Optimal. Leaf size=107 \[ \frac {b x}{a^2+b^2}+\frac {b \cot (c+d x)}{a^2 d}-\frac {\cot ^2(c+d x)}{2 a d}-\frac {\left (a^2-b^2\right ) \log (\sin (c+d x))}{a^3 d}-\frac {b^4 \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right ) d} \]
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Rubi [A]
time = 0.21, antiderivative size = 107, 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 = {3650, 3730,
3733, 3611, 3556} \begin {gather*} \frac {b x}{a^2+b^2}+\frac {b \cot (c+d x)}{a^2 d}-\frac {\left (a^2-b^2\right ) \log (\sin (c+d x))}{a^3 d}-\frac {b^4 \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )}-\frac {\cot ^2(c+d x)}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3611
Rule 3650
Rule 3730
Rule 3733
Rubi steps
\begin {align*} \int \frac {\cot ^3(c+d x)}{a+b \tan (c+d x)} \, dx &=-\frac {\cot ^2(c+d x)}{2 a d}-\frac {\int \frac {\cot ^2(c+d x) \left (2 b+2 a \tan (c+d x)+2 b \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a}\\ &=\frac {b \cot (c+d x)}{a^2 d}-\frac {\cot ^2(c+d x)}{2 a d}+\frac {\int \frac {\cot (c+d x) \left (-2 \left (a^2-b^2\right )+2 b^2 \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a^2}\\ &=\frac {b x}{a^2+b^2}+\frac {b \cot (c+d x)}{a^2 d}-\frac {\cot ^2(c+d x)}{2 a d}-\frac {\left (a^2-b^2\right ) \int \cot (c+d x) \, dx}{a^3}-\frac {b^4 \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )}\\ &=\frac {b x}{a^2+b^2}+\frac {b \cot (c+d x)}{a^2 d}-\frac {\cot ^2(c+d x)}{2 a d}-\frac {\left (a^2-b^2\right ) \log (\sin (c+d x))}{a^3 d}-\frac {b^4 \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.68, size = 106, normalized size = 0.99 \begin {gather*} -\frac {-\frac {2 b \cot (c+d x)}{a^2}+\frac {\cot ^2(c+d x)}{a}-\frac {\log (i-\cot (c+d x))}{a-i b}-\frac {\log (i+\cot (c+d x))}{a+i b}+\frac {2 b^4 \log (b+a \cot (c+d x))}{a^3 \left (a^2+b^2\right )}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 114, normalized size = 1.07
method | result | size |
derivativedivides | \(\frac {-\frac {1}{2 a \tan \left (d x +c \right )^{2}}+\frac {\left (-a^{2}+b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {b}{a^{2} \tan \left (d x +c \right )}-\frac {b^{4} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a^{3}}+\frac {\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+b \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}}{d}\) | \(114\) |
default | \(\frac {-\frac {1}{2 a \tan \left (d x +c \right )^{2}}+\frac {\left (-a^{2}+b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {b}{a^{2} \tan \left (d x +c \right )}-\frac {b^{4} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a^{3}}+\frac {\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+b \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}}{d}\) | \(114\) |
norman | \(\frac {\frac {b x \left (\tan ^{2}\left (d x +c \right )\right )}{a^{2}+b^{2}}+\frac {b \tan \left (d x +c \right )}{a^{2} d}-\frac {1}{2 d a}}{\tan \left (d x +c \right )^{2}}+\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{2}+b^{2}\right )}-\frac {\left (a^{2}-b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}-\frac {b^{4} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a^{3} d}\) | \(135\) |
