Optimal. Leaf size=73 \[ (A c-c C-B d) x-\frac {(B c+(A-C) d) \log (\cos (e+f x))}{f}+\frac {B d \tan (e+f x)}{f}+\frac {C (c+d \tan (e+f x))^2}{2 d f} \]
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Rubi [A]
time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3711, 3606,
3556} \begin {gather*} -\frac {(d (A-C)+B c) \log (\cos (e+f x))}{f}+x (A c-B d-c C)+\frac {B d \tan (e+f x)}{f}+\frac {C (c+d \tan (e+f x))^2}{2 d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3606
Rule 3711
Rubi steps
\begin {align*} \int (c+d \tan (e+f x)) \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac {C (c+d \tan (e+f x))^2}{2 d f}+\int (A-C+B \tan (e+f x)) (c+d \tan (e+f x)) \, dx\\ &=(A c-c C-B d) x+\frac {B d \tan (e+f x)}{f}+\frac {C (c+d \tan (e+f x))^2}{2 d f}+(B c+(A-C) d) \int \tan (e+f x) \, dx\\ &=(A c-c C-B d) x-\frac {(B c+(A-C) d) \log (\cos (e+f x))}{f}+\frac {B d \tan (e+f x)}{f}+\frac {C (c+d \tan (e+f x))^2}{2 d f}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 76, normalized size = 1.04 \begin {gather*} \frac {2 A c f x-2 (c C+B d) \text {ArcTan}(\tan (e+f x))-2 (B c+(A-C) d) \log (\cos (e+f x))+2 (c C+B d) \tan (e+f x)+C d \tan ^2(e+f x)}{2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 80, normalized size = 1.10
method | result | size |
norman | \(\left (A c -B d -c C \right ) x +\frac {\left (B d +c C \right ) \tan \left (f x +e \right )}{f}+\frac {C d \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}+\frac {\left (A d +B c -C d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f}\) | \(75\) |
derivativedivides | \(\frac {\frac {C d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+B \tan \left (f x +e \right ) d +C \tan \left (f x +e \right ) c +\frac {\left (A d +B c -C d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (A c -B d -c C \right ) \arctan \left (\tan \left (f x +e \right )\right )}{f}\) | \(80\) |
default | \(\frac {\frac {C d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+B \tan \left (f x +e \right ) d +C \tan \left (f x +e \right ) c +\frac {\left (A d +B c -C d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (A c -B d -c C \right ) \arctan \left (\tan \left (f x +e \right )\right )}{f}\) | \(80\) |
risch | \(-\frac {2 i C d e}{f}-i C d x +\frac {2 i A d e}{f}+A c x -B d x -C c x +i B c x +\frac {2 i B c e}{f}+i A d x +\frac {2 i \left (-i C d \,{\mathrm e}^{2 i \left (f x +e \right )}+B d \,{\mathrm e}^{2 i \left (f x +e \right )}+C c \,{\mathrm e}^{2 i \left (f x +e \right )}+B d +c C \right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) A d}{f}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) B c}{f}+\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) C d}{f}\) | \(181\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 78, normalized size = 1.07 \begin {gather*} \frac {C d \tan \left (f x + e\right )^{2} + 2 \, {\left ({\left (A - C\right )} c - B d\right )} {\left (f x + e\right )} + {\left (B c + {\left (A - C\right )} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 2 \, {\left (C c + B d\right )} \tan \left (f x + e\right )}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.15, size = 77, normalized size = 1.05 \begin {gather*} \frac {C d \tan \left (f x + e\right )^{2} + 2 \, {\left ({\left (A - C\right )} c - B d\right )} f x - {\left (B c + {\left (A - C\right )} d\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \, {\left (C c + B d\right )} \tan \left (f x + e\right )}{2 \, f} \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.
\(2 (60) = 120\).
time = 0.10, size = 131, normalized size = 1.79 \begin {gather*} \begin {cases} A c x + \frac {A d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {B c \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - B d x + \frac {B d \tan {\left (e + f x \right )}}{f} - C c x + \frac {C c \tan {\left (e + f x \right )}}{f} - \frac {C d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {C d \tan ^{2}{\left (e + f x \right )}}{2 f} & \text {for}\: f \neq 0 \\x \left (c + d \tan {\left (e \right )}\right ) \left (A + B \tan {\left (e \right )} + C \tan ^{2}{\left (e \right )}\right ) & \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. 918 vs.
