Optimal. Leaf size=46 \[ \frac {b \log \left (b+a \cos ^2(e+f x)\right )}{2 a (a+b) f}+\frac {\log (\sin (e+f x))}{(a+b) f} \]
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Rubi [A]
time = 0.06, antiderivative size = 46, 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 = {4223, 457, 78}
\begin {gather*} \frac {\log (\sin (e+f x))}{f (a+b)}+\frac {b \log \left (a \cos ^2(e+f x)+b\right )}{2 a f (a+b)} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rule 4223
Rubi steps
\begin {align*} \int \frac {\cot (e+f x)}{a+b \sec ^2(e+f x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {x^3}{\left (1-x^2\right ) \left (b+a x^2\right )} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {\text {Subst}\left (\int \frac {x}{(1-x) (b+a x)} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {\text {Subst}\left (\int \left (\frac {1}{(-a-b) (-1+x)}-\frac {b}{(a+b) (b+a x)}\right ) \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac {b \log \left (b+a \cos ^2(e+f x)\right )}{2 a (a+b) f}+\frac {\log (\sin (e+f x))}{(a+b) f}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 43, normalized size = 0.93 \begin {gather*} \frac {2 a \log (\sin (e+f x))+b \log \left (a+b-a \sin ^2(e+f x)\right )}{2 a^2 f+2 a b f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 68, normalized size = 1.48
method | result | size |
derivativedivides | \(\frac {\frac {b \ln \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )}{2 \left (a +b \right ) a}+\frac {\ln \left (\cos \left (f x +e \right )-1\right )}{2 a +2 b}+\frac {\ln \left (1+\cos \left (f x +e \right )\right )}{2 a +2 b}}{f}\) | \(68\) |
default | \(\frac {\frac {b \ln \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )}{2 \left (a +b \right ) a}+\frac {\ln \left (\cos \left (f x +e \right )-1\right )}{2 a +2 b}+\frac {\ln \left (1+\cos \left (f x +e \right )\right )}{2 a +2 b}}{f}\) | \(68\) |
risch | \(\frac {i x}{a}-\frac {2 i x}{a +b}-\frac {2 i e}{f \left (a +b \right )}-\frac {2 i b x}{a \left (a +b \right )}-\frac {2 i b e}{a f \left (a +b \right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}{f \left (a +b \right )}+\frac {b \ln \left ({\mathrm e}^{4 i \left (f x +e \right )}+\frac {2 \left (a +2 b \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{a}+1\right )}{2 a f \left (a +b \right )}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 52, normalized size = 1.13 \begin {gather*} \frac {\frac {b \log \left (a \sin \left (f x + e\right )^{2} - a - b\right )}{a^{2} + a b} + \frac {\log \left (\sin \left (f x + e\right )^{2}\right )}{a + b}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.91, size = 44, normalized size = 0.96 \begin {gather*} \frac {b \log \left (a \cos \left (f x + e\right )^{2} + b\right ) + 2 \, a \log \left (\frac {1}{2} \, \sin \left (f x + e\right )\right )}{2 \, {\left (a^{2} + a b\right )} f} \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 {\left (e + f x \right )}}{a + b \sec ^{2}{\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. 166 vs.
\(2 (44) = 88\).
time = 0.49, size = 166, normalized size = 3.61 \begin {gather*} \frac {\frac {b \log \left (a + b + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {2 \, b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}{a^{2} + a b} + \frac {\log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right )}{a + b} - \frac {2 \, \log \left ({\left | -\frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 1 \right |}\right )}{a}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.70, size = 65, normalized size = 1.41 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )\right )}{f\,\left (a+b\right )}-\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )}{2\,a\,f}+\frac {b\,\ln \left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a+b\right )}{2\,f\,\left (a^2+b\,a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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