Optimal. Leaf size=128 \[ -\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+2 a d e \log (x)-\frac {1}{2} b c^2 d^2 \tan ^{-1}(c x)+\frac {b e^2 \tan ^{-1}(c x)}{2 c^2}-\frac {b c d^2}{2 x}+i b d e \text {Li}_2(-i c x)-i b d e \text {Li}_2(i c x)-\frac {b e^2 x}{2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4980, 4852, 325, 203, 4848, 2391, 321} \[ i b d e \text {PolyLog}(2,-i c x)-i b d e \text {PolyLog}(2,i c x)-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+2 a d e \log (x)-\frac {1}{2} b c^2 d^2 \tan ^{-1}(c x)+\frac {b e^2 \tan ^{-1}(c x)}{2 c^2}-\frac {b c d^2}{2 x}-\frac {b e^2 x}{2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 321
Rule 325
Rule 2391
Rule 4848
Rule 4852
Rule 4980
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )}{x^3} \, dx &=\int \left (\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{x^3}+\frac {2 d e \left (a+b \tan ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \tan ^{-1}(c x)\right )\right ) \, dx\\ &=d^2 \int \frac {a+b \tan ^{-1}(c x)}{x^3} \, dx+(2 d e) \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx+e^2 \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx\\ &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+2 a d e \log (x)+\frac {1}{2} \left (b c d^2\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx+(i b d e) \int \frac {\log (1-i c x)}{x} \, dx-(i b d e) \int \frac {\log (1+i c x)}{x} \, dx-\frac {1}{2} \left (b c e^2\right ) \int \frac {x^2}{1+c^2 x^2} \, dx\\ &=-\frac {b c d^2}{2 x}-\frac {b e^2 x}{2 c}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+2 a d e \log (x)+i b d e \text {Li}_2(-i c x)-i b d e \text {Li}_2(i c x)-\frac {1}{2} \left (b c^3 d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx+\frac {\left (b e^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 c}\\ &=-\frac {b c d^2}{2 x}-\frac {b e^2 x}{2 c}-\frac {1}{2} b c^2 d^2 \tan ^{-1}(c x)+\frac {b e^2 \tan ^{-1}(c x)}{2 c^2}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+2 a d e \log (x)+i b d e \text {Li}_2(-i c x)-i b d e \text {Li}_2(i c x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.12, size = 118, normalized size = 0.92 \[ \frac {1}{2} \left (-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{x^2}+e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+4 a d e \log (x)-\frac {b c d^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-c^2 x^2\right )}{x}-\frac {b e^2 \left (c x-\tan ^{-1}(c x)\right )}{c^2}+2 i b d e \text {Li}_2(-i c x)-2 i b d e \text {Li}_2(i c x)\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} + {\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \arctan \left (c x\right )}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 178, normalized size = 1.39 \[ \frac {a \,x^{2} e^{2}}{2}+2 a e d \ln \left (c x \right )-\frac {a \,d^{2}}{2 x^{2}}+\frac {b \arctan \left (c x \right ) x^{2} e^{2}}{2}+2 b \arctan \left (c x \right ) e d \ln \left (c x \right )-\frac {b \arctan \left (c x \right ) d^{2}}{2 x^{2}}-\frac {b \,e^{2} x}{2 c}-\frac {b c \,d^{2}}{2 x}-\frac {b \,c^{2} d^{2} \arctan \left (c x \right )}{2}+\frac {b \,e^{2} \arctan \left (c x \right )}{2 c^{2}}+i b e d \ln \left (c x \right ) \ln \left (i c x +1\right )-i b e d \ln \left (c x \right ) \ln \left (-i c x +1\right )+i b e d \dilog \left (i c x +1\right )-i b e d \dilog \left (-i c x +1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.62, size = 153, normalized size = 1.20 \[ \frac {1}{2} \, a e^{2} x^{2} - \frac {1}{2} \, {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b d^{2} + 2 \, a d e \log \relax (x) - \frac {a d^{2}}{2 \, x^{2}} - \frac {\pi b c^{2} d e \log \left (c^{2} x^{2} + 1\right ) - 4 \, b c^{2} d e \arctan \left (c x\right ) \log \left (c x\right ) + 2 i \, b c^{2} d e {\rm Li}_2\left (i \, c x + 1\right ) - 2 i \, b c^{2} d e {\rm Li}_2\left (-i \, c x + 1\right ) + b c e^{2} x - {\left (b c^{2} e^{2} x^{2} + b e^{2}\right )} \arctan \left (c x\right )}{2 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.68, size = 157, normalized size = 1.23 \[ \left \{\begin {array}{cl} \frac {a\,\left (e^2\,x^4-d^2+4\,d\,e\,x^2\,\ln \relax (x)\right )}{2\,x^2} & \text {\ if\ \ }c=0\\ \frac {a\,\left (e^2\,x^4-d^2+4\,d\,e\,x^2\,\ln \relax (x)\right )}{2\,x^2}-b\,e^2\,\left (\frac {x}{2\,c}-\mathrm {atan}\left (c\,x\right )\,\left (\frac {1}{2\,c^2}+\frac {x^2}{2}\right )\right )-\frac {b\,d^2\,\left (c^3\,\mathrm {atan}\left (c\,x\right )+\frac {c^2}{x}\right )}{2\,c}-\frac {b\,d^2\,\mathrm {atan}\left (c\,x\right )}{2\,x^2}-b\,d\,e\,\left ({\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )-{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\right )\,1{}\mathrm {i} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________