Optimal. Leaf size=56 \[ -\frac {a^2 \cot ^2(e+f x)}{2 f}-\frac {(a-b)^2 \log (\cos (e+f x))}{f}-\frac {a (a-2 b) \log (\tan (e+f x))}{f} \]
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Rubi [A]
time = 0.06, antiderivative size = 56, 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 = {3751, 457, 90}
\begin {gather*} -\frac {a^2 \cot ^2(e+f x)}{2 f}-\frac {a (a-2 b) \log (\tan (e+f x))}{f}-\frac {(a-b)^2 \log (\cos (e+f x))}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 457
Rule 3751
Rubi steps
\begin {align*} \int \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^2}{x^3 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {(a+b x)^2}{x^2 (1+x)} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^2}{x^2}-\frac {a (a-2 b)}{x}+\frac {(a-b)^2}{1+x}\right ) \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=-\frac {a^2 \cot ^2(e+f x)}{2 f}-\frac {(a-b)^2 \log (\cos (e+f x))}{f}-\frac {a (a-2 b) \log (\tan (e+f x))}{f}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 51, normalized size = 0.91 \begin {gather*} -\frac {a^2 \cot ^2(e+f x)+2 (a-b)^2 \log (\cos (e+f x))+2 a (a-2 b) \log (\tan (e+f x))}{2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 53, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {-b^{2} \ln \left (\cos \left (f x +e \right )\right )+2 a b \ln \left (\sin \left (f x +e \right )\right )+a^{2} \left (-\frac {\left (\cot ^{2}\left (f x +e \right )\right )}{2}-\ln \left (\sin \left (f x +e \right )\right )\right )}{f}\) | \(53\) |
default | \(\frac {-b^{2} \ln \left (\cos \left (f x +e \right )\right )+2 a b \ln \left (\sin \left (f x +e \right )\right )+a^{2} \left (-\frac {\left (\cot ^{2}\left (f x +e \right )\right )}{2}-\ln \left (\sin \left (f x +e \right )\right )\right )}{f}\) | \(53\) |
norman | \(-\frac {a^{2}}{2 f \tan \left (f x +e \right )^{2}}+\frac {\left (a^{2}-2 a b +b^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f}-\frac {a \left (a -2 b \right ) \ln \left (\tan \left (f x +e \right )\right )}{f}\) | \(63\) |
risch | \(i a^{2} x -2 i a b x +i b^{2} x +\frac {2 i b^{2} e}{f}+\frac {2 i a^{2} e}{f}-\frac {4 i a b e}{f}+\frac {2 a^{2} {\mathrm e}^{2 i \left (f x +e \right )}}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) b^{2}}{f}-\frac {a^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}{f}+\frac {2 a \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right ) b}{f}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 54, normalized size = 0.96 \begin {gather*} -\frac {b^{2} \log \left (\sin \left (f x + e\right )^{2} - 1\right ) + {\left (a^{2} - 2 \, a b\right )} \log \left (\sin \left (f x + e\right )^{2}\right ) + \frac {a^{2}}{\sin \left (f x + e\right )^{2}}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.98, size = 100, normalized size = 1.79 \begin {gather*} -\frac {b^{2} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{2} + a^{2} \tan \left (f x + e\right )^{2} + {\left (a^{2} - 2 \, a b\right )} \log \left (\frac {\tan \left (f x + e\right )^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{2} + a^{2}}{2 \, f \tan \left (f x + e\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 131 vs.
\(2 (48) = 96\).
time = 1.80, size = 131, normalized size = 2.34 \begin {gather*} \begin {cases} \tilde {\infty } a^{2} x & \text {for}\: \left (e = 0 \vee e = - f x\right ) \wedge \left (e = - f x \vee f = 0\right ) \\x \left (a + b \tan ^{2}{\left (e \right )}\right )^{2} \cot ^{3}{\left (e \right )} & \text {for}\: f = 0 \\\frac {a^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - \frac {a^{2} \log {\left (\tan {\left (e + f x \right )} \right )}}{f} - \frac {a^{2}}{2 f \tan ^{2}{\left (e + f x \right )}} - \frac {a b \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} + \frac {2 a b \log {\left (\tan {\left (e + f x \right )} \right )}}{f} + \frac {b^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} & \text {otherwise} \end {cases} \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. 167 vs.
\(2 (57) = 114\).
time = 1.15, size = 167, normalized size = 2.98 \begin {gather*} \frac {a^{2} {\left (\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} + \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}\right )} - 4 \, b^{2} \log \left ({\left | -\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} - \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} - 2 \right |}\right ) + 4 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \log \left ({\left | -\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} - \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 2 \right |}\right )}{8 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 12.09, size = 68, normalized size = 1.21 \begin {gather*} \frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )\,\left (\frac {a^2}{2}-a\,b+\frac {b^2}{2}\right )}{f}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )\right )\,\left (2\,a\,b-a^2\right )}{f}-\frac {a^2\,{\mathrm {cot}\left (e+f\,x\right )}^2}{2\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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