Integrand size = 25, antiderivative size = 222 \[ \int \tan ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=-\frac {\sqrt {a-b} \arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}+\frac {\left (a^3+2 a^2 b+8 a b^2-16 b^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{16 b^{5/2} f}-\frac {(a-2 b) (a+4 b) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{16 b^2 f}+\frac {(a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{24 b f}+\frac {\tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{6 f} \]
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Time = 0.39 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3751, 489, 596, 537, 223, 212, 385, 209} \[ \int \tan ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\frac {\left (a^3+2 a^2 b+8 a b^2-16 b^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{16 b^{5/2} f}-\frac {\sqrt {a-b} \arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}-\frac {(a-2 b) (a+4 b) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{16 b^2 f}+\frac {\tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{6 f}+\frac {(a-6 b) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{24 b f} \]
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Rule 209
Rule 212
Rule 223
Rule 385
Rule 489
Rule 537
Rule 596
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \frac {\text {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 {\text {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 {\text {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 {\text {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) \text {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 ) \text {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) \text {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 ) \text {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} \arctan \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 ) \text {arctanh}\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*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 6.35 (sec) , antiderivative size = 823, normalized size of antiderivative = 3.71 \[ \int \tan ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\frac {-\frac {b \left (a^3+2 a^2 b-8 b^3\right ) \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (1+\cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right ) \sin ^4(e+f x)}{a (a+b+(a-b) \cos (2 (e+f x)))}-\frac {4 b \left (-8 a b^2+8 b^3\right ) \sqrt {1+\cos (2 (e+f x))} \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \left (\frac {\sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (1+\cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right ) \sin ^4(e+f x)}{4 a \sqrt {1+\cos (2 (e+f x))} \sqrt {a+b+(a-b) \cos (2 (e+f x))}}-\frac {\sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (1+\cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \operatorname {EllipticPi}\left (-\frac {b}{a-b},\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right ) \sin ^4(e+f x)}{2 (a-b) \sqrt {1+\cos (2 (e+f x))} \sqrt {a+b+(a-b) \cos (2 (e+f x))}}\right )}{\sqrt {a+b+(a-b) \cos (2 (e+f x))}}}{8 b^2 f}+\frac {\sqrt {\frac {a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \left (\frac {\sec ^3(e+f x) (a \sin (e+f x)-14 b \sin (e+f x))}{24 b}+\frac {\sec (e+f x) \left (-3 a^2 \sin (e+f x)-8 a b \sin (e+f x)+44 b^2 \sin (e+f x)\right )}{48 b^2}+\frac {1}{6} \sec ^4(e+f x) \tan (e+f x)\right )}{f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(428\) vs. \(2(196)=392\).
Time = 0.08 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.93
method | result | size |
derivativedivides | \(\frac {\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}\, \tan \left (f x +e \right )}{2}+\frac {a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 \sqrt {b}}+\frac {\tan \left (f x +e \right )^{3} \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{6 b}-\frac {a \left (\frac {\tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}\, \tan \left (f x +e \right )}{2}+\frac {a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 \sqrt {b}}\right )}{4 b}\right )}{2 b}-\frac {\tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{4 b}+\frac {a \left (\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}\, \tan \left (f x +e \right )}{2}+\frac {a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 \sqrt {b}}\right )}{4 b}-b \left (\frac {\ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{\sqrt {b}}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{b^{2} \left (a -b \right )}\right )-\frac {a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{b^{2} \left (a -b \right )}}{f}\) | \(429\) |
default | \(\frac {\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}\, \tan \left (f x +e \right )}{2}+\frac {a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 \sqrt {b}}+\frac {\tan \left (f x +e \right )^{3} \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{6 b}-\frac {a \left (\frac {\tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}\, \tan \left (f x +e \right )}{2}+\frac {a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 \sqrt {b}}\right )}{4 b}\right )}{2 b}-\frac {\tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{4 b}+\frac {a \left (\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}\, \tan \left (f x +e \right )}{2}+\frac {a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 \sqrt {b}}\right )}{4 b}-b \left (\frac {\ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{\sqrt {b}}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{b^{2} \left (a -b \right )}\right )-\frac {a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{b^{2} \left (a -b \right )}}{f}\) | \(429\) |
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Time = 1.21 (sec) , antiderivative size = 826, normalized size of antiderivative = 3.72 \[ \int \tan ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\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 ] \]
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\[ \int \tan ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\int \sqrt {a + b \tan ^{2}{\left (e + f x \right )}} \tan ^{6}{\left (e + f x \right )}\, dx \]
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\[ \int \tan ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\int { \sqrt {b \tan \left (f x + e\right )^{2} + a} \tan \left (f x + e\right )^{6} \,d x } \]
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Timed out. \[ \int \tan ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \tan ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^6\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a} \,d x \]
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