Optimal. Leaf size=441 \[ -\frac {2 b (4 a+3 b) \sin (e+f x) \cos ^2(e+f x)}{3 a^2 f (a+b)^2 \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}-\frac {b \left (a^2-16 a b-16 b^2\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} F\left (\sin ^{-1}(\sin (e+f x))|\frac {a}{a+b}\right )}{3 a^4 f (a+b) \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}+\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \left (-a \sin ^2(e+f x)+a+b\right ) E\left (\sin ^{-1}(\sin (e+f x))|\frac {a}{a+b}\right )}{3 a^4 f (a+b)^2 \sqrt {\cos ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}+\frac {\left (a^2+11 a b+8 b^2\right ) \sin (e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{3 a^3 f (a+b)^2 \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}-\frac {b \sin (e+f x) \cos ^4(e+f x)}{3 a f (a+b) \left (-a \sin ^2(e+f x)+a+b\right ) \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}} \]
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Rubi [A] time = 0.73, antiderivative size = 512, normalized size of antiderivative = 1.16, number of steps used = 11, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {4148, 6722, 1974, 413, 526, 528, 524, 426, 424, 421, 419} \[ \frac {\left (a^2+11 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a \cos ^2(e+f x)+b}}{3 a^3 f (a+b)^2 \sqrt {a+b \sec ^2(e+f x)}}-\frac {b \left (a^2-16 a b-16 b^2\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \sqrt {a \cos ^2(e+f x)+b} F\left (\sin ^{-1}(\sin (e+f x))|\frac {a}{a+b}\right )}{3 a^4 f (a+b) \sqrt {\cos ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a+b \sec ^2(e+f x)}}+\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a \cos ^2(e+f x)+b} E\left (\sin ^{-1}(\sin (e+f x))|\frac {a}{a+b}\right )}{3 a^4 f (a+b)^2 \sqrt {\cos ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \sqrt {a+b \sec ^2(e+f x)}}-\frac {2 b (4 a+3 b) \sin (e+f x) \cos ^2(e+f x) \sqrt {a \cos ^2(e+f x)+b}}{3 a^2 f (a+b)^2 \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a+b \sec ^2(e+f x)}}-\frac {b \sin (e+f x) \cos ^4(e+f x) \sqrt {a \cos ^2(e+f x)+b}}{3 a f (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2} \sqrt {a+b \sec ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 413
Rule 419
Rule 421
Rule 424
Rule 426
Rule 524
Rule 526
Rule 528
Rule 1974
Rule 4148
Rule 6722
Rubi steps
\begin {align*} \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-x^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 )^{7/2}}{\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 )^{7/2}}{\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 ^4(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 )^{3/2} \left (-3 a-b+3 (a+2 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 ^4(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 (4 a+3 b) \cos ^2(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 {\sqrt {1-x^2} \left (-3 (a+b) (a+2 b)+3 \left (a^2+11 a b+8 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 ^4(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 (4 a+3 b) \cos ^2(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 (a^2+11 a b+8 b^2\right ) \sqrt {b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt {a+b-a \sin ^2(e+f x)}}{3 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 (2 a^2-5 a b-8 b^2\right )-6 (a+2 b) \left (a^2-4 a b-4 b^2\right ) x^2}{\sqrt {1-x^2} \sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{9 a^3 (a+b)^2 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)}{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 (4 a+3 b) \cos ^2(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 (a^2+11 a b+8 b^2\right ) \sqrt {b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt {a+b-a \sin ^2(e+f x)}}{3 a^3 (a+b)^2 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\left (b \left (a^2-16 a b-16 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 )}{3 a^4 (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {a+b \sec ^2(e+f x)}}+\frac {\left (2 (a+2 b) \left (a^2-4 a b-4 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 )}{3 a^4 (a+b)^2 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)}{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 (4 a+3 b) \cos ^2(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 (a^2+11 a b+8 b^2\right ) \sqrt {b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt {a+b-a \sin ^2(e+f x)}}{3 a^3 (a+b)^2 f \sqrt {a+b \sec ^2(e+f x)}}+\frac {\left (2 (a+2 b) \left (a^2-4 a b-4 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 )}{3 a^4 (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 (a^2-16 a b-16 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 )}{3 a^4 (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 ^4(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 (4 a+3 b) \cos ^2(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 (a^2+11 a b+8 b^2\right ) \sqrt {b+a \cos ^2(e+f x)} \sin (e+f x) \sqrt {a+b-a \sin ^2(e+f x)}}{3 a^3 (a+b)^2 f \sqrt {a+b \sec ^2(e+f x)}}+\frac {2 (a+2 b) \left (a^2-4 a b-4 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)}}{3 a^4 (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 (a^2-16 a b-16 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}}}{3 a^4 (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*}
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Mathematica [F] time = 12.19, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (f x + e\right )^{2} + a} \cos \left (f x + e\right )^{3}}{b^{3} \sec \left (f x + e\right )^{6} + 3 \, a b^{2} \sec \left (f x + e\right )^{4} + 3 \, a^{2} b \sec \left (f x + e\right )^{2} + a^{3}}, x\right ) \]
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 \[ \int \frac {\cos \left (f x + e\right )^{3}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.52, size = 20922, normalized size = 47.44 \[ \text {output too large to display} \]
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 \[ \int \frac {\cos \left (f x + e\right )^{3}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (e+f\,x\right )}^3}{{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{5/2}} \,d x \]
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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