Optimal. Leaf size=51 \[ -a x-\frac {a \cot (e+f x)}{f}+\frac {a \cot ^3(e+f x)}{3 f}-\frac {(a+b) \cot ^5(e+f x)}{5 f} \]
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Rubi [A]
time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4226, 1816,
209} \begin {gather*} -\frac {(a+b) \cot ^5(e+f x)}{5 f}+\frac {a \cot ^3(e+f x)}{3 f}-\frac {a \cot (e+f x)}{f}-a x \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 1816
Rule 4226
Rubi steps
\begin {align*} \int \cot ^6(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \left (1+x^2\right )}{x^6 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a+b}{x^6}-\frac {a}{x^4}+\frac {a}{x^2}-\frac {a}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {a \cot (e+f x)}{f}+\frac {a \cot ^3(e+f x)}{3 f}-\frac {(a+b) \cot ^5(e+f x)}{5 f}-\frac {a \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-a x-\frac {a \cot (e+f x)}{f}+\frac {a \cot ^3(e+f x)}{3 f}-\frac {(a+b) \cot ^5(e+f x)}{5 f}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.04, size = 51, normalized size = 1.00 \begin {gather*} -\frac {b \cot ^5(e+f x)}{5 f}-\frac {a \cot ^5(e+f x) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\tan ^2(e+f x)\right )}{5 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 63, normalized size = 1.24
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\left (\cot ^{5}\left (f x +e \right )\right )}{5}+\frac {\left (\cot ^{3}\left (f x +e \right )\right )}{3}-\cot \left (f x +e \right )-f x -e \right )-\frac {b \left (\cos ^{5}\left (f x +e \right )\right )}{5 \sin \left (f x +e \right )^{5}}}{f}\) | \(63\) |
default | \(\frac {a \left (-\frac {\left (\cot ^{5}\left (f x +e \right )\right )}{5}+\frac {\left (\cot ^{3}\left (f x +e \right )\right )}{3}-\cot \left (f x +e \right )-f x -e \right )-\frac {b \left (\cos ^{5}\left (f x +e \right )\right )}{5 \sin \left (f x +e \right )^{5}}}{f}\) | \(63\) |
risch | \(-a x -\frac {2 i \left (45 a \,{\mathrm e}^{8 i \left (f x +e \right )}+15 b \,{\mathrm e}^{8 i \left (f x +e \right )}-90 a \,{\mathrm e}^{6 i \left (f x +e \right )}+140 a \,{\mathrm e}^{4 i \left (f x +e \right )}+30 b \,{\mathrm e}^{4 i \left (f x +e \right )}-70 a \,{\mathrm e}^{2 i \left (f x +e \right )}+23 a +3 b \right )}{15 f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{5}}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 56, normalized size = 1.10 \begin {gather*} -\frac {15 \, {\left (f x + e\right )} a + \frac {15 \, a \tan \left (f x + e\right )^{4} - 5 \, a \tan \left (f x + e\right )^{2} + 3 \, a + 3 \, b}{\tan \left (f x + e\right )^{5}}}{15 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs.
\(2 (50) = 100\).
time = 3.40, size = 119, normalized size = 2.33 \begin {gather*} -\frac {{\left (23 \, a + 3 \, b\right )} \cos \left (f x + e\right )^{5} - 35 \, a \cos \left (f x + e\right )^{3} + 15 \, a \cos \left (f x + e\right ) + 15 \, {\left (a f x \cos \left (f x + e\right )^{4} - 2 \, a f x \cos \left (f x + e\right )^{2} + a f x\right )} \sin \left (f x + e\right )}{15 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f\right )} \sin \left (f x + e\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 \left (a + b \sec ^{2}{\left (e + f x \right )}\right ) \cot ^{6}{\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 170 vs.
\(2 (47) = 94\).
time = 0.48, size = 170, normalized size = 3.33 \begin {gather*} \frac {3 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 35 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 15 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 480 \, {\left (f x + e\right )} a + 330 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 30 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \frac {330 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 35 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, a + 3 \, b}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5}}}{480 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.89, size = 46, normalized size = 0.90 \begin {gather*} -a\,x-\frac {a\,{\mathrm {tan}\left (e+f\,x\right )}^4-\frac {a\,{\mathrm {tan}\left (e+f\,x\right )}^2}{3}+\frac {a}{5}+\frac {b}{5}}{f\,{\mathrm {tan}\left (e+f\,x\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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