Optimal. Leaf size=436 \[ \frac {(a+6 b) \sin (e+f x) \cos ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{5 a^2 f (a+b) \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}-\frac {4 b \left (a^2-2 a b+12 b^2\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} F\left (\sin ^{-1}(\sin (e+f x))|\frac {a}{a+b}\right )}{15 a^4 f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}+\frac {\left (4 a^2-5 a b-24 b^2\right ) \sin (e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{15 a^3 f (a+b) \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}+\frac {\left (8 a^3-9 a^2 b+16 a b^2+48 b^3\right ) \left (-a \sin ^2(e+f x)+a+b\right ) E\left (\sin ^{-1}(\sin (e+f x))|\frac {a}{a+b}\right )}{15 a^4 f (a+b) \sqrt {\cos ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}-\frac {b \sin (e+f x) \cos ^4(e+f x)}{a f (a+b) \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}} \]
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Rubi [A] time = 0.75, antiderivative size = 509, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4148, 6722, 1974, 413, 528, 524, 426, 424, 421, 419} \[ \frac {\left (4 a^2-5 a b-24 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a \cos ^2(e+f x)+b}}{15 a^3 f (a+b) \sqrt {a+b \sec ^2(e+f x)}}-\frac {4 b \left (a^2-2 a b+12 b^2\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \sqrt {a \cos ^2(e+f x)+b} F\left (\sin ^{-1}(\sin (e+f x))|\frac {a}{a+b}\right )}{15 a^4 f \sqrt {\cos ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a+b \sec ^2(e+f x)}}+\frac {\left (-9 a^2 b+8 a^3+16 a b^2+48 b^3\right ) \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a \cos ^2(e+f x)+b} E\left (\sin ^{-1}(\sin (e+f x))|\frac {a}{a+b}\right )}{15 a^4 f (a+b) \sqrt {\cos ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \sqrt {a+b \sec ^2(e+f x)}}+\frac {(a+6 b) \sin (e+f x) \cos ^2(e+f x) \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a \cos ^2(e+f x)+b}}{5 a^2 f (a+b) \sqrt {a+b \sec ^2(e+f x)}}-\frac {b \sin (e+f x) \cos ^4(e+f x) \sqrt {a \cos ^2(e+f x)+b}}{a f (a+b) \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a+b \sec ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 413
Rule 419
Rule 421
Rule 424
Rule 426
Rule 524
Rule 528
Rule 1974
Rule 4148
Rule 6722
Rubi steps
\begin {align*} \int \frac {\cos ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\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 {\sqrt {b+a \cos ^2(e+f x)} \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^{7/2}}{\left (b+a \left (1-x^2\right )\right )^{3/2}} \, 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 )^{7/2}}{\left (a+b-a x^2\right )^{3/2}} \, 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 ^4(e+f x) \sqrt {b+a \cos ^2(e+f x)} \sin (e+f x)}{a (a+b) 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+(a+6 b) x^2\right )}{\sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{a (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {a+b \sec ^2(e+f x)}}\\ &=-\frac {b \cos ^4(e+f x) \sqrt {b+a \cos ^2(e+f x)} \sin (e+f x)}{a (a+b) f \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)}}+\frac {(a+6 b) \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^2 (a+b) 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 (2 (2 a-3 b) (a+b)+\left (-4 a^2+5 a b+24 b^2\right ) x^2\right )}{\sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{5 a^2 (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {a+b \sec ^2(e+f x)}}\\ &=-\frac {b \cos ^4(e+f x) \sqrt {b+a \cos ^2(e+f x)} \sin (e+f x)}{a (a+b) f \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)}}+\frac {\left (4 a^2-5 a b-24 b^2\right ) \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) f \sqrt {a+b \sec ^2(e+f x)}}+\frac {(a+6 b) \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^2 (a+b) f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\sqrt {b+a \cos ^2(e+f x)} \operatorname {Subst}\left (\int \frac {-(a+b) \left (8 a^2-13 a b+24 b^2\right )+\left (8 a^3-9 a^2 b+16 a b^2+48 b^3\right ) x^2}{\sqrt {1-x^2} \sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{15 a^3 (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {a+b \sec ^2(e+f x)}}\\ &=-\frac {b \cos ^4(e+f x) \sqrt {b+a \cos ^2(e+f x)} \sin (e+f x)}{a (a+b) f \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)}}+\frac {\left (4 a^2-5 a b-24 b^2\right ) \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) f \sqrt {a+b \sec ^2(e+f x)}}+\frac {(a+6 b) \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^2 (a+b) f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\left (4 b \left (a^2-2 a b+12 b^2\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^4 f \sqrt {\cos ^2(e+f x)} \sqrt {a+b \sec ^2(e+f x)}}+\frac {\left (\left (8 a^3-9 a^2 b+16 a b^2+48 b^3\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^4 (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {a+b \sec ^2(e+f x)}}\\ &=-\frac {b \cos ^4(e+f x) \sqrt {b+a \cos ^2(e+f x)} \sin (e+f x)}{a (a+b) f \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)}}+\frac {\left (4 a^2-5 a b-24 b^2\right ) \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) f \sqrt {a+b \sec ^2(e+f x)}}+\frac {(a+6 b) \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^2 (a+b) f \sqrt {a+b \sec ^2(e+f x)}}+\frac {\left (\left (8 a^3-9 a^2 b+16 a b^2+48 b^3\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^4 (a+b) 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 (4 b \left (a^2-2 a b+12 b^2\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^4 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 ^4(e+f x) \sqrt {b+a \cos ^2(e+f x)} \sin (e+f x)}{a (a+b) f \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)}}+\frac {\left (4 a^2-5 a b-24 b^2\right ) \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) f \sqrt {a+b \sec ^2(e+f x)}}+\frac {(a+6 b) \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^2 (a+b) f \sqrt {a+b \sec ^2(e+f x)}}+\frac {\left (8 a^3-9 a^2 b+16 a b^2+48 b^3\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^4 (a+b) 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 {4 b \left (a^2-2 a b+12 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^4 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*}
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Mathematica [F] time = 15.47, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (f x + e\right )^{2} + a} \cos \left (f x + e\right )^{5}}{b^{2} \sec \left (f x + e\right )^{4} + 2 \, a b \sec \left (f x + e\right )^{2} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (f x + e\right )^{5}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.76, size = 15199, normalized size = 34.86 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (f x + e\right )^{5}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (e+f\,x\right )}^5}{{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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