Optimal. Leaf size=222 \[ \frac {\left (a^3+2 a^2 b+8 a b^2-16 b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{16 b^{5/2} f}-\frac {(a-2 b) (a+4 b) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{16 b^2 f}-\frac {\sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}+\frac {\tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{6 f}+\frac {(a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{24 b f} \]
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Rubi [A] time = 0.34, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3670, 478, 582, 523, 217, 206, 377, 203} \[ \frac {\left (2 a^2 b+a^3+8 a b^2-16 b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{16 b^{5/2} f}-\frac {(a-2 b) (a+4 b) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{16 b^2 f}+\frac {\tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{6 f}+\frac {(a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{24 b f}-\frac {\sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 217
Rule 377
Rule 478
Rule 523
Rule 582
Rule 3670
Rubi steps
\begin {align*} \int \tan ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^6 \sqrt {a+b x^2}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{6 f}-\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (5 a+(-a+6 b) x^2\right )}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{6 f}\\ &=\frac {(a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{24 b f}+\frac {\tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{6 f}+\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (-3 a (a-6 b)-3 (a-2 b) (a+4 b) x^2\right )}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{24 b f}\\ &=-\frac {(a-2 b) (a+4 b) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{16 b^2 f}+\frac {(a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{24 b f}+\frac {\tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{6 f}-\frac {\operatorname {Subst}\left (\int \frac {-3 a (a-2 b) (a+4 b)-3 \left (a^3+2 a^2 b+8 a b^2-16 b^3\right ) x^2}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{48 b^2 f}\\ &=-\frac {(a-2 b) (a+4 b) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{16 b^2 f}+\frac {(a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{24 b f}+\frac {\tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{6 f}-\frac {(a-b) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}+\frac {\left (a^3+2 a^2 b+8 a b^2-16 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{16 b^2 f}\\ &=-\frac {(a-2 b) (a+4 b) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{16 b^2 f}+\frac {(a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{24 b f}+\frac {\tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{6 f}-\frac {(a-b) \operatorname {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}+\frac {\left (a^3+2 a^2 b+8 a b^2-16 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{16 b^2 f}\\ &=-\frac {\sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}+\frac {\left (a^3+2 a^2 b+8 a b^2-16 b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{16 b^{5/2} f}-\frac {(a-2 b) (a+4 b) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{16 b^2 f}+\frac {(a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{24 b f}+\frac {\tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{6 f}\\ \end {align*}
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Mathematica [C] time = 6.33, size = 823, normalized size = 3.71 \[ \frac {-\frac {b \left (a^3+2 b a^2-8 b^3\right ) \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left (\left .\sin ^{-1}\left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right )\right |1\right ) \sin ^4(e+f x)}{a (a+b+(a-b) \cos (2 (e+f x)))}-\frac {4 b \left (8 b^3-8 a b^2\right ) \sqrt {\cos (2 (e+f x))+1} \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left (\frac {\sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left (\left .\sin ^{-1}\left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right )\right |1\right ) \sin ^4(e+f x)}{4 a \sqrt {\cos (2 (e+f x))+1} \sqrt {a+b+(a-b) \cos (2 (e+f x))}}-\frac {\sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \Pi \left (-\frac {b}{a-b};\left .