Optimal. Leaf size=249 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{(a-b)^{5/2} f}-\frac {b \cot ^3(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {(3 a-2 b) b \cot ^3(e+f x)}{a^2 (a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}}+\frac {(a-2 b) \left (3 a^2+8 a b-8 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^4 (a-b)^2 f}-\frac {\left (a^2-12 a b+8 b^2\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^3 (a-b)^2 f} \]
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Rubi [A]
time = 0.24, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3751, 483, 593,
597, 12, 385, 209} \begin {gather*} -\frac {b (3 a-2 b) \cot ^3(e+f x)}{a^2 f (a-b)^2 \sqrt {a+b \tan ^2(e+f x)}}+\frac {(a-2 b) \left (3 a^2+8 a b-8 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^4 f (a-b)^2}-\frac {\left (a^2-12 a b+8 b^2\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^3 f (a-b)^2}+\frac {\text {ArcTan}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f (a-b)^{5/2}}-\frac {b \cot ^3(e+f x)}{3 a f (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 209
Rule 385
Rule 483
Rule 593
Rule 597
Rule 3751
Rubi steps
\begin {align*} \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^4 \left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {b \cot ^3(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {3 (a-2 b)-6 b x^2}{x^4 \left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 a (a-b) f}\\ &=-\frac {b \cot ^3(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {(3 a-2 b) b \cot ^3(e+f x)}{a^2 (a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {3 \left (a^2-12 a b+8 b^2\right )-12 (3 a-2 b) b x^2}{x^4 \left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{3 a^2 (a-b)^2 f}\\ &=-\frac {b \cot ^3(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {(3 a-2 b) b \cot ^3(e+f x)}{a^2 (a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\left (a^2-12 a b+8 b^2\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^3 (a-b)^2 f}-\frac {\text {Subst}\left (\int \frac {3 (a-2 b) \left (3 a^2+8 a b-8 b^2\right )+6 b \left (a^2-12 a b+8 b^2\right ) x^2}{x^2 \left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{9 a^3 (a-b)^2 f}\\ &=-\frac {b \cot ^3(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {(3 a-2 b) b \cot ^3(e+f x)}{a^2 (a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}}+\frac {(a-2 b) \left (3 a^2+8 a b-8 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^4 (a-b)^2 f}-\frac {\left (a^2-12 a b+8 b^2\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^3 (a-b)^2 f}+\frac {\text {Subst}\left (\int \frac {9 a^4}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{9 a^4 (a-b)^2 f}\\ &=-\frac {b \cot ^3(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {(3 a-2 b) b \cot ^3(e+f x)}{a^2 (a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}}+\frac {(a-2 b) \left (3 a^2+8 a b-8 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^4 (a-b)^2 f}-\frac {\left (a^2-12 a b+8 b^2\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^3 (a-b)^2 f}+\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{(a-b)^2 f}\\ &=-\frac {b \cot ^3(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {(3 a-2 b) b \cot ^3(e+f x)}{a^2 (a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}}+\frac {(a-2 b) \left (3 a^2+8 a b-8 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^4 (a-b)^2 f}-\frac {\left (a^2-12 a b+8 b^2\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^3 (a-b)^2 f}+\frac {\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 )}{(a-b)^2 f}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{(a-b)^{5/2} f}-\frac {b \cot ^3(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {(3 a-2 b) b \cot ^3(e+f x)}{a^2 (a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}}+\frac {(a-2 b) \left (3 a^2+8 a b-8 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^4 (a-b)^2 f}-\frac {\left (a^2-12 a b+8 b^2\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^3 (a-b)^2 f}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 16.39, size = 871, normalized size = 3.50 \begin {gather*} \frac {-\frac {b \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)) F\left (\left .