Optimal. Leaf size=210 \[ -\frac {b^3 (4 a-3 b)}{2 a^4 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )}+\frac {(a+3 b) \cot ^2(e+f x)}{2 a^4 f}-\frac {b^3}{4 a^3 f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}-\frac {\cot ^4(e+f x)}{4 a^3 f}+\frac {\left (a^2+3 a b+6 b^2\right ) \log (\tan (e+f x))}{a^5 f}+\frac {b^3 \left (10 a^2-15 a b+6 b^2\right ) \log \left (a+b \tan ^2(e+f x)\right )}{2 a^5 f (a-b)^3}+\frac {\log (\cos (e+f x))}{f (a-b)^3} \]
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Rubi [A] time = 0.24, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3670, 446, 88} \[ -\frac {b^3 (4 a-3 b)}{2 a^4 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )}-\frac {b^3}{4 a^3 f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}+\frac {b^3 \left (10 a^2-15 a b+6 b^2\right ) \log \left (a+b \tan ^2(e+f x)\right )}{2 a^5 f (a-b)^3}+\frac {\left (a^2+3 a b+6 b^2\right ) \log (\tan (e+f x))}{a^5 f}+\frac {(a+3 b) \cot ^2(e+f x)}{2 a^4 f}-\frac {\cot ^4(e+f x)}{4 a^3 f}+\frac {\log (\cos (e+f x))}{f (a-b)^3} \]
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rule 3670
Rubi steps
\begin {align*} \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^5 \left (1+x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^3 (1+x) (a+b x)^3} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^3 x^3}+\frac {-a-3 b}{a^4 x^2}+\frac {a^2+3 a b+6 b^2}{a^5 x}-\frac {1}{(a-b)^3 (1+x)}+\frac {b^4}{a^3 (a-b) (a+b x)^3}+\frac {(4 a-3 b) b^4}{a^4 (a-b)^2 (a+b x)^2}+\frac {b^4 \left (10 a^2-15 a b+6 b^2\right )}{a^5 (a-b)^3 (a+b x)}\right ) \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac {(a+3 b) \cot ^2(e+f x)}{2 a^4 f}-\frac {\cot ^4(e+f x)}{4 a^3 f}+\frac {\log (\cos (e+f x))}{(a-b)^3 f}+\frac {\left (a^2+3 a b+6 b^2\right ) \log (\tan (e+f x))}{a^5 f}+\frac {b^3 \left (10 a^2-15 a b+6 b^2\right ) \log \left (a+b \tan ^2(e+f x)\right )}{2 a^5 (a-b)^3 f}-\frac {b^3}{4 a^3 (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {(4 a-3 b) b^3}{2 a^4 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )}\\ \end {align*}
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Mathematica [A] time = 2.65, size = 178, normalized size = 0.85 \[ \frac {\frac {(a+3 b) \cot ^2(e+f x)}{a^4}-\frac {\cot ^4(e+f x)}{2 a^3}+\frac {4 \left (a^2+3 a b+6 b^2\right ) \log (\tan (e+f x))+\frac {b^3 \left (2 \left (10 a^2-15 a b+6 b^2\right ) \log \left (a+b \tan ^2(e+f x)\right )-\frac {a (a-b) \left (2 b (4 a-3 b) \tan ^2(e+f x)+a (9 a-7 b)\right )}{\left (a+b \tan ^2(e+f x)\right )^2}\right )}{(a-b)^3}}{2 a^5}+\frac {2 \log (\cos (e+f x))}{(a-b)^3}}{2 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 611, normalized size = 2.91 \[ \frac {3 \, {\left (a^{5} b^{2} - a^{4} b^{3} - 3 \, a^{3} b^{4} + 8 \, a^{2} b^{5} - 4 \, a b^{6}\right )} \tan \left (f x + e\right )^{8} - a^{7} + 3 \, a^{6} b - 3 \, a^{5} b^{2} + a^{4} b^{3} + 2 \, {\left (3 \, a^{6} b - 2 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 14 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - 6 \, a b^{6}\right )} \tan \left (f x + e\right )^{6} + {\left (3 \, a^{7} + a^{6} b - 10 \, a^{5} b^{2} - 6 \, a^{4} b^{3} + 33 \, a^{3} b^{4} - 18 \, a^{2} b^{5}\right )} \tan \left (f x + e\right )^{4} + 2 \, {\left (a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left ({\left (a^{5} b^{2} - 10 \, a^{2} b^{5} + 15 \, a b^{6} - 6 \, b^{7}\right )} \tan \left (f x + e\right )^{8} + 2 \, {\left (a^{6} b - 10 \, a^{3} b^{4} + 15 \, a^{2} b^{5} - 6 \, a b^{6}\right )} \tan \left (f x + e\right )^{6} + {\left (a^{7} - 10 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5}\right )} \tan \left (f x + e\right )^{4}\right )} \log \left (\frac {\tan \left (f x + e\right )^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \, {\left ({\left (10 \, a^{2} b^{5} - 15 \, a b^{6} + 6 \, b^{7}\right )} \tan \left (f x + e\right )^{8} + 2 \, {\left (10 \, a^{3} b^{4} - 15 \, a^{2} b^{5} + 6 \, a b^{6}\right )} \tan \left (f x + e\right )^{6} + {\left (10 \, a^{4} b^{3} - 15 \, a^{3} b^{4} + 6 \, a^{2} b^{5}\right )} \tan \left (f x + e\right )^{4}\right )} \log \left (\frac {b \tan \left (f x + e\right )^{2} + a}{\tan \left (f x + e\right )^{2} + 1}\right )}{4 \, {\left ({\left (a^{8} b^{2} - 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} - a^{5} b^{5}\right )} f \tan \left (f x + e\right )^{8} + 2 \, {\left (a^{9} b - 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} - a^{6} b^{4}\right )} f \tan \left (f x + e\right )^{6} + {\left (a^{10} - 3 \, a^{9} b + 3 \, a^{8} b^{2} - a^{7} b^{3}\right )} f \tan \left (f x + e\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.92, size = 477, normalized size = 2.27 \[ \frac {5 b^{4}}{2 f \,a^{3} \left (a -b \right )^{3} \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )}-\frac {3 b^{5}}{2 f \,a^{4} \left (a -b \right )^{3} \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )}+\frac {5 b^{3} \ln \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )}{f \,a^{3} \left (a -b \right )^{3}}-\frac {15 b^{4} \ln \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )}{2 f \,a^{4} \left (a -b \right )^{3}}+\frac {3 b^{5} \ln \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )}{f \,a^{5} \left (a -b \right )^{3}}-\frac {b^{5}}{4 f \,a^{3} \left (a -b \right )^{3} \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )^{2}}-\frac {1}{16 f \,a^{3} \left (-1+\cos \left (f x +e \right )\right )^{2}}-\frac {7}{16 f \,a^{3} \left (-1+\cos \left (f x +e \right )\right )}-\frac {3 b}{4 f \,a^{4} \left (-1+\cos \left (f x +e \right )\right )}+\frac {\ln \left (-1+\cos \left (f x +e \right )\right )}{2 f \,a^{3}}+\frac {3 \ln \left (-1+\cos \left (f x +e \right )\right ) b}{2 f \,a^{4}}+\frac {3 \ln \left (-1+\cos \left (f x +e \right )\right ) b^{2}}{f \,a^{5}}-\frac {1}{16 f \,a^{3} \left (1+\cos \left (f x +e \right )\right )^{2}}+\frac {7}{16 f \,a^{3} \left (1+\cos \left (f x +e \right )\right )}+\frac {3 b}{4 f \,a^{4} \left (1+\cos \left (f x +e \right )\right )}+\frac {\ln \left (1+\cos \left (f x +e \right )\right )}{2 f \,a^{3}}+\frac {3 \ln \left (1+\cos \left (f x +e \right )\right ) b}{2 f \,a^{4}}+\frac {3 \ln \left (1+\cos \left (f x +e \right )\right ) b^{2}}{f \,a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 416, normalized size = 1.98 \[ \frac {\frac {2 \, {\left (10 \, a^{2} b^{3} - 15 \, a b^{4} + 6 \, b^{5}\right )} \log \left (-{\left (a - b\right )} \sin \left (f x + e\right )^{2} + a\right )}{a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}} + \frac {2 \, {\left (2 \, a^{6} - 7 \, a^{5} b + 5 \, a^{4} b^{2} + 10 \, a^{3} b^{3} - 25 \, a^{2} b^{4} + 21 \, a b^{5} - 6 \, b^{6}\right )} \sin \left (f x + e\right )^{6} - a^{6} + 3 \, a^{5} b - 3 \, a^{4} b^{2} + a^{3} b^{3} - {\left (9 \, a^{6} - 25 \, a^{5} b + 10 \, a^{4} b^{2} + 30 \, a^{3} b^{3} - 45 \, a^{2} b^{4} + 18 \, a b^{5}\right )} \sin \left (f x + e\right )^{4} + 2 \, {\left (3 \, a^{6} - 7 \, a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - 2 \, a^{2} b^{4}\right )} \sin \left (f x + e\right )^{2}}{{\left (a^{9} - 5 \, a^{8} b + 10 \, a^{7} b^{2} - 10 \, a^{6} b^{3} + 5 \, a^{5} b^{4} - a^{4} b^{5}\right )} \sin \left (f x + e\right )^{8} - 2 \, {\left (a^{9} - 4 \, a^{8} b + 6 \, a^{7} b^{2} - 4 \, a^{6} b^{3} + a^{5} b^{4}\right )} \sin \left (f x + e\right )^{6} + {\left (a^{9} - 3 \, a^{8} b + 3 \, a^{7} b^{2} - a^{6} b^{3}\right )} \sin \left (f x + e\right )^{4}} + \frac {2 \, {\left (a^{2} + 3 \, a b + 6 \, b^{2}\right )} \log \left (\sin \left (f x + e\right )^{2}\right )}{a^{5}}}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.51, size = 269, normalized size = 1.28 \[ \frac {\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (a+2\,b\right )}{2\,a^2}-\frac {1}{4\,a}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (a^3\,b^2+a^2\,b^3-9\,a\,b^4+6\,b^5\right )}{2\,a^4\,\left (a^2-2\,a\,b+b^2\right )}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (4\,a^3\,b+3\,a^2\,b^2-27\,a\,b^3+18\,b^4\right )}{4\,a^3\,\left (a^2-2\,a\,b+b^2\right )}}{f\,\left (a^2\,{\mathrm {tan}\left (e+f\,x\right )}^4+2\,a\,b\,{\mathrm {tan}\left (e+f\,x\right )}^6+b^2\,{\mathrm {tan}\left (e+f\,x\right )}^8\right )}-\frac {\ln \left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )\,\left (\frac {3\,b}{2\,a^4}+\frac {1}{2\,a^3}-\frac {1}{2\,{\left (a-b\right )}^3}+\frac {3\,b^2}{a^5}\right )}{f}-\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )}{2\,f\,{\left (a-b\right )}^3}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )\right )\,\left (a^2+3\,a\,b+6\,b^2\right )}{a^5\,f} \]
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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