Optimal. Leaf size=129 \[ \frac {b \left (3 a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {a \left (a^2-3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a^2-b^2}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
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Rubi [A]
time = 0.11, antiderivative size = 129, 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 = {3610, 3612,
3611} \begin {gather*} \frac {a}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {a^2-b^2}{d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {a \left (a^2-3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {b x \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3610
Rule 3611
Rule 3612
Rubi steps
\begin {align*} \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac {a}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {b+a \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx}{a^2+b^2}\\ &=\frac {a}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a^2-b^2}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {2 a b+\left (a^2-b^2\right ) \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\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 {a}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a^2-b^2}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left (a \left (a^2-3 b^2\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\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 {a \left (a^2-3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a^2-b^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 = 3.56, size = 234, normalized size = 1.81 \begin {gather*} \frac {-\frac {i \log (i-\tan (c+d x))}{(a+i b)^2}+\frac {i \log (i+\tan (c+d x))}{(a-i b)^2}+\frac {4 a b \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^2}-\frac {2 b}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}+a \left (\frac {i \log (i-\tan (c+d x))}{(a+i b)^3}-\frac {\log (i+\tan (c+d x))}{(i a+b)^3}+\frac {b \left (\left (-6 a^2+2 b^2\right ) \log (a+b \tan (c+d x))+\frac {\left (a^2+b^2\right ) \left (5 a^2+b^2+4 a b \tan (c+d x)\right )}{(a+b \tan (c+d x))^2}\right )}{\left (a^2+b^2\right )^3}\right )}{2 b d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 147, normalized size = 1.14
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (a^{3}-3 b^{2} a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (3 a^{2} b -b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {a}{2 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{2}-b^{2}}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(147\) |
default | \(\frac {\frac {\frac {\left (a^{3}-3 b^{2} a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (3 a^{2} b -b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {a}{2 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{2}-b^{2}}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(147\) |
norman | \(\frac {\frac {\left (a^{2} b^{2}-b^{4}\right ) \tan \left (d x +c \right )}{d b \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {b^{3} \left (3 a^{2}-b^{2}\right ) x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {\left (3 a^{2}-b^{2}\right ) a^{2} b x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {a \left (3 a^{2} b^{2}-b^{4}\right )}{2 d \,b^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b^{2} \left (3 a^{2}-b^{2}\right ) a x \tan \left (d x +c \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(344\) |
risch | \(\frac {i x}{3 i b \,a^{2}-i b^{3}-a^{3}+3 b^{2} a}+\frac {2 i a^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {6 i a \,b^{2} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 i a^{3} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {6 i a \,b^{2} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {2 i b \left (-i a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2 i b \,a^{2}-i b^{3}+2 a^{3}-b^{2} a \right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )^{2} \left (i b +a \right )^{2} d \left (-i b +a \right )^{3}}-\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{2}}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(419\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 253, normalized size = 1.96 \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 (a^{3} - 3 \, a b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\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 {3 \, a^{3} - a b^{2} + 2 \, {\left (a^{2} b - b^{3}\right )} \tan \left (d x + c\right )}{a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4} + {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (d x + c\right )}}{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. 328 vs.
\(2 (127) = 254\).
time = 1.52, size = 328, normalized size = 2.54 \begin {gather*} \frac {5 \, a^{3} b^{2} - a b^{4} + 2 \, {\left (3 \, a^{4} b - a^{2} b^{3}\right )} d x - {\left (3 \, a^{3} b^{2} - 3 \, a b^{4} - 2 \, {\left (3 \, a^{2} b^{3} - b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} - {\left (a^{5} - 3 \, a^{3} b^{2} + {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{4} b - 3 \, a^{2} b^{3}\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 (2 \, a^{4} b - 3 \, a^{2} b^{3} + b^{5} - 2 \, {\left (3 \, a^{3} b^{2} - a b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} d \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 )} d \tan \left (d x + c\right ) + {\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 275 vs.
\(2 (127) = 254\).
time = 0.71, size = 275, normalized size = 2.13 \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 (a^{3} b - 3 \, a b^{3}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} + \frac {3 \, a^{3} b^{2} \tan \left (d x + c\right )^{2} - 9 \, a b^{4} \tan \left (d x + c\right )^{2} + 8 \, a^{4} b \tan \left (d x + c\right ) - 18 \, a^{2} b^{3} \tan \left (d x + c\right ) - 2 \, b^{5} \tan \left (d x + c\right ) + 6 \, a^{5} - 7 \, a^{3} b^{2} - a b^{4}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.01, size = 224, normalized size = 1.74 \begin {gather*} -\frac {\frac {a\,b^2-3\,a^3}{2\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (a^2\,b-b^3\right )}{a^4+2\,a^2\,b^2+b^4}}{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 {\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 {a\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2-3\,b^2\right )}{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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