Optimal. Leaf size=99 \[ \frac {\left (A d^2-B c d+c^2 C\right ) \log (c+d \tan (e+f x))}{d f \left (c^2+d^2\right )}-\frac {(B c-d (A-C)) \log (\cos (e+f x))}{f \left (c^2+d^2\right )}+\frac {x (A c+B d-c C)}{c^2+d^2} \]
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Rubi [A] time = 0.10, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3626, 3617, 31, 3475} \[ \frac {\left (A d^2-B c d+c^2 C\right ) \log (c+d \tan (e+f x))}{d f \left (c^2+d^2\right )}-\frac {(B c-d (A-C)) \log (\cos (e+f x))}{f \left (c^2+d^2\right )}+\frac {x (A c+B d-c C)}{c^2+d^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3475
Rule 3617
Rule 3626
Rubi steps
\begin {align*} \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{c+d \tan (e+f x)} \, dx &=\frac {(A c-c C+B d) x}{c^2+d^2}-\frac {(-B c+A d-C d) \int \tan (e+f x) \, dx}{c^2+d^2}+\frac {\left (c^2 C-B c d+A d^2\right ) \int \frac {1+\tan ^2(e+f x)}{c+d \tan (e+f x)} \, dx}{c^2+d^2}\\ &=\frac {(A c-c C+B d) x}{c^2+d^2}-\frac {(B c-(A-C) d) \log (\cos (e+f x))}{\left (c^2+d^2\right ) f}+\frac {\left (c^2 C-B c d+A d^2\right ) \operatorname {Subst}\left (\int \frac {1}{c+x} \, dx,x,d \tan (e+f x)\right )}{d \left (c^2+d^2\right ) f}\\ &=\frac {(A c-c C+B d) x}{c^2+d^2}-\frac {(B c-(A-C) d) \log (\cos (e+f x))}{\left (c^2+d^2\right ) f}+\frac {\left (c^2 C-B c d+A d^2\right ) \log (c+d \tan (e+f x))}{d \left (c^2+d^2\right ) f}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 117, normalized size = 1.18 \[ \frac {\frac {2 \left (A d^2-B c d+c^2 C\right ) \log (c+d \tan (e+f x))}{d \left (c^2+d^2\right )}+\frac {(-i A+B+i C) \log (-\tan (e+f x)+i)}{c+i d}+\frac {(i A+B-i C) \log (\tan (e+f x)+i)}{c-i d}}{2 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 118, normalized size = 1.19 \[ \frac {2 \, {\left ({\left (A - C\right )} c d + B d^{2}\right )} f x + {\left (C c^{2} - B c d + A d^{2}\right )} \log \left (\frac {d^{2} \tan \left (f x + e\right )^{2} + 2 \, c d \tan \left (f x + e\right ) + c^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) - {\left (C c^{2} + C d^{2}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right )}{2 \, {\left (c^{2} d + d^{3}\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.06, size = 109, normalized size = 1.10 \[ \frac {\frac {2 \, {\left (A c - C c + B d\right )} {\left (f x + e\right )}}{c^{2} + d^{2}} + \frac {{\left (B c - A d + C d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{2} + d^{2}} + \frac {2 \, {\left (C c^{2} - B c d + A d^{2}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{c^{2} d + d^{3}}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.26, size = 234, normalized size = 2.36 \[ \frac {d \ln \left (c +d \tan \left (f x +e \right )\right ) A}{f \left (c^{2}+d^{2}\right )}-\frac {\ln \left (c +d \tan \left (f x +e \right )\right ) B c}{f \left (c^{2}+d^{2}\right )}+\frac {\ln \left (c +d \tan \left (f x +e \right )\right ) c^{2} C}{f \left (c^{2}+d^{2}\right ) d}-\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) A d}{2 f \left (c^{2}+d^{2}\right )}+\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) B c}{2 f \left (c^{2}+d^{2}\right )}+\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) C d}{2 f \left (c^{2}+d^{2}\right )}+\frac {A \arctan \left (\tan \left (f x +e \right )\right ) c}{f \left (c^{2}+d^{2}\right )}+\frac {B \arctan \left (\tan \left (f x +e \right )\right ) d}{f \left (c^{2}+d^{2}\right )}-\frac {C \arctan \left (\tan \left (f x +e \right )\right ) c}{f \left (c^{2}+d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 106, normalized size = 1.07 \[ \frac {\frac {2 \, {\left ({\left (A - C\right )} c + B d\right )} {\left (f x + e\right )}}{c^{2} + d^{2}} + \frac {2 \, {\left (C c^{2} - B c d + A d^{2}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{2} d + d^{3}} + \frac {{\left (B c - {\left (A - C\right )} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{2} + d^{2}}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.90, size = 109, normalized size = 1.10 \[ \frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (C-A+B\,1{}\mathrm {i}\right )}{2\,f\,\left (d+c\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (B-A\,1{}\mathrm {i}+C\,1{}\mathrm {i}\right )}{2\,f\,\left (c+d\,1{}\mathrm {i}\right )}+\frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (C\,c^2-B\,c\,d+A\,d^2\right )}{d\,f\,\left (c^2+d^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.31, size = 984, normalized size = 9.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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