risch | \(\frac {i x}{i b -a}+\frac {2 i x}{a}+\frac {2 i c}{d a}-\frac {2 i b^{2} x}{a^{3}}-\frac {2 i b^{2} c}{a^{3} d}+\frac {2 i b^{4} x}{\left (a^{2}+b^{2}\right ) a^{3}}+\frac {2 i b^{4} c}{\left (a^{2}+b^{2}\right ) a^{3} d}+\frac {2 i \left (-i a \,{\mathrm e}^{2 i \left (d x +c \right )}+b \,{\mathrm e}^{2 i \left (d x +c \right )}-b \right )}{d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{a^{3} d}-\frac {b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{2}+b^{2}\right ) a^{3} d}\) | \(238\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.67, size = 122, normalized size = 1.14 \begin {gather*} -\frac {\frac {2 \, b^{4} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{5} + a^{3} b^{2}} - \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 \, {\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{3}} - \frac {2 \, b \tan \left (d x + c\right ) - a}{a^{2} \tan \left (d x + c\right )^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.10, size = 180, normalized size = 1.68 \begin {gather*} -\frac {b^{4} \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 ) \tan \left (d x + c\right )^{2} + a^{4} + a^{2} b^{2} + {\left (a^{4} - b^{4}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{2} - {\left (2 \, a^{3} b d x - a^{4} - a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} - 2 \, {\left (a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{5} + a^{3} b^{2}\right )} d \tan \left (d x + c\right )^{2}} \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 = 2.38, size = 1336, normalized size = 12.49 \begin {gather*} \begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \wedge c = 0 \wedge d = 0 \\\frac {\frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {\log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {1}{2 d \tan ^{2}{\left (c + d x \right )}}}{a} & \text {for}\: b = 0 \\\frac {x + \frac {1}{d \tan {\left (c + d x \right )}} - \frac {1}{3 d \tan ^{3}{\left (c + d x \right )}}}{b} & \text {for}\: a = 0 \\- \frac {3 d x \tan ^{3}{\left (c + d x \right )}}{2 b d \tan ^{3}{\left (c + d x \right )} - 2 i b d \tan ^{2}{\left (c + d x \right )}} + \frac {3 i d x \tan ^{2}{\left (c + d x \right )}}{2 b d \tan ^{3}{\left (c + d x \right )} - 2 i b d \tan ^{2}{\left (c + d x \right )}} + \frac {2 i \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan ^{3}{\left (c + d x \right )}}{2 b d \tan ^{3}{\left (c + d x \right )} - 2 i b d \tan ^{2}{\left (c + d x \right )}} + \frac {2 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan ^{2}{\left (c + d x \right )}}{2 b d \tan ^{3}{\left (c + d x \right )} - 2 i b d \tan ^{2}{\left (c + d x \right )}} - \frac {4 i \log {\left (\tan {\left (c + d x \right )} \right )} \tan ^{3}{\left (c + d x \right )}}{2 b d \tan ^{3}{\left (c + d x \right )} - 2 i b d \tan ^{2}{\left (c + d x \right )}} - \frac {4 \log {\left (\tan {\left (c + d x \right )} \right )} \tan ^{2}{\left (c + d x \right )}}{2 b d \tan ^{3}{\left (c + d x \right )} - 2 i b d \tan ^{2}{\left (c + d x \right )}} - \frac {3 \tan ^{2}{\left (c + d x \right )}}{2 b d \tan ^{3}{\left (c + d x \right )} - 2 i b d \tan ^{2}{\left (c + d x \right )}} + \frac {i \tan {\left (c + d x \right )}}{2 b d \tan ^{3}{\left (c + d x \right )} - 2 i b d \tan ^{2}{\left (c + d x \right )}} - \frac {1}{2 b d \tan ^{3}{\left (c + d x \right )} - 2 i b d \tan ^{2}{\left (c + d x \right )}} & \text {for}\: a = - i b \\- \frac {3 d x \tan ^{3}{\left (c + d x \right )}}{2 b d \tan ^{3}{\left (c + d x \right )} + 2 i b d \tan ^{2}{\left (c + d x \right )}} - \frac {3 i d x \tan ^{2}{\left (c + d x \right )}}{2 b d \tan ^{3}{\left (c + d x \right )} + 2 i b d \tan ^{2}{\left (c + d x \right )}} - \frac {2 i \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan ^{3}{\left (c + d x \right )}}{2 b d \tan ^{3}{\left (c + d x \right )} + 2 i b d \tan ^{2}{\left (c + d x \right )}} + \frac {2 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan ^{2}{\left (c + d x \right )}}{2 b d \tan ^{3}{\left (c + d x \right )} + 2 i b d \tan ^{2}{\left (c + d x \right )}} + \frac {4 i \log {\left (\tan {\left (c + d x \right )} \right )} \tan ^{3}{\left (c + d x \right )}}{2 b d \tan ^{3}{\left (c + d x \right )} + 2 i b d \tan ^{2}{\left (c + d x \right )}} - \frac {4 \log {\left (\tan {\left (c + d x \right )} \right )} \tan ^{2}{\left (c + d x \right )}}{2 b d \tan ^{3}{\left (c + d x \right )} + 2 i b d \tan ^{2}{\left (c + d x \right )}} - \frac {3 \tan ^{2}{\left (c + d x \right )}}{2 b d \tan ^{3}{\left (c + d x \right )} + 2 i b d \tan ^{2}{\left (c + d x \right )}} - \frac {i \tan {\left (c + d x \right )}}{2 b d \tan ^{3}{\left (c + d x \right )} + 2 i b d \tan ^{2}{\left (c + d x \right )}} - \frac {1}{2 b d \tan ^{3}{\left (c + d x \right )} + 2 i b d \tan ^{2}{\left (c + d x \right )}} & \text {for}\: a = i b \\\frac {\tilde {\infty } x}{a} & \text {for}\: c = - d x \\\frac {x \cot ^{3}{\left (c \right )}}{a + b \tan {\left (c \right )}} & \text {for}\: d = 0 \\\frac {a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan ^{2}{\left (c + d x \right )}}{2 a^{5} d \tan ^{2}{\left (c + d x \right )} + 2 a^{3} b^{2} d \tan ^{2}{\left (c + d x \right )}} - \frac {2 a^{4} \log {\left (\tan {\left (c + d x \right )} \right )} \tan ^{2}{\left (c + d x \right )}}{2 a^{5} d \tan ^{2}{\left (c + d x \right )} + 2 a^{3} b^{2} d \tan ^{2}{\left (c + d x \right )}} - \frac {a^{4}}{2 a^{5} d \tan ^{2}{\left (c + d x \right )} + 2 a^{3} b^{2} d \tan ^{2}{\left (c + d x \right )}} + \frac {2 a^{3} b d x \tan ^{2}{\left (c + d x \right )}}{2 a^{5} d \tan ^{2}{\left (c + d x \right )} + 2 a^{3} b^{2} d \tan ^{2}{\left (c + d x \right )}} + \frac {2 a^{3} b \tan {\left (c + d x \right )}}{2 a^{5} d \tan ^{2}{\left (c + d x \right )} + 2 a^{3} b^{2} d \tan ^{2}{\left (c + d x \right )}} - \frac {a^{2} b^{2}}{2 a^{5} d \tan ^{2}{\left (c + d x \right )} + 2 a^{3} b^{2} d \tan ^{2}{\left (c + d x \right )}} + \frac {2 a b^{3} \tan {\left (c + d x \right )}}{2 a^{5} d \tan ^{2}{\left (c + d x \right )} + 2 a^{3} b^{2} d \tan ^{2}{\left (c + d x \right )}} - \frac {2 b^{4} \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )} \tan ^{2}{\left (c + d x \right )}}{2 a^{5} d \tan ^{2}{\left (c + d x \right )} + 2 a^{3} b^{2} d \tan ^{2}{\left (c + d x \right )}} + \frac {2 b^{4} \log {\left (\tan {\left (c + d x \right )} \right )} \tan ^{2}{\left (c + d x \right )}}{2 a^{5} d \tan ^{2}{\left (c + d x \right )} + 2 a^{3} b^{2} d \tan ^{2}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.75, size = 155, normalized size = 1.45 \begin {gather*} -\frac {\frac {2 \, b^{5} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{5} b + a^{3} b^{3}} - \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 \, {\left (a^{2} - b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac {3 \, a^{2} \tan \left (d x + c\right )^{2} - 3 \, b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) - a^{2}}{a^{3} \tan \left (d x + c\right )^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.20, size = 137, normalized size = 1.28 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (a-b\,1{}\mathrm {i}\right )}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^2\,\left (\frac {1}{2\,a}-\frac {b\,\mathrm {tan}\left (c+d\,x\right )}{a^2}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2-b^2\right )}{a^3\,d}-\frac {b^4\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d\,\left (a^5+a^3\,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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