Optimal. Leaf size=168 \[ -\frac {b \left (3 a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac {\log (\sin (c+d x))}{a^3 d}-\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right )^3 d}+\frac {b^2}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b^2 \left (3 a^2+b^2\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
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Rubi [A]
time = 0.29, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3650, 3730,
3732, 3611, 3556} \begin {gather*} \frac {\log (\sin (c+d x))}{a^3 d}+\frac {b^2 \left (3 a^2+b^2\right )}{a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {b^2}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {b x \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3}-\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3611
Rule 3650
Rule 3730
Rule 3732
Rubi steps
\begin {align*} \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac {b^2}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {\cot (c+d x) \left (2 \left (a^2+b^2\right )-2 a b \tan (c+d x)+2 b^2 \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 a \left (a^2+b^2\right )}\\ &=\frac {b^2}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b^2 \left (3 a^2+b^2\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\cot (c+d x) \left (2 \left (a^2+b^2\right )^2-4 a^3 b \tan (c+d x)+2 b^2 \left (3 a^2+b^2\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a^2 \left (a^2+b^2\right )^2}\\ &=-\frac {b \left (3 a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac {b^2}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b^2 \left (3 a^2+b^2\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \cot (c+d x) \, dx}{a^3}-\frac {\left (b^2 \left (6 a^4+3 a^2 b^2+b^4\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )^3}\\ &=-\frac {b \left (3 a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac {\log (\sin (c+d x))}{a^3 d}-\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right )^3 d}+\frac {b^2}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b^2 \left (3 a^2+b^2\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 4.38, size = 209, normalized size = 1.24 \begin {gather*} \frac {-\frac {a (a-i b) \log (i-\tan (c+d x))}{(a+i b)^2}+\frac {2 \left (a^2+b^2\right ) \log (\tan (c+d x))}{a^2}-\frac {a (a+i b) \log (i+\tan (c+d x))}{(a-i b)^2}-\frac {2 b^2 \left (6 a^4+3 a^2 b^2+b^4\right ) \log (a+b \tan (c+d x))}{a^2 \left (a^2+b^2\right )^2}+\frac {b^2}{(a+b \tan (c+d x))^2}+\frac {4 a b^2}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {2 b^2}{a^2+a b \tan (c+d x)}}{2 a \left (a^2+b^2\right ) d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 182, normalized size = 1.08
method | result | size |
derivativedivides | \(\frac {\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {b^{2}}{2 \left (a^{2}+b^{2}\right ) a \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{2} \left (3 a^{2}+b^{2}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{2} \left (6 a^{4}+3 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{3}}+\frac {\frac {\left (-a^{3}+3 b^{2} a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-3 a^{2} b +b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(182\) |
default | \(\frac {\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {b^{2}}{2 \left (a^{2}+b^{2}\right ) a \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{2} \left (3 a^{2}+b^{2}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{2} \left (6 a^{4}+3 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{3}}+\frac {\frac {\left (-a^{3}+3 b^{2} a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-3 a^{2} b +b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(182\) |
norman | \(\frac {-\frac {b^{3} \left (3 a^{2}-b^{2}\right ) x \left (\tan ^{2}\left (d x +c \right )\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {\left (3 a^{2}-b^{2}\right ) a^{2} b x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 b^{2} \left (3 a^{2}-b^{2}\right ) a x \tan \left (d x +c \right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 b \left (-2 a^{2} b^{2}-b^{4}\right ) \tan \left (d x +c \right )}{d \,a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {b^{2} \left (-7 a^{2} b^{2}-3 b^{4}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d \,a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}-\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b^{2} \left (6 a^{4}+3 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d \,a^{3}}\) | \(383\) |
risch | \(-\frac {i x}{3 i b \,a^{2}-i b^{3}-a^{3}+3 b^{2} a}+\frac {12 i a \,b^{2} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {12 i a \,b^{2} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {6 i b^{4} x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a}+\frac {6 i b^{4} c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d a}+\frac {2 i b^{6} x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a^{3}}+\frac {2 i b^{6} c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d \,a^{3}}-\frac {2 i x}{a^{3}}-\frac {2 i c}{a^{3} d}-\frac {2 i b^{3} \left (-3 i a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-i b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+4 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+4 i b \,a^{2}+i b^{3}+4 a^{3}+b^{2} a \right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )^{2} a^{2} \left (i b +a \right )^{2} d \left (-i b +a \right )^{3}}-\frac {6 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{2}}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {3 b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d a}-\frac {b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d \,a^{3}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{3} d}\) | \(621\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 290, normalized size = 1.73 \begin {gather*} -\frac {\frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (6 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}} + \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {7 \, a^{3} b^{2} + 3 \, a b^{4} + 2 \, {\left (3 \, a^{2} b^{3} + b^{5}\right )} \tan \left (d x + c\right )}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4} + {\left (a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \tan \left (d x + c\right )} - \frac {2 \, \log \left (\tan \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 494 vs.
\(2 (166) = 332\).
time = 1.04, size = 494, normalized size = 2.94 \begin {gather*} \frac {9 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - 2 \, {\left (3 \, a^{7} b - a^{5} b^{3}\right )} d x - {\left (7 \, a^{4} b^{4} + a^{2} b^{6} + 2 \, {\left (3 \, a^{5} b^{3} - a^{3} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6} + {\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (6 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6} + {\left (6 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )\right )} \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 ) - 2 \, {\left (4 \, a^{5} b^{3} - 3 \, a^{3} b^{5} - a b^{7} + 2 \, {\left (3 \, a^{6} b^{2} - a^{4} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{9} b^{2} + 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} + a^{3} b^{8}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{10} b + 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} + a^{4} b^{7}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} + a^{5} b^{6}\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.93, size = 328, normalized size = 1.95 \begin {gather*} -\frac {\frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (6 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{9} b + 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} + a^{3} b^{7}} - \frac {18 \, a^{4} b^{4} \tan \left (d x + c\right )^{2} + 9 \, a^{2} b^{6} \tan \left (d x + c\right )^{2} + 3 \, b^{8} \tan \left (d x + c\right )^{2} + 42 \, a^{5} b^{3} \tan \left (d x + c\right ) + 26 \, a^{3} b^{5} \tan \left (d x + c\right ) + 8 \, a b^{7} \tan \left (d x + c\right ) + 25 \, a^{6} b^{2} + 19 \, a^{4} b^{4} + 6 \, a^{2} b^{6}}{{\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}} - \frac {2 \, \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.68, size = 256, normalized size = 1.52 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^3\,d}+\frac {\frac {7\,a^2\,b^2+3\,b^4}{2\,a\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (3\,a^2\,b^3+b^5\right )}{a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}-\frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (6\,a^4+3\,a^2\,b^2+b^4\right )}{a^3\,d\,{\left (a^2+b^2\right )}^3}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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