\text {ArcSin}\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 \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)) F\left (\left .\text {ArcSin}\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 {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)) \Pi \left (-\frac {b}{a-b};\left .\text {ArcSin}\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 {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))}}}{(a-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 {4 (a \cos (e+f x)+2 b \cos (e+f x)) \csc (e+f x)}{3 a^4}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a^3}+\frac {2 b^4 \sin (2 (e+f x))}{3 a^3 (a-b)^2 (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))^2}-\frac {4 \left (3 a b^3 \sin (2 (e+f x))-2 b^4 \sin (2 (e+f x))\right )}{3 a^4 (a-b)^2 (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))}\right )}{f} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.34, size = 0, normalized size = 0.00 \[\int \frac {\cot ^{4}\left (f x +e \right )}{\left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.61, size = 909, normalized size = 3.65 \begin {gather*} \left [-\frac {3 \, {\left (a^{4} b^{2} \tan \left (f x + e\right )^{7} + 2 \, a^{5} b \tan \left (f x + e\right )^{5} + a^{6} \tan \left (f x + e\right )^{3}\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 ) - 4 \, {\left ({\left (3 \, a^{4} b^{2} - a^{3} b^{3} - 26 \, a^{2} b^{4} + 40 \, a b^{5} - 16 \, b^{6}\right )} \tan \left (f x + e\right )^{6} - a^{6} + 3 \, a^{5} b - 3 \, a^{4} b^{2} + a^{3} b^{3} + 3 \, {\left (2 \, a^{5} b - a^{4} b^{2} - 13 \, a^{3} b^{3} + 20 \, a^{2} b^{4} - 8 \, a b^{5}\right )} \tan \left (f x + e\right )^{4} + 3 \, {\left (a^{6} - a^{5} b - 3 \, a^{4} b^{2} + 5 \, a^{3} b^{3} - 2 \, a^{2} b^{4}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{12 \, {\left ({\left (a^{7} b^{2} - 3 \, a^{6} b^{3} + 3 \, a^{5} b^{4} - a^{4} b^{5}\right )} f \tan \left (f x + e\right )^{7} + 2 \, {\left (a^{8} b - 3 \, a^{7} b^{2} + 3 \, a^{6} b^{3} - a^{5} b^{4}\right )} f \tan \left (f x + e\right )^{5} + {\left (a^{9} - 3 \, a^{8} b + 3 \, a^{7} b^{2} - a^{6} b^{3}\right )} f \tan \left (f x + e\right )^{3}\right )}}, \frac {3 \, {\left (a^{4} b^{2} \tan \left (f x + e\right )^{7} + 2 \, a^{5} b \tan \left (f x + e\right )^{5} + a^{6} \tan \left (f x + e\right )^{3}\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 ) + 2 \, {\left ({\left (3 \, a^{4} b^{2} - a^{3} b^{3} - 26 \, a^{2} b^{4} + 40 \, a b^{5} - 16 \, b^{6}\right )} \tan \left (f x + e\right )^{6} - a^{6} + 3 \, a^{5} b - 3 \, a^{4} b^{2} + a^{3} b^{3} + 3 \, {\left (2 \, a^{5} b - a^{4} b^{2} - 13 \, a^{3} b^{3} + 20 \, a^{2} b^{4} - 8 \, a b^{5}\right )} \tan \left (f x + e\right )^{4} + 3 \, {\left (a^{6} - a^{5} b - 3 \, a^{4} b^{2} + 5 \, a^{3} b^{3} - 2 \, a^{2} b^{4}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{6 \, {\left ({\left (a^{7} b^{2} - 3 \, a^{6} b^{3} + 3 \, a^{5} b^{4} - a^{4} b^{5}\right )} f \tan \left (f x + e\right )^{7} + 2 \, {\left (a^{8} b - 3 \, a^{7} b^{2} + 3 \, a^{6} b^{3} - a^{5} b^{4}\right )} f \tan \left (f x + e\right )^{5} + {\left (a^{9} - 3 \, a^{8} b + 3 \, a^{7} b^{2} - a^{6} b^{3}\right )} f \tan \left (f x + e\right )^{3}\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\cot ^{4}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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