Optimal. Leaf size=186 \[ \frac {\left (a^2-2 a b+5 b^2\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)+b}}{16 b^2 f}+\frac {(a+b) \left (a^2-2 a b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)+b}}\right )}{16 b^{5/2} f}-\frac {(3 a-5 b) \tan (e+f x) \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}{24 b^2 f}+\frac {\tan (e+f x) \sec ^2(e+f x) \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}{6 b f} \]
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Rubi [A] time = 0.17, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4146, 416, 388, 195, 217, 206} \[ \frac {\left (a^2-2 a b+5 b^2\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)+b}}{16 b^2 f}+\frac {(a+b) \left (a^2-2 a b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)+b}}\right )}{16 b^{5/2} f}-\frac {(3 a-5 b) \tan (e+f x) \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}{24 b^2 f}+\frac {\tan (e+f x) \sec ^2(e+f x) \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}{6 b f} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 388
Rule 416
Rule 4146
Rubi steps
\begin {align*} \int \sec ^6(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \left (1+x^2\right )^2 \sqrt {a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\sec ^2(e+f x) \tan (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{6 b f}+\frac {\operatorname {Subst}\left (\int \left (-a+5 b-(3 a-5 b) x^2\right ) \sqrt {a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{6 b f}\\ &=-\frac {(3 a-5 b) \tan (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{24 b^2 f}+\frac {\sec ^2(e+f x) \tan (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{6 b f}+\frac {\left (a^2-2 a b+5 b^2\right ) \operatorname {Subst}\left (\int \sqrt {a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{8 b^2 f}\\ &=\frac {\left (a^2-2 a b+5 b^2\right ) \tan (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{16 b^2 f}-\frac {(3 a-5 b) \tan (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{24 b^2 f}+\frac {\sec ^2(e+f x) \tan (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{6 b f}+\frac {\left ((a+b) \left (a^2-2 a b+5 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{16 b^2 f}\\ &=\frac {\left (a^2-2 a b+5 b^2\right ) \tan (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{16 b^2 f}-\frac {(3 a-5 b) \tan (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{24 b^2 f}+\frac {\sec ^2(e+f x) \tan (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{6 b f}+\frac {\left ((a+b) \left (a^2-2 a b+5 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{16 b^2 f}\\ &=\frac {(a+b) \left (a^2-2 a b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{16 b^{5/2} f}+\frac {\left (a^2-2 a b+5 b^2\right ) \tan (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{16 b^2 f}-\frac {(3 a-5 b) \tan (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{24 b^2 f}+\frac {\sec ^2(e+f x) \tan (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{6 b f}\\ \end {align*}
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Mathematica [C] time = 11.40, size = 968, normalized size = 5.20 \[ -\frac {i e^{i (e+f x)} \sqrt {a e^{-2 i (e+f x)} \left (1+e^{2 i (e+f x)}\right )^2+4 b} \left (\frac {15 (a+b) \tan ^{-1}\left (\frac {\sqrt {b} \left (-1+e^{2 i (e+f x)}\right )}{\sqrt {a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}}}\right ) a^2}{8 b^{3/2}}+\frac {15 \left (-1+e^{2 i (e+f x)}\right ) \sqrt {a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}} a^2}{8 b \left (1+e^{2 i (e+f x)}\right )^2}-\frac {15 (a+b) \tan ^{-1}\left (\frac {\sqrt {b} \left (-1+e^{2 i (e+f x)}\right )}{\sqrt {a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}}}\right ) a}{4 \sqrt {b}}-\frac {15 \left (-1+e^{2 i (e+f x)}\right ) \sqrt {a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}} a}{4 \left (1+e^{2 i (e+f x)}\right )^2}+\frac {7 (7 a+15 b) \left (a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}\right )^{3/2}}{4 b \left (1+e^{2 i (e+f x)}\right )^3}-\frac {2 (8 a+35 b) \left (a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}\right )^{3/2}}{b \left (1+e^{2 i (e+f x)}\right )^3}+\frac {50 \left (a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}\right )^{3/2}}{\left (1+e^{2 i (e+f x)}\right )^3}+\frac {3 (5 a+21 b) \left (a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}\right )^{3/2}}{2 b \left (1+e^{2 i (e+f x)}\right )^4}-\frac {24 \left (a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}\right )^{3/2}}{\left (1+e^{2 i (e+f x)}\right )^4}-\frac {60 \left (a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}\right )^{3/2}}{\left (1+e^{2 i (e+f x)}\right )^5}+\frac {40 \left (a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}\right )^{3/2}}{\left (1+e^{2 i (e+f x)}\right )^6}+\frac {75}{8} \sqrt {b} (a+b) \tan ^{-1}\left (\frac {\sqrt {b} \left (-1+e^{2 i (e+f x)}\right )}{\sqrt {a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}}}\right )+\frac {75 b \left (-1+e^{2 i (e+f x)}\right ) \sqrt {a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}}}{8 \left (1+e^{2 i (e+f x)}\right )^2}\right ) \cos (e+f x) \sqrt {b \sec ^2(e+f x)+a}}{15 \sqrt {2} b \sqrt {a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}} f \sqrt {\cos (2 e+2 f x) a+a+2 b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.98, size = 468, normalized size = 2.52 \[ \left [\frac {3 \, {\left (a^{3} - a^{2} b + 3 \, a b^{2} + 5 \, b^{3}\right )} \sqrt {b} \cos \left (f x + e\right )^{5} \log \left (\frac {{\left (a^{2} - 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 8 \, {\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a - b\right )} \cos \left (f x + e\right )^{3} + 2 \, b \cos \left (f x + e\right )\right )} \sqrt {b} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right ) + 8 \, b^{2}}{\cos \left (f x + e\right )^{4}}\right ) - 4 \, {\left ({\left (3 \, a^{2} b - 4 \, a b^{2} - 15 \, b^{3}\right )} \cos \left (f x + e\right )^{4} - 8 \, b^{3} - 2 \, {\left (a b^{2} + 5 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right )}{192 \, b^{3} f \cos \left (f x + e\right )^{5}}, \frac {3 \, {\left (a^{3} - a^{2} b + 3 \, a b^{2} + 5 \, b^{3}\right )} \sqrt {-b} \arctan \left (-\frac {{\left ({\left (a - b\right )} \cos \left (f x + e\right )^{3} + 2 \, b \cos \left (f x + e\right )\right )} \sqrt {-b} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{2 \, {\left (a b \cos \left (f x + e\right )^{2} + b^{2}\right )} \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right )^{5} - 2 \, {\left ({\left (3 \, a^{2} b - 4 \, a b^{2} - 15 \, b^{3}\right )} \cos \left (f x + e\right )^{4} - 8 \, b^{3} - 2 \, {\left (a b^{2} + 5 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right )}{96 \, b^{3} f \cos \left (f x + e\right )^{5}}\right ] \]
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 \[ \int \sqrt {b \sec \left (f x + e\right )^{2} + a} \sec \left (f x + e\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.12, size = 2518, normalized size = 13.54 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 317, normalized size = 1.70 \[ \frac {\frac {8 \, {\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{3}}{b} + \frac {3 \, {\left (a + b\right )}^{2} a \operatorname {arsinh}\left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{b^{\frac {5}{2}}} + \frac {3 \, {\left (a + b\right )}^{2} \operatorname {arsinh}\left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{b^{\frac {3}{2}}} - \frac {12 \, {\left (a + b\right )} a \operatorname {arsinh}\left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{b^{\frac {3}{2}}} - \frac {12 \, {\left (a + b\right )} \operatorname {arsinh}\left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {b}} + \frac {24 \, a \operatorname {arsinh}\left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {b}} + 24 \, \sqrt {b} \operatorname {arsinh}\left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right ) + 24 \, \sqrt {b \tan \left (f x + e\right )^{2} + a + b} \tan \left (f x + e\right ) - \frac {6 \, {\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} {\left (a + b\right )} \tan \left (f x + e\right )}{b^{2}} + \frac {3 \, \sqrt {b \tan \left (f x + e\right )^{2} + a + b} {\left (a + b\right )}^{2} \tan \left (f x + e\right )}{b^{2}} + \frac {24 \, {\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} \tan \left (f x + e\right )}{b} - \frac {12 \, \sqrt {b \tan \left (f x + e\right )^{2} + a + b} {\left (a + b\right )} \tan \left (f x + e\right )}{b}}{48 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a+\frac {b}{{\cos \left (e+f\,x\right )}^2}}}{{\cos \left (e+f\,x\right )}^6} \,d x \]
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 \[ \int \sqrt {a + b \sec ^{2}{\left (e + f x \right )}} \sec ^{6}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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