Optimal. Leaf size=167 \[ -\frac {\left (15 a^2-5 a b-2 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^2 f}-\frac {\sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}-\frac {\cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}+\frac {(5 a-b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f} \]
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Rubi [A] time = 0.23, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3670, 475, 583, 12, 377, 203} \[ -\frac {\left (15 a^2-5 a b-2 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^2 f}-\frac {\sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}-\frac {\cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}+\frac {(5 a-b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f} \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 377
Rule 475
Rule 583
Rule 3670
Rubi steps
\begin {align*} \int \cot ^6(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2}}{x^6 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}+\frac {\operatorname {Subst}\left (\int \frac {-5 a+b-4 b x^2}{x^4 \left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{5 f}\\ &=\frac {(5 a-b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}-\frac {\cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}-\frac {\operatorname {Subst}\left (\int \frac {-15 a^2+5 a b+2 b^2-2 (5 a-b) b x^2}{x^2 \left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{15 a f}\\ &=-\frac {\left (15 a^2-5 a b-2 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^2 f}+\frac {(5 a-b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}-\frac {\cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}+\frac {\operatorname {Subst}\left (\int -\frac {15 a^2 (a-b)}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{15 a^2 f}\\ &=-\frac {\left (15 a^2-5 a b-2 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^2 f}+\frac {(5 a-b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}-\frac {\cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 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 (15 a^2-5 a b-2 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^2 f}+\frac {(5 a-b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}-\frac {\cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 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 {\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 (15 a^2-5 a b-2 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^2 f}+\frac {(5 a-b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}-\frac {\cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}\\ \end {align*}
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Mathematica [C] time = 14.39, size = 339, normalized size = 2.03 \[ -\frac {\cos ^4(e+f x) \cot ^5(e+f x) \left (\frac {b \tan ^2(e+f x)}{a}+1\right ) \left (8 (a-b) \tan ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^3 \, _3F_2\left (2,2,2;1,\frac {3}{2};\frac {(a-b) \sin ^2(e+f x)}{a}\right )+8 \tan ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \left (-2 a^2+a b \left (3 \tan ^2(e+f x)+2\right )-3 b^2 \tan ^2(e+f x)\right ) \, _2F_1\left (2,2;\frac {3}{2};\frac {(a-b) \sin ^2(e+f x)}{a}\right )+\frac {a^2 \sec ^4(e+f x) \left (3 a^2-4 a b \tan ^2(e+f x)+8 b^2 \tan ^4(e+f x)\right ) \left (\sqrt {\frac {(a-b) \sin ^2(e+f x)}{a}} \sin ^{-1}\left (\sqrt {\frac {(a-b) \sin ^2(e+f x)}{a}}\right )+\sqrt {\frac {\cos ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )}{a}}\right )}{\sqrt {\frac {\cos ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )}{a}}}\right )}{15 a^3 f \sqrt {a+b \tan ^2(e+f x)}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.54, size = 375, normalized size = 2.25 \[ \left [\frac {15 \, a^{2} \sqrt {-a + b} \log \left (-\frac {{\left (a^{2} - 8 \, a b + 8 \, b^{2}\right )} \tan \left (f x + e\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (f x + e\right )^{2} + a^{2} - 4 \, {\left ({\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{3} - a \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{5} - 4 \, {\left ({\left (15 \, a^{2} - 5 \, a b - 2 \, b^{2}\right )} \tan \left (f x + e\right )^{4} - {\left (5 \, a^{2} - a b\right )} \tan \left (f x + e\right )^{2} + 3 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{60 \, a^{2} f \tan \left (f x + e\right )^{5}}, -\frac {15 \, \sqrt {a - b} a^{2} \arctan \left (-\frac {2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} \tan \left (f x + e\right )}{{\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{2} - a}\right ) \tan \left (f x + e\right )^{5} + 2 \, {\left ({\left (15 \, a^{2} - 5 \, a b - 2 \, b^{2}\right )} \tan \left (f x + e\right )^{4} - {\left (5 \, a^{2} - a b\right )} \tan \left (f x + e\right )^{2} + 3 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{30 \, a^{2} f \tan \left (f x + e\right )^{5}}\right ] \]
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 \[ \int \sqrt {b \tan \left (f x + e\right )^{2} + a} \cot \left (f x + e\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.36, size = 6894, normalized size = 41.28 \[ \text {output too large to display} \]
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 \[ \int \sqrt {b \tan \left (f x + e\right )^{2} + a} \cot \left (f x + e\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {cot}\left (e+f\,x\right )}^6\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a} \,d x \]
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 \[ \int \sqrt {a + b \tan ^{2}{\left (e + f x \right )}} \cot ^{6}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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