Optimal. Leaf size=165 \[ -\frac {\left (15 a^2-20 a b+3 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}-\frac {(a-b)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}-\frac {a \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}+\frac {(5 a-6 b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 f} \]
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Rubi [A] time = 0.24, antiderivative size = 165, 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, 474, 583, 12, 377, 203} \[ -\frac {\left (15 a^2-20 a b+3 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}-\frac {(a-b)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}-\frac {a \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}+\frac {(5 a-6 b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 f} \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 377
Rule 474
Rule 583
Rule 3670
Rubi steps
\begin {align*} \int \cot ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/2}}{x^6 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {a \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}+\frac {\operatorname {Subst}\left (\int \frac {-a (5 a-6 b)-(4 a-5 b) 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-6 b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 f}-\frac {a \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}-\frac {\operatorname {Subst}\left (\int \frac {-a \left (15 a^2-20 a b+3 b^2\right )-2 a (5 a-6 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^2-20 a b+3 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}+\frac {(5 a-6 b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 f}-\frac {a \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)^2}{\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^2-20 a b+3 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}+\frac {(5 a-6 b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 f}-\frac {a \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}-\frac {(a-b)^2 \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^2-20 a b+3 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}+\frac {(5 a-6 b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 f}-\frac {a \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}-\frac {(a-b)^2 \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 {(a-b)^{3/2} \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^2-20 a b+3 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}+\frac {(5 a-6 b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 f}-\frac {a \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}\\ \end {align*}
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Mathematica [C] time = 9.50, size = 140, normalized size = 0.85 \[ -\frac {\sin (e+f x) \cos (e+f x) \sqrt {a+b \tan ^2(e+f x)} \left (a \cot ^2(e+f x)+b\right )^2 \left (2 (a-b) ((a-b) \cos (2 (e+f x))+a+b) \, _2F_1\left (2,2;\frac {1}{2};\frac {(a-b) \sin ^2(e+f x)}{a}\right )+a \left (3 a \cot ^2(e+f x)-2 b\right ) \, _2F_1\left (1,1;-\frac {1}{2};\frac {(a-b) \sin ^2(e+f x)}{a}\right )\right )}{15 a^3 f} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.57, size = 385, normalized size = 2.33 \[ \left [-\frac {15 \, {\left (a^{2} - a b\right )} \sqrt {-a + b} \log \left (-\frac {{\left (a^{2} - 8 \, a b + 8 \, b^{2}\right )} \tan \left (f x + e\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (f x + e\right )^{2} + a^{2} + 4 \, {\left ({\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{3} - a \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{5} + 4 \, {\left ({\left (15 \, a^{2} - 20 \, a b + 3 \, b^{2}\right )} \tan \left (f x + e\right )^{4} - {\left (5 \, a^{2} - 6 \, a b\right )} \tan \left (f x + e\right )^{2} + 3 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{60 \, a f \tan \left (f x + e\right )^{5}}, -\frac {15 \, {\left (a^{2} - a b\right )} \sqrt {a - b} \arctan \left (-\frac {2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} \tan \left (f x + e\right )}{{\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{2} - a}\right ) \tan \left (f x + e\right )^{5} + 2 \, {\left ({\left (15 \, a^{2} - 20 \, a b + 3 \, b^{2}\right )} \tan \left (f x + e\right )^{4} - {\left (5 \, a^{2} - 6 \, a b\right )} \tan \left (f x + e\right )^{2} + 3 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{30 \, a f \tan \left (f x + e\right )^{5}}\right ] \]
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 \[ \int {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cot \left (f x + e\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.41, size = 10026, normalized size = 60.76 \[ \text {output too large to display} \]
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 \[ \int {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cot \left (f x + e\right )^{6}\,{d x} \]
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 \[ \int {\mathrm {cot}\left (e+f\,x\right )}^6\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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