Optimal. Leaf size=372 \[ -\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}{15 b^2 f}+\frac {\left (2 a^2-3 a b-8 b^2\right ) \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )} E\left (\sin ^{-1}(\sin (e+f x))|\frac {a}{a+b}\right )}{15 b^2 f \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}+\frac {(a+4 b) \tan (e+f x) \sec (e+f x) \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}{15 b f}+\frac {\tan (e+f x) \sec ^3(e+f x) \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}{5 f}-\frac {(a-8 b) (a+b) \sqrt {\cos ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )} F\left (\sin ^{-1}(\sin (e+f x))|\frac {a}{a+b}\right )}{15 b f \left (-a \sin ^2(e+f x)+a+b\right )} \]
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Rubi [A] time = 0.69, antiderivative size = 471, normalized size of antiderivative = 1.27, 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, 412, 527, 524, 426, 424, 421, 419} \[ -\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a+b \sec ^2(e+f x)}}{15 b^2 f \sqrt {a \cos ^2(e+f x)+b}}+\frac {\left (2 a^2-3 a b-8 b^2\right ) \sqrt {\cos ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a+b \sec ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|\frac {a}{a+b}\right )}{15 b^2 f \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \sqrt {a \cos ^2(e+f x)+b}}+\frac {\tan (e+f x) \sec ^3(e+f x) \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a+b \sec ^2(e+f x)}}{5 f \sqrt {a \cos ^2(e+f x)+b}}+\frac {(a+4 b) \tan (e+f x) \sec (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a+b \sec ^2(e+f x)}}{15 b f \sqrt {a \cos ^2(e+f x)+b}}-\frac {(a-8 b) (a+b) \sqrt {\cos ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \sqrt {a+b \sec ^2(e+f x)} F\left (\sin ^{-1}(\sin (e+f x))|\frac {a}{a+b}\right )}{15 b f \sqrt {-a \sin ^2(e+f x)+a+b} \sqrt {a \cos ^2(e+f x)+b}} \]
Antiderivative was successfully verified.
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Rule 412
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) \sqrt {a+b \sec ^2(e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+\frac {b}{1-x^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 {\sqrt {b+a \left (1-x^2\right )}}{\left (1-x^2\right )^{7/2}} \, 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 {\sqrt {a+b-a x^2}}{\left (1-x^2\right )^{7/2}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {b+a \cos ^2(e+f x)}}\\ &=\frac {\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)}{5 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 {-4 (a+b)+3 a x^2}{\left (1-x^2\right )^{5/2} \sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{5 f \sqrt {b+a \cos ^2(e+f x)}}\\ &=\frac {(a+4 b) \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)}{15 b f \sqrt {b+a \cos ^2(e+f x)}}+\frac {\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)}{5 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-8 b) (a+b)+a (a+4 b) x^2}{\left (1-x^2\right )^{3/2} \sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{15 b f \sqrt {b+a \cos ^2(e+f x)}}\\ &=-\frac {\left (2 a^2-3 a b-8 b^2\right ) \sqrt {a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt {a+b-a \sin ^2(e+f x)}}{15 b^2 f \sqrt {b+a \cos ^2(e+f x)}}+\frac {(a+4 b) \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)}{15 b f \sqrt {b+a \cos ^2(e+f x)}}+\frac {\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)}{5 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 {-2 a (a-2 b) (a+b)+a \left (2 a^2-3 a b-8 b^2\right ) x^2}{\sqrt {1-x^2} \sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{15 b^2 f \sqrt {b+a \cos ^2(e+f x)}}\\ &=-\frac {\left (2 a^2-3 a b-8 b^2\right ) \sqrt {a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt {a+b-a \sin ^2(e+f x)}}{15 b^2 f \sqrt {b+a \cos ^2(e+f x)}}+\frac {(a+4 b) \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)}{15 b f \sqrt {b+a \cos ^2(e+f x)}}+\frac {\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)}{5 f \sqrt {b+a \cos ^2(e+f x)}}-\frac {\left ((a-8 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 )}{15 b f \sqrt {b+a \cos ^2(e+f x)}}-\frac {\left (\left (-2 a^2+3 a b+8 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 )}{15 b^2 f \sqrt {b+a \cos ^2(e+f x)}}\\ &=-\frac {\left (2 a^2-3 a b-8 b^2\right ) \sqrt {a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt {a+b-a \sin ^2(e+f x)}}{15 b^2 f \sqrt {b+a \cos ^2(e+f x)}}+\frac {(a+4 b) \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)}{15 b f \sqrt {b+a \cos ^2(e+f x)}}+\frac {\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)}{5 f \sqrt {b+a \cos ^2(e+f x)}}-\frac {\left (\left (-2 a^2+3 a b+8 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 )}{15 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-8 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 )}{15 b f \sqrt {b+a \cos ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)}}\\ &=-\frac {\left (2 a^2-3 a b-8 b^2\right ) \sqrt {a+b \sec ^2(e+f x)} \sin (e+f x) \sqrt {a+b-a \sin ^2(e+f x)}}{15 b^2 f \sqrt {b+a \cos ^2(e+f x)}}+\frac {\left (2 a^2-3 a b-8 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)}}{15 b^2 f \sqrt {b+a \cos ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {(a-8 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}}}{15 b f \sqrt {b+a \cos ^2(e+f x)} \sqrt {a+b-a \sin ^2(e+f x)}}+\frac {(a+4 b) \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)}{15 b f \sqrt {b+a \cos ^2(e+f x)}}+\frac {\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)}{5 f \sqrt {b+a \cos ^2(e+f x)}}\\ \end {align*}
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Mathematica [F] time = 27.03, size = 0, normalized size = 0.00 \[ \int \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sec \left (f x + e\right )^{2} + a} \sec \left (f x + e\right )^{5}, x\right ) \]
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 \[ \int \sqrt {b \sec \left (f x + e\right )^{2} + a} \sec \left (f x + e\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.29, size = 6562, normalized size = 17.64 \[ \text {output too large to display} \]
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 \[ \int \sqrt {b \sec \left (f x + e\right )^{2} + a} \sec \left (f x + e\right )^{5}\,{d x} \]
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 \[ \int \frac {\sqrt {a+\frac {b}{{\cos \left (e+f\,x\right )}^2}}}{{\cos \left (e+f\,x\right )}^5} \,d x \]
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 \[ \int \sqrt {a + b \sec ^{2}{\left (e + f x \right )}} \sec ^{5}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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