^5 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}+\frac{2 \left (-3 a^2 b+2 a^3-42 a b^2-32 b^3\right ) \sin (e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{15 a^4 f (a+b)^2 \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}+\frac{\left (3 a^2+61 a b+48 b^2\right ) \sin (e+f x) \cos ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{15 a^3 f (a+b)^2 \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}+\frac{\left (27 a^2 b^2-11 a^3 b+8 a^4+184 a b^3+128 b^4\right ) \left (-a \sin ^2(e+f x)+a+b\right ) E\left (\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right )}{15 a^5 f (a+b)^2 \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}-\frac{2 b (5 a+4 b) \sin (e+f x) \cos ^4(e+f x)}{3 a^2 f (a+b)^2 \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}-\frac{b \sin (e+f x) \cos ^6(e+f x)}{3 a f (a+b) \left (-a \sin ^2(e+f x)+a+b\right ) \sqrt{\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}} \]
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Rubi [A] time = 0.905856, antiderivative size = 639, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {4148, 6722, 1974, 413, 526, 528, 524, 426, 424, 421, 419} \[ \frac{\left (3 a^2+61 a b+48 b^2\right ) \sin (e+f x) \cos ^2(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}{15 a^3 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}+\frac{2 \left (-3 a^2 b+2 a^3-42 a b^2-32 b^3\right ) \sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}{15 a^4 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}-\frac{b \left (-9 a^2 b+4 a^3+120 a b^2+128 b^3\right ) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left (\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right )}{15 a^5 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}+\frac{\left (27 a^2 b^2-11 a^3 b+8 a^4+184 a b^3+128 b^4\right ) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left (\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right )}{15 a^5 f (a+b)^2 \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}-\frac{2 b (5 a+4 b) \sin (e+f x) \cos ^4(e+f x) \sqrt{a \cos ^2(e+f x)+b}}{3 a^2 f (a+b)^2 \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}-\frac{b \sin (e+f x) \cos ^6(e+f x) \sqrt{a \cos ^2(e+f x)+b}}{3 a f (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2} \sqrt{a+b \sec ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 4148
Rule 6722
Rule 1974
Rule 413
Rule 526
Rule 528
Rule 524
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{\cos ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{\left (a+\frac{b}{1-x^2}\right )^{5/2}} \, 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 )^{9/2}}{\left (b+a \left (1-x^2\right )\right )^{5/2}} \, 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 )^{9/2}}{\left (a+b-a x^2\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}}\\ &=-\frac{b \cos ^6(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x)}{3 a (a+b) f \sqrt{a+b \sec ^2(e+f x)} \left (a+b-a \sin ^2(e+f x)\right )^{3/2}}-\frac{\sqrt{b+a \cos ^2(e+f x)} \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^{5/2} \left (-3 a-b+(3 a+8 b) x^2\right )}{\left (a+b-a x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 a (a+b) f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}}\\ &=-\frac{b \cos ^6(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x)}{3 a (a+b) f \sqrt{a+b \sec ^2(e+f x)} \left (a+b-a \sin ^2(e+f x)\right )^{3/2}}-\frac{2 b (5 a+4 b) \cos ^4(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt{a+b \sec ^2(e+f x)} \sqrt{a+b-a \sin ^2(e+f x)}}-\frac{\sqrt{b+a \cos ^2(e+f x)} \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^{3/2} \left (-(a+b) (3 a+8 b)+\left (3 a^2+61 a b+48 b^2\right ) x^2\right )}{\sqrt{a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 (a+b)^2 f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}}\\ &=-\frac{b \cos ^6(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x)}{3 a (a+b) f \sqrt{a+b \sec ^2(e+f x)} \left (a+b-a \sin ^2(e+f x)\right )^{3/2}}-\frac{2 b (5 a+4 b) \cos ^4(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt{a+b \sec ^2(e+f x)} \sqrt{a+b-a \sin ^2(e+f x)}}+\frac{\left (3 a^2+61 a b+48 b^2\right ) \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)}}{15 a^3 (a+b)^2 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 (3 (a+b) \left (4 a^2-7 a b-16 b^2\right )-6 \left (2 a^3-3 a^2 b-42 a b^2-32 b^3\right ) x^2\right )}{\sqrt{a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{15 a^3 (a+b)^2 f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}}\\ &=-\frac{b \cos ^6(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x)}{3 a (a+b) f \sqrt{a+b \sec ^2(e+f x)} \left (a+b-a \sin ^2(e+f x)\right )^{3/2}}-\frac{2 b (5 a+4 b) \cos ^4(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt{a+b \sec ^2(e+f x)} \sqrt{a+b-a \sin ^2(e+f x)}}+\frac{2 \left (2 a^3-3 a^2 b-42 a b^2-32 b^3\right ) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{15 a^4 (a+b)^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\left (3 a^2+61 a b+48 b^2\right ) \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)}}{15 a^3 (a+b)^2 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\sqrt{b+a \cos ^2(e+f x)} \operatorname{Subst}\left (\int \frac{-3 (a+b) \left (8 a^3-15 a^2 b+36 a b^2+64 b^3\right )+3 \left (8 a^4-11 a^3 b+27 a^2 b^2+184 a b^3+128 b^4\right ) x^2}{\sqrt{1-x^2} \sqrt{a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{45 a^4 (a+b)^2 f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}}\\ &=-\frac{b \cos ^6(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x)}{3 a (a+b) f \sqrt{a+b \sec ^2(e+f x)} \left (a+b-a \sin ^2(e+f x)\right )^{3/2}}-\frac{2 b (5 a+4 b) \cos ^4(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt{a+b \sec ^2(e+f x)} \sqrt{a+b-a \sin ^2(e+f x)}}+\frac{2 \left (2 a^3-3 a^2 b-42 a b^2-32 b^3\right ) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{15 a^4 (a+b)^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\left (3 a^2+61 a b+48 b^2\right ) \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)}}{15 a^3 (a+b)^2 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\left (b \left (4 a^3-9 a^2 b+120 a b^2+128 b^3\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^5 (a+b) f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}}+\frac{\left (\left (8 a^4-11 a^3 b+27 a^2 b^2+184 a b^3+128 b^4\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^5 (a+b)^2 f \sqrt{\cos ^2(e+f x)} \sqrt{a+b \sec ^2(e+f x)}}\\ &=-\frac{b \cos ^6(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x)}{3 a (a+b) f \sqrt{a+b \sec ^2(e+f x)} \left (a+b-a \sin ^2(e+f x)\right )^{3/2}}-\frac{2 b (5 a+4 b) \cos ^4(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt{a+b \sec ^2(e+f x)} \sqrt{a+b-a \sin ^2(e+f x)}}+\frac{2 \left (2 a^3-3 a^2 b-42 a b^2-32 b^3\right ) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{15 a^4 (a+b)^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\left (3 a^2+61 a b+48 b^2\right ) \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)}}{15 a^3 (a+b)^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\left (\left (8 a^4-11 a^3 b+27 a^2 b^2+184 a b^3+128 b^4\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^5 (a+b)^2 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 (b \left (4 a^3-9 a^2 b+120 a b^2+128 b^3\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^5 (a+b) 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{b \cos ^6(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x)}{3 a (a+b) f \sqrt{a+b \sec ^2(e+f x)} \left (a+b-a \sin ^2(e+f x)\right )^{3/2}}-\frac{2 b (5 a+4 b) \cos ^4(e+f x) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt{a+b \sec ^2(e+f x)} \sqrt{a+b-a \sin ^2(e+f x)}}+\frac{2 \left (2 a^3-3 a^2 b-42 a b^2-32 b^3\right ) \sqrt{b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt{a+b-a \sin ^2(e+f x)}}{15 a^4 (a+b)^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\left (3 a^2+61 a b+48 b^2\right ) \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)}}{15 a^3 (a+b)^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\left (8 a^4-11 a^3 b+27 a^2 b^2+184 a b^3+128 b^4\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^5 (a+b)^2 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^3-9 a^2 b+120 a b^2+128 b^3\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^5 (a+b) 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 = 23.4481, size = 0, normalized size = 0. \[ \int \frac{\cos ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [C] time = 2.248, size = 26983, normalized size = 48.3 \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}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{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{\sqrt{b \sec \left (f x + e\right )^{2} + a} \cos \left (f x + e\right )^{5}}{b^{3} \sec \left (f x + e\right )^{6} + 3 \, a b^{2} \sec \left (f x + e\right )^{4} + 3 \, a^{2} b \sec \left (f x + e\right )^{2} + a^{3}}, 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}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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