\(2 (74) = 148\).
time = 0.86, size = 918, normalized size = 12.58 \begin {gather*} \frac {2 \, A c f x \tan \left (f x\right )^{2} \tan \left (e\right )^{2} - 2 \, C c f x \tan \left (f x\right )^{2} \tan \left (e\right )^{2} - 2 \, B d f x \tan \left (f x\right )^{2} \tan \left (e\right )^{2} - B c \log \left (\frac {4 \, {\left (\tan \left (f x\right )^{4} \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right )^{3} \tan \left (e\right ) + \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + \tan \left (f x\right )^{2} - 2 \, \tan \left (f x\right ) \tan \left (e\right ) + 1\right )}}{\tan \left (e\right )^{2} + 1}\right ) \tan \left (f x\right )^{2} \tan \left (e\right )^{2} - A d \log \left (\frac {4 \, {\left (\tan \left (f x\right )^{4} \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right )^{3} \tan \left (e\right ) + \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + \tan \left (f x\right )^{2} - 2 \, \tan \left (f x\right ) \tan \left (e\right ) + 1\right )}}{\tan \left (e\right )^{2} + 1}\right ) \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + C d \log \left (\frac {4 \, {\left (\tan \left (f x\right )^{4} \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right )^{3} \tan \left (e\right ) + \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + \tan \left (f x\right )^{2} - 2 \, \tan \left (f x\right ) \tan \left (e\right ) + 1\right )}}{\tan \left (e\right )^{2} + 1}\right ) \tan \left (f x\right )^{2} \tan \left (e\right )^{2} - 4 \, A c f x \tan \left (f x\right ) \tan \left (e\right ) + 4 \, C c f x \tan \left (f x\right ) \tan \left (e\right ) + 4 \, B d f x \tan \left (f x\right ) \tan \left (e\right ) + C d \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + 2 \, B c \log \left (\frac {4 \, {\left (\tan \left (f x\right )^{4} \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right )^{3} \tan \left (e\right ) + \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + \tan \left (f x\right )^{2} - 2 \, \tan \left (f x\right ) \tan \left (e\right ) + 1\right )}}{\tan \left (e\right )^{2} + 1}\right ) \tan \left (f x\right ) \tan \left (e\right ) + 2 \, A d \log \left (\frac {4 \, {\left (\tan \left (f x\right )^{4} \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right )^{3} \tan \left (e\right ) + \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + \tan \left (f x\right )^{2} - 2 \, \tan \left (f x\right ) \tan \left (e\right ) + 1\right )}}{\tan \left (e\right )^{2} + 1}\right ) \tan \left (f x\right ) \tan \left (e\right ) - 2 \, C d \log \left (\frac {4 \, {\left (\tan \left (f x\right )^{4} \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right )^{3} \tan \left (e\right ) + \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + \tan \left (f x\right )^{2} - 2 \, \tan \left (f x\right ) \tan \left (e\right ) + 1\right )}}{\tan \left (e\right )^{2} + 1}\right ) \tan \left (f x\right ) \tan \left (e\right ) - 2 \, C c \tan \left (f x\right )^{2} \tan \left (e\right ) - 2 \, B d \tan \left (f x\right )^{2} \tan \left (e\right ) - 2 \, C c \tan \left (f x\right ) \tan \left (e\right )^{2} - 2 \, B d \tan \left (f x\right ) \tan \left (e\right )^{2} + 2 \, A c f x - 2 \, C c f x - 2 \, B d f x + C d \tan \left (f x\right )^{2} + C d \tan \left (e\right )^{2} - B c \log \left (\frac {4 \, {\left (\tan \left (f x\right )^{4} \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right )^{3} \tan \left (e\right ) + \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + \tan \left (f x\right )^{2} - 2 \, \tan \left (f x\right ) \tan \left (e\right ) + 1\right )}}{\tan \left (e\right )^{2} + 1}\right ) - A d \log \left (\frac {4 \, {\left (\tan \left (f x\right )^{4} \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right )^{3} \tan \left (e\right ) + \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + \tan \left (f x\right )^{2} - 2 \, \tan \left (f x\right ) \tan \left (e\right ) + 1\right )}}{\tan \left (e\right )^{2} + 1}\right ) + C d \log \left (\frac {4 \, {\left (\tan \left (f x\right )^{4} \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right )^{3} \tan \left (e\right ) + \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + \tan \left (f x\right )^{2} - 2 \, \tan \left (f x\right ) \tan \left (e\right ) + 1\right )}}{\tan \left (e\right )^{2} + 1}\right ) + 2 \, C c \tan \left (f x\right ) + 2 \, B d \tan \left (f x\right ) + 2 \, C c \tan \left (e\right ) + 2 \, B d \tan \left (e\right ) + C d}{2 \, {\left (f \tan \left (f x\right )^{2} \tan \left (e\right )^{2} - 2 \, f \tan \left (f x\right ) \tan \left (e\right ) + f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.68, size = 75, normalized size = 1.03 \begin {gather*} \frac {\mathrm {tan}\left (e+f\,x\right )\,\left (B\,d+C\,c\right )}{f}-x\,\left (B\,d-A\,c+C\,c\right )+\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )\,\left (\frac {A\,d}{2}+\frac {B\,c}{2}-\frac {C\,d}{2}\right )}{f}+\frac {C\,d\,{\mathrm {tan}\left (e+f\,x\right )}^2}{2\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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