Integrand size = 16, antiderivative size = 57 \[ \int \frac {x}{a+b \tan \left (c+d x^2\right )} \, dx=\frac {a x^2}{2 \left (a^2+b^2\right )}+\frac {b \log \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )}{2 \left (a^2+b^2\right ) d} \]
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Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3832, 3565, 3611} \[ \int \frac {x}{a+b \tan \left (c+d x^2\right )} \, dx=\frac {b \log \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )}{2 d \left (a^2+b^2\right )}+\frac {a x^2}{2 \left (a^2+b^2\right )} \]
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Rule 3565
Rule 3611
Rule 3832
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{a+b \tan (c+d x)} \, dx,x,x^2\right ) \\ & = \frac {a x^2}{2 \left (a^2+b^2\right )}+\frac {b \text {Subst}\left (\int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx,x,x^2\right )}{2 \left (a^2+b^2\right )} \\ & = \frac {a x^2}{2 \left (a^2+b^2\right )}+\frac {b \log \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )}{2 \left (a^2+b^2\right ) d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.44 \[ \int \frac {x}{a+b \tan \left (c+d x^2\right )} \, dx=\frac {(-i a-b) \log \left (i-\tan \left (c+d x^2\right )\right )+i (a+i b) \log \left (i+\tan \left (c+d x^2\right )\right )+2 b \log \left (a+b \tan \left (c+d x^2\right )\right )}{4 \left (a^2+b^2\right ) d} \]
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Time = 0.13 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \(\frac {2 a d \,x^{2}+2 b \ln \left (a +b \tan \left (d \,x^{2}+c \right )\right )-b \ln \left (1+\tan ^{2}\left (d \,x^{2}+c \right )\right )}{4 d \left (a^{2}+b^{2}\right )}\) | \(55\) |
derivativedivides | \(\frac {\frac {-\frac {b \ln \left (1+\tan ^{2}\left (d \,x^{2}+c \right )\right )}{2}+a \arctan \left (\tan \left (d \,x^{2}+c \right )\right )}{a^{2}+b^{2}}+\frac {b \ln \left (a +b \tan \left (d \,x^{2}+c \right )\right )}{a^{2}+b^{2}}}{2 d}\) | \(69\) |
default | \(\frac {\frac {-\frac {b \ln \left (1+\tan ^{2}\left (d \,x^{2}+c \right )\right )}{2}+a \arctan \left (\tan \left (d \,x^{2}+c \right )\right )}{a^{2}+b^{2}}+\frac {b \ln \left (a +b \tan \left (d \,x^{2}+c \right )\right )}{a^{2}+b^{2}}}{2 d}\) | \(69\) |
norman | \(\frac {a \,x^{2}}{2 a^{2}+2 b^{2}}-\frac {b \ln \left (1+\tan ^{2}\left (d \,x^{2}+c \right )\right )}{4 d \left (a^{2}+b^{2}\right )}+\frac {b \ln \left (a +b \tan \left (d \,x^{2}+c \right )\right )}{2 d \left (a^{2}+b^{2}\right )}\) | \(73\) |
risch | \(-\frac {x^{2}}{2 \left (i b -a \right )}-\frac {i b \,x^{2}}{a^{2}+b^{2}}-\frac {i b c}{d \left (a^{2}+b^{2}\right )}+\frac {b \ln \left ({\mathrm e}^{2 i \left (d \,x^{2}+c \right )}-\frac {i b +a}{i b -a}\right )}{2 d \left (a^{2}+b^{2}\right )}\) | \(96\) |
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Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.23 \[ \int \frac {x}{a+b \tan \left (c+d x^2\right )} \, dx=\frac {2 \, a d x^{2} + b \log \left (\frac {b^{2} \tan \left (d x^{2} + c\right )^{2} + 2 \, a b \tan \left (d x^{2} + c\right ) + a^{2}}{\tan \left (d x^{2} + c\right )^{2} + 1}\right )}{4 \, {\left (a^{2} + b^{2}\right )} d} \]
