Optimal. Leaf size=60 \[ b x+\frac {a \log (\cos (c+d x))}{d}-\frac {b \tan (c+d x)}{d}+\frac {a \tan ^2(c+d x)}{2 d}+\frac {b \tan ^3(c+d x)}{3 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 60, 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 = {3609, 3606,
3556} \begin {gather*} \frac {a \tan ^2(c+d x)}{2 d}+\frac {a \log (\cos (c+d x))}{d}+\frac {b \tan ^3(c+d x)}{3 d}-\frac {b \tan (c+d x)}{d}+b x \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3606
Rule 3609
Rubi steps
\begin {align*} \int \tan ^3(c+d x) (a+b \tan (c+d x)) \, dx &=\frac {b \tan ^3(c+d x)}{3 d}+\int \tan ^2(c+d x) (-b+a \tan (c+d x)) \, dx\\ &=\frac {a \tan ^2(c+d x)}{2 d}+\frac {b \tan ^3(c+d x)}{3 d}+\int \tan (c+d x) (-a-b \tan (c+d x)) \, dx\\ &=b x-\frac {b \tan (c+d x)}{d}+\frac {a \tan ^2(c+d x)}{2 d}+\frac {b \tan ^3(c+d x)}{3 d}-a \int \tan (c+d x) \, dx\\ &=b x+\frac {a \log (\cos (c+d x))}{d}-\frac {b \tan (c+d x)}{d}+\frac {a \tan ^2(c+d x)}{2 d}+\frac {b \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 67, normalized size = 1.12 \begin {gather*} \frac {b \text {ArcTan}(\tan (c+d x))}{d}-\frac {b \tan (c+d x)}{d}+\frac {b \tan ^3(c+d x)}{3 d}+\frac {a \left (2 \log (\cos (c+d x))+\tan ^2(c+d x)\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 60, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2}-b \tan \left (d x +c \right )-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+b \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(60\) |
default | \(\frac {\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2}-b \tan \left (d x +c \right )-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+b \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(60\) |
norman | \(b x +\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {b \tan \left (d x +c \right )}{d}+\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(62\) |
risch | \(b x -i a x -\frac {2 i a c}{d}-\frac {2 i \left (3 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+6 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+6 b \,{\mathrm e}^{2 i \left (d x +c \right )}+4 b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 59, normalized size = 0.98 \begin {gather*} \frac {2 \, b \tan \left (d x + c\right )^{3} + 3 \, a \tan \left (d x + c\right )^{2} + 6 \, {\left (d x + c\right )} b - 3 \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, b \tan \left (d x + c\right )}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.90, size = 58, normalized size = 0.97 \begin {gather*} \frac {2 \, b \tan \left (d x + c\right )^{3} + 6 \, b d x + 3 \, a \tan \left (d x + c\right )^{2} + 3 \, a \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 6 \, b \tan \left (d x + c\right )}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 70, normalized size = 1.17 \begin {gather*} \begin {cases} - \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \tan ^{2}{\left (c + d x \right )}}{2 d} + b x + \frac {b \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {b \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right ) \tan ^{3}{\left (c \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. 515 vs.
\(2 (56) = 112\).
time = 1.09, size = 515, normalized size = 8.58 \begin {gather*} \frac {6 \, b d x \tan \left (d x\right )^{3} \tan \left (c\right )^{3} + 3 \, a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 18 \, b d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 3 \, a \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 9 \, a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 6 \, b \tan \left (d x\right )^{3} \tan \left (c\right )^{2} + 6 \, b \tan \left (d x\right )^{2} \tan \left (c\right )^{3} + 18 \, b d x \tan \left (d x\right ) \tan \left (c\right ) + 3 \, a \tan \left (d x\right )^{3} \tan \left (c\right ) - 3 \, a \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 3 \, a \tan \left (d x\right ) \tan \left (c\right )^{3} - 2 \, b \tan \left (d x\right )^{3} + 9 \, a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) - 18 \, b \tan \left (d x\right )^{2} \tan \left (c\right ) - 18 \, b \tan \left (d x\right ) \tan \left (c\right )^{2} - 2 \, b \tan \left (c\right )^{3} - 6 \, b d x - 3 \, a \tan \left (d x\right )^{2} + 3 \, a \tan \left (d x\right ) \tan \left (c\right ) - 3 \, a \tan \left (c\right )^{2} - 3 \, a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (c\right )^{2} + 1}\right ) + 6 \, b \tan \left (d x\right ) + 6 \, b \tan \left (c\right ) - 3 \, a}{6 \, {\left (d \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 3 \, d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 3 \, d \tan \left (d x\right ) \tan \left (c\right ) - d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.04, size = 54, normalized size = 0.90 \begin {gather*} \frac {\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}-\frac {a\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2}-b\,\mathrm {tan}\left (c+d\,x\right )+\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}+b\,d\,x}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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