Optimal. Leaf size=79 \[ -\frac {b x}{a^2+b^2}+\frac {a \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}-\frac {a^3 \log (a+b \tan (c+d x))}{b^2 \left (a^2+b^2\right ) d}+\frac {\tan (c+d x)}{b d} \]
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Rubi [A]
time = 0.09, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3647, 3707,
3698, 31, 3556} \begin {gather*} \frac {a \log (\cos (c+d x))}{d \left (a^2+b^2\right )}-\frac {b x}{a^2+b^2}-\frac {a^3 \log (a+b \tan (c+d x))}{b^2 d \left (a^2+b^2\right )}+\frac {\tan (c+d x)}{b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 3556
Rule 3647
Rule 3698
Rule 3707
Rubi steps
\begin {align*} \int \frac {\tan ^3(c+d x)}{a+b \tan (c+d x)} \, dx &=\frac {\tan (c+d x)}{b d}+\frac {\int \frac {-a-b \tan (c+d x)-a \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b}\\ &=-\frac {b x}{a^2+b^2}+\frac {\tan (c+d x)}{b d}-\frac {a \int \tan (c+d x) \, dx}{a^2+b^2}-\frac {a^3 \int \frac {1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac {b x}{a^2+b^2}+\frac {a \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}+\frac {\tan (c+d x)}{b d}-\frac {a^3 \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^2 \left (a^2+b^2\right ) d}\\ &=-\frac {b x}{a^2+b^2}+\frac {a \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}-\frac {a^3 \log (a+b \tan (c+d x))}{b^2 \left (a^2+b^2\right ) d}+\frac {\tan (c+d x)}{b d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.42, size = 91, normalized size = 1.15 \begin {gather*} -\frac {\frac {\log (i-\tan (c+d x))}{a+i b}+\frac {\log (i+\tan (c+d x))}{a-i b}+\frac {2 a^3 \log (a+b \tan (c+d x))}{b^2 \left (a^2+b^2\right )}-\frac {2 \tan (c+d x)}{b}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 79, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {\frac {\tan \left (d x +c \right )}{b}+\frac {-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-b \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {a^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{2} \left (a^{2}+b^{2}\right )}}{d}\) | \(79\) |
default | \(\frac {\frac {\tan \left (d x +c \right )}{b}+\frac {-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-b \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {a^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{2} \left (a^{2}+b^{2}\right )}}{d}\) | \(79\) |
norman | \(\frac {\tan \left (d x +c \right )}{b d}-\frac {b x}{a^{2}+b^{2}}-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{2}+b^{2}\right )}-\frac {a^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{2} \left (a^{2}+b^{2}\right ) d}\) | \(85\) |
risch | \(-\frac {i x}{i b -a}+\frac {2 i a^{3} x}{b^{2} \left (a^{2}+b^{2}\right )}+\frac {2 i a^{3} c}{b^{2} d \left (a^{2}+b^{2}\right )}-\frac {2 i a x}{b^{2}}-\frac {2 i a c}{b^{2} d}+\frac {2 i}{b d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b^{2} d \left (a^{2}+b^{2}\right )}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{2} d}\) | \(167\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 85, normalized size = 1.08 \begin {gather*} -\frac {\frac {2 \, a^{3} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (d x + c\right )} b}{a^{2} + b^{2}} + \frac {a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, \tan \left (d x + c\right )}{b}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.30, size = 111, normalized size = 1.41 \begin {gather*} -\frac {2 \, b^{3} d x + a^{3} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (a^{3} + a b^{2}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (a^{2} b + b^{3}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{2} b^{2} + b^{4}\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.57, size = 554, normalized size = 7.01 \begin {gather*} \begin {cases} \tilde {\infty } x \tan ^{2}{\left (c \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\- \frac {3 d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {3 i d x}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {i \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {2 \tan ^{2}{\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {3}{2 b d \tan {\left (c + d x \right )} - 2 i b d} & \text {for}\: a = - i b \\- \frac {3 d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {3 i d x}{2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {i \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {2 \tan ^{2}{\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {3}{2 b d \tan {\left (c + d x \right )} + 2 i b d} & \text {for}\: a = i b \\\frac {- \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {\tan ^{2}{\left (c + d x \right )}}{2 d}}{a} & \text {for}\: b = 0 \\\frac {x \tan ^{3}{\left (c \right )}}{a + b \tan {\left (c \right )}} & \text {for}\: d = 0 \\- \frac {2 a^{3} \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )}}{2 a^{2} b^{2} d + 2 b^{4} d} + \frac {2 a^{2} b \tan {\left (c + d x \right )}}{2 a^{2} b^{2} d + 2 b^{4} d} - \frac {a b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{2} b^{2} d + 2 b^{4} d} - \frac {2 b^{3} d x}{2 a^{2} b^{2} d + 2 b^{4} d} + \frac {2 b^{3} \tan {\left (c + d x \right )}}{2 a^{2} b^{2} d + 2 b^{4} d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.90, size = 86, normalized size = 1.09 \begin {gather*} -\frac {\frac {2 \, a^{3} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (d x + c\right )} b}{a^{2} + b^{2}} + \frac {a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, \tan \left (d x + c\right )}{b}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.11, size = 94, normalized size = 1.19 \begin {gather*} \frac {\mathrm {tan}\left (c+d\,x\right )}{b\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (a-b\,1{}\mathrm {i}\right )}-\frac {a^3\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{b^2\,d\,\left (a^2+b^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-b+a\,1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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