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Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 359, normalized size of antiderivative = 6.30 \[ \int \frac {x}{a+b \tan \left (c+d x^2\right )} \, dx=\begin {cases} \frac {\tilde {\infty } x^{2}}{\tan {\left (c \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {x^{2}}{2 a} & \text {for}\: b = 0 \\\frac {i \left (\operatorname {atan}{\left (\tan {\left (c + d x^{2} \right )} \right )} + \pi \left \lfloor {\frac {c + d x^{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan {\left (c + d x^{2} \right )}}{4 b d \tan {\left (c + d x^{2} \right )} - 4 i b d} + \frac {\operatorname {atan}{\left (\tan {\left (c + d x^{2} \right )} \right )} + \pi \left \lfloor {\frac {c + d x^{2} - \frac {\pi }{2}}{\pi }}\right \rfloor }{4 b d \tan {\left (c + d x^{2} \right )} - 4 i b d} + \frac {i}{4 b d \tan {\left (c + d x^{2} \right )} - 4 i b d} & \text {for}\: a = - i b \\- \frac {i \left (\operatorname {atan}{\left (\tan {\left (c + d x^{2} \right )} \right )} + \pi \left \lfloor {\frac {c + d x^{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan {\left (c + d x^{2} \right )}}{4 b d \tan {\left (c + d x^{2} \right )} + 4 i b d} + \frac {\operatorname {atan}{\left (\tan {\left (c + d x^{2} \right )} \right )} + \pi \left \lfloor {\frac {c + d x^{2} - \frac {\pi }{2}}{\pi }}\right \rfloor }{4 b d \tan {\left (c + d x^{2} \right )} + 4 i b d} - \frac {i}{4 b d \tan {\left (c + d x^{2} \right )} + 4 i b d} & \text {for}\: a = i b \\\frac {x^{2}}{2 \left (a + b \tan {\left (c \right )}\right )} & \text {for}\: d = 0 \\\frac {2 a d x^{2}}{4 a^{2} d + 4 b^{2} d} + \frac {2 b \log {\left (\frac {a}{b} + \tan {\left (c + d x^{2} \right )} \right )}}{4 a^{2} d + 4 b^{2} d} - \frac {b \log {\left (\tan ^{2}{\left (c + d x^{2} \right )} + 1 \right )}}{4 a^{2} d + 4 b^{2} d} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (53) = 106\).
Time = 0.30 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.51 \[ \int \frac {x}{a+b \tan \left (c+d x^2\right )} \, dx=\frac {2 \, a d x^{2} + b \log \left (\frac {{\left (a^{2} + b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + 4 \, a b \sin \left (2 \, d x^{2} + 2 \, c\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )}{{\left (a^{2} + b^{2}\right )} \cos \left (2 \, c\right )^{2} + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, c\right )^{2}}\right )}{4 \, {\left (a^{2} + b^{2}\right )} d} \]
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Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.51 \[ \int \frac {x}{a+b \tan \left (c+d x^2\right )} \, dx=\frac {b^{2} \log \left ({\left | b \tan \left (d x^{2} + c\right ) + a \right |}\right )}{2 \, {\left (a^{2} b d + b^{3} d\right )}} + \frac {{\left (d x^{2} + c\right )} a}{2 \, {\left (a^{2} d + b^{2} d\right )}} - \frac {b \log \left (\tan \left (d x^{2} + c\right )^{2} + 1\right )}{4 \, {\left (a^{2} d + b^{2} d\right )}} \]
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Time = 3.54 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.14 \[ \int \frac {x}{a+b \tan \left (c+d x^2\right )} \, dx=\frac {\frac {b\,\ln \left (a+b\,\mathrm {tan}\left (d\,x^2+c\right )\right )}{2}-\frac {b\,\ln \left ({\mathrm {tan}\left (d\,x^2+c\right )}^2+1\right )}{4}}{d\,\left (a^2+b^2\right )}+\frac {a\,x^2}{2\,\left (a^2+b^2\right )} \]
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