\sin ^{-1}\left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right )\right |1\right ) \sin ^4(e+f x)}{2 (a-b) \sqrt {\cos (2 (e+f x))+1} \sqrt {a+b+(a-b) \cos (2 (e+f x))}}\right )}{\sqrt {a+b+(a-b) \cos (2 (e+f x))}}}{8 b^2 f}+\frac {\sqrt {\frac {\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left (\frac {1}{6} \tan (e+f x) \sec ^4(e+f x)+\frac {(a \sin (e+f x)-14 b \sin (e+f x)) \sec ^3(e+f x)}{24 b}+\frac {\left (-3 \sin (e+f x) a^2-8 b \sin (e+f x) a+44 b^2 \sin (e+f x)\right ) \sec (e+f x)}{48 b^2}\right )}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.10, size = 826, normalized size = 3.72 \[ \left [\frac {48 \, \sqrt {-a + b} b^{3} \log \left (-\frac {{\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b} \tan \left (f x + e\right ) - a}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 \, {\left (a^{3} + 2 \, a^{2} b + 8 \, a b^{2} - 16 \, b^{3}\right )} \sqrt {b} \log \left (2 \, b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {b} \tan \left (f x + e\right ) + a\right ) + 2 \, {\left (8 \, b^{3} \tan \left (f x + e\right )^{5} + 2 \, {\left (a b^{2} - 6 \, b^{3}\right )} \tan \left (f x + e\right )^{3} - 3 \, {\left (a^{2} b + 2 \, a b^{2} - 8 \, b^{3}\right )} \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{96 \, b^{3} f}, -\frac {96 \, \sqrt {a - b} b^{3} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{\sqrt {a - b} \tan \left (f x + e\right )}\right ) + 3 \, {\left (a^{3} + 2 \, a^{2} b + 8 \, a b^{2} - 16 \, b^{3}\right )} \sqrt {b} \log \left (2 \, b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {b} \tan \left (f x + e\right ) + a\right ) - 2 \, {\left (8 \, b^{3} \tan \left (f x + e\right )^{5} + 2 \, {\left (a b^{2} - 6 \, b^{3}\right )} \tan \left (f x + e\right )^{3} - 3 \, {\left (a^{2} b + 2 \, a b^{2} - 8 \, b^{3}\right )} \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{96 \, b^{3} f}, \frac {24 \, \sqrt {-a + b} b^{3} \log \left (-\frac {{\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b} \tan \left (f x + e\right ) - a}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 \, {\left (a^{3} + 2 \, a^{2} b + 8 \, a b^{2} - 16 \, b^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-b}}{b \tan \left (f x + e\right )}\right ) + {\left (8 \, b^{3} \tan \left (f x + e\right )^{5} + 2 \, {\left (a b^{2} - 6 \, b^{3}\right )} \tan \left (f x + e\right )^{3} - 3 \, {\left (a^{2} b + 2 \, a b^{2} - 8 \, b^{3}\right )} \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{48 \, b^{3} f}, -\frac {48 \, \sqrt {a - b} b^{3} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{\sqrt {a - b} \tan \left (f x + e\right )}\right ) + 3 \, {\left (a^{3} + 2 \, a^{2} b + 8 \, a b^{2} - 16 \, b^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-b}}{b \tan \left (f x + e\right )}\right ) - {\left (8 \, b^{3} \tan \left (f x + e\right )^{5} + 2 \, {\left (a b^{2} - 6 \, b^{3}\right )} \tan \left (f x + e\right )^{3} - 3 \, {\left (a^{2} b + 2 \, a b^{2} - 8 \, b^{3}\right )} \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{48 \, b^{3} f}\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 \[ \int \sqrt {b \tan \left (f x + e\right )^{2} + a} \tan \left (f x + e\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.37, size = 451, normalized size = 2.03 \[ \frac {\left (\tan ^{3}\left (f x +e \right )\right ) \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}{6 f b}-\frac {a \tan \left (f x +e \right ) \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}{8 f \,b^{2}}+\frac {a^{2} \tan \left (f x +e \right ) \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{16 f \,b^{2}}+\frac {a^{3} \ln \left (\tan \left (f x +e \right ) \sqrt {b}+\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}\right )}{16 f \,b^{\frac {5}{2}}}-\frac {\tan \left (f x +e \right ) \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}{4 f b}+\frac {a \tan \left (f x +e \right ) \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{8 f b}+\frac {a^{2} \ln \left (\tan \left (f x +e \right ) \sqrt {b}+\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}\right )}{8 f \,b^{\frac {3}{2}}}+\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}\, \tan \left (f x +e \right )}{2 f}+\frac {a \ln \left (\tan \left (f x +e \right ) \sqrt {b}+\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}\right )}{2 f \sqrt {b}}-\frac {\sqrt {b}\, \ln \left (\tan \left (f x +e \right ) \sqrt {b}+\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}\right )}{f}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {\left (a -b \right ) b^{2} \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}\right )}{f b \left (a -b \right )}-\frac {a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {\left (a -b \right ) b^{2} \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}\right )}{f \,b^{2} \left (a -b \right )} \]
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 \[ \int \sqrt {b \tan \left (f x + e\right )^{2} + a} \tan \left (f x + e\right )^{6}\,{d x} \]
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 \[ \int {\mathrm {tan}\left (e+f\,x\right )}^6\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a} \,d x \]
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 \[ \int \sqrt {a + b \tan ^{2}{\left (e + f x \right )}} \tan ^{6}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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