Optimal. Leaf size=450 \[ -\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sin (e+f x) \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}{35 b^2 f}+\frac {\left (a^2+11 a b+8 b^2\right ) \tan (e+f x) \sec (e+f x) \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}{35 b f}-\frac {(a+b) \left (a^2-16 a b-16 b^2\right ) \sqrt {\cos ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )} F\left (\sin ^{-1}(\sin (e+f x))|\frac {a}{a+b}\right )}{35 b f \left (-a \sin ^2(e+f x)+a+b\right )}+\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )} E\left (\sin ^{-1}(\sin (e+f x))|\frac {a}{a+b}\right )}{35 b^2 f \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}+\frac {b \tan (e+f x) \sec ^5(e+f x) \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}{7 f}+\frac {2 (4 a+3 b) \tan (e+f x) \sec ^3(e+f x) \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}{35 f} \]
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Rubi [A] time = 0.89, antiderivative size = 572, normalized size of antiderivative = 1.27, number of steps used = 12, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4148, 6722, 1974, 413, 527, 524, 426, 424, 421, 419} \[ -\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a+b \sec ^2(e+f x)}}{35 b^2 f \sqrt {a \cos ^2(e+f x)+b}}+\frac {\left (a^2+11 a b+8 b^2\right ) \tan (e+f x) \sec (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a+b \sec ^2(e+f x)}}{35 b f \sqrt {a \cos ^2(e+f x)+b}}-\frac {(a+b) \left (a^2-16 a b-16 b^2\right ) \sqrt {\cos ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \sqrt {a+b \sec ^2(e+f x)} F\left (\sin ^{-1}(\sin (e+f x))|\frac {a}{a+b}\right )}{35 b f \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a \cos ^2(e+f x)+b}}+\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt {\cos ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a+b \sec ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|\frac {a}{a+b}\right )}{35 b^2 f \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \sqrt {a \cos ^2(e+f x)+b}}+\frac {b \tan (e+f x) \sec ^5(e+f x) \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a+b \sec ^2(e+f x)}}{7 f \sqrt {a \cos ^2(e+f x)+b}}+\frac {2 (4 a+3 b) \tan (e+f x) \sec ^3(e+f x) \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a+b \sec ^2(e+f x)}}{35 f \sqrt {a \cos ^2(e+f x)+b}} \]
Antiderivative was successfully verified.
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Rule 413
Rule 419
Rule 421
Rule 424
Rule 426
Rule 524
Rule 527
Rule 1974
Rule 4148
Rule 6722
Rubi steps
\begin {align*} \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+\frac {b}{1-x^2}\right )^{3/2}}{\left (1-x^2\right )^3} \, 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}}{\left (1-x^2\right )^{9/2}} \, 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}}{\left (1-x^2\right )^{9/2}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {b+a \cos ^2(e+f x)}}\\ &=\frac {b \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{7 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) (7 a+6 b)+a (7 a+5 b) x^2}{\left (1-x^2\right )^{7/2} \sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{7 f \sqrt {b+a \cos ^2(e+f x)}}\\ &=\frac {2 (4 a+3 b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{35 f \sqrt {b+a \cos ^2(e+f x)}}+\frac {b \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{7 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 {-3 b (a+b) (9 a+8 b)+6 a b (4 a+3 b) x^2}{\left (1-x^2\right )^{5/2} \sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{35 b f \sqrt {b+a \cos ^2(e+f x)}}\\ &=\frac {\left (a^2+11 a b+8 b^2\right ) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{35 b f \sqrt {b+a \cos ^2(e+f x)}}+\frac {2 (4 a+3 b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{35 f \sqrt {b+a \cos ^2(e+f x)}}+\frac {b \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{7 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 {3 b (a+b) \left (a^2-16 a b-16 b^2\right )+3 a b \left (a^2+11 a b+8 b^2\right ) x^2}{\left (1-x^2\right )^{3/2} \sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{105 b^2 f \sqrt {b+a \cos ^2(e+f x)}}\\ &=-\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt {a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt {a+b-a \sin ^2(e+f x)}}{35 b^2 f \sqrt {b+a \cos ^2(e+f x)}}+\frac {\left (a^2+11 a b+8 b^2\right ) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{35 b f \sqrt {b+a \cos ^2(e+f x)}}+\frac {2 (4 a+3 b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{35 f \sqrt {b+a \cos ^2(e+f x)}}+\frac {b \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{7 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 {-3 a b (a+b) \left (2 a^2-5 a b-8 b^2\right )+6 a b (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 )}{105 b^3 f \sqrt {b+a \cos ^2(e+f x)}}\\ &=-\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt {a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt {a+b-a \sin ^2(e+f x)}}{35 b^2 f \sqrt {b+a \cos ^2(e+f x)}}+\frac {\left (a^2+11 a b+8 b^2\right ) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{35 b f \sqrt {b+a \cos ^2(e+f x)}}+\frac {2 (4 a+3 b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{35 f \sqrt {b+a \cos ^2(e+f x)}}+\frac {b \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{7 f \sqrt {b+a \cos ^2(e+f x)}}-\frac {\left ((a+b) \left (a^2-16 a b-16 b^2\right ) \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 )}{35 b f \sqrt {b+a \cos ^2(e+f x)}}+\frac {\left (2 (a+2 b) \left (a^2-4 a b-4 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 )}{35 b^2 f \sqrt {b+a \cos ^2(e+f x)}}\\ &=-\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt {a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt {a+b-a \sin ^2(e+f x)}}{35 b^2 f \sqrt {b+a \cos ^2(e+f x)}}+\frac {\left (a^2+11 a b+8 b^2\right ) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{35 b f \sqrt {b+a \cos ^2(e+f x)}}+\frac {2 (4 a+3 b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{35 f \sqrt {b+a \cos ^2(e+f x)}}+\frac {b \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{7 f \sqrt {b+a \cos ^2(e+f x)}}+\frac {\left (2 (a+2 b) \left (a^2-4 a b-4 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 )}{35 b^2 f \sqrt {b+a \cos ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {\left ((a+b) \left (a^2-16 a b-16 b^2\right ) \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 )}{35 b f \sqrt {b+a \cos ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)}}\\ &=-\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt {a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt {a+b-a \sin ^2(e+f x)}}{35 b^2 f \sqrt {b+a \cos ^2(e+f x)}}+\frac {2 (a+2 b) \left (a^2-4 a b-4 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)}}{35 b^2 f \sqrt {b+a \cos ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {(a+b) \left (a^2-16 a b-16 b^2\right ) \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}}}{35 b f \sqrt {b+a \cos ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)}}+\frac {\left (a^2+11 a b+8 b^2\right ) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{35 b f \sqrt {b+a \cos ^2(e+f x)}}+\frac {2 (4 a+3 b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{35 f \sqrt {b+a \cos ^2(e+f x)}}+\frac {b \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)} \tan (e+f x)}{7 f \sqrt {b+a \cos ^2(e+f x)}}\\ \end {align*}
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Mathematica [F] time = 9.68, size = 0, normalized size = 0.00 \[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sec \left (f x + e\right )^{7} + a \sec \left (f x + e\right )^{5}\right )} \sqrt {b \sec \left (f x + e\right )^{2} + a}, x\right ) \]
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 \[ \int {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.83, size = 7996, normalized size = 17.77 \[ \text {output too large to display} \]
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 \[ \int {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )^{5}\,{d x} \]
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 \[ \int \frac {{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2}}{{\cos \left (e+f\,x\right )}^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \sec ^{5}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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