Optimal. Leaf size=28 \[ \frac {(a+b) \log (\sin (e+f x))}{f}-\frac {b \log (\cos (e+f x))}{f} \]
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Rubi [A] time = 0.05, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4138, 446, 72} \[ \frac {(a+b) \log (\sin (e+f x))}{f}-\frac {b \log (\cos (e+f x))}{f} \]
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rule 4138
Rubi steps
\begin {align*} \int \cot (e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {b+a x^2}{x \left (1-x^2\right )} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {b+a x}{(1-x) x} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {-a-b}{-1+x}+\frac {b}{x}\right ) \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {b \log (\cos (e+f x))}{f}+\frac {(a+b) \log (\sin (e+f x))}{f}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 44, normalized size = 1.57 \[ \frac {a (\log (\tan (e+f x))+\log (\cos (e+f x)))}{f}-\frac {b (\log (\cos (e+f x))-\log (\sin (e+f x)))}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 35, normalized size = 1.25 \[ -\frac {b \log \left (\cos \left (f x + e\right )^{2}\right ) - {\left (a + b\right )} \log \left (-\frac {1}{4} \, \cos \left (f x + e\right )^{2} + \frac {1}{4}\right )}{2 \, f} \]
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 [A] time = 0.60, size = 26, normalized size = 0.93 \[ \frac {b \ln \left (\tan \left (f x +e \right )\right )}{f}+\frac {a \ln \left (\sin \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 33, normalized size = 1.18 \[ -\frac {b \log \left (\sin \left (f x + e\right )^{2} - 1\right ) - {\left (a + b\right )} \log \left (\sin \left (f x + e\right )^{2}\right )}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.87, size = 32, normalized size = 1.14 \[ \frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )\right )\,\left (a+b\right )}{f}-\frac {a\,\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )}{2\,f} \]
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 \[ \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right ) \cot {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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