Optimal. Leaf size=128 \[ -\frac {b c d^2}{2 x}-\frac {b e^2 x}{2 c}-\frac {1}{2} b c^2 d^2 \text {ArcTan}(c x)+\frac {b e^2 \text {ArcTan}(c x)}{2 c^2}-\frac {d^2 (a+b \text {ArcTan}(c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \text {ArcTan}(c x))+2 a d e \log (x)+i b d e \text {PolyLog}(2,-i c x)-i b d e \text {PolyLog}(2,i c x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, 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 = {5100, 4946,
331, 209, 4940, 2438, 327} \begin {gather*} -\frac {d^2 (a+b \text {ArcTan}(c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \text {ArcTan}(c x))+2 a d e \log (x)-\frac {1}{2} b c^2 d^2 \text {ArcTan}(c x)+\frac {b e^2 \text {ArcTan}(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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 327
Rule 331
Rule 2438
Rule 4940
Rule 4946
Rule 5100
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 [A]
time = 0.08, size = 114, normalized size = 0.89 \begin {gather*} \frac {1}{2} \left (-\frac {a d^2}{x^2}+a e^2 x^2-\frac {b d^2 \text {ArcTan}(c x)}{x^2}-\frac {b c d^2 (1+c x \text {ArcTan}(c x))}{x}+\frac {b e^2 \left (-c x+\left (1+c^2 x^2\right ) \text {ArcTan}(c x)\right )}{c^2}+4 a d e \log (x)+2 i b d e (\text {PolyLog}(2,-i c x)-\text {PolyLog}(2,i c x))\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.17, size = 211, normalized size = 1.65
method | result | size |
derivativedivides | \(c^{2} \left (\frac {a \,e^{2} x^{2}}{2 c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}+\frac {b \arctan \left (c x \right ) e^{2} x^{2}}{2 c^{2}}-\frac {b \arctan \left (c x \right ) d^{2}}{2 c^{2} x^{2}}+\frac {2 b \arctan \left (c x \right ) d e \ln \left (c x \right )}{c^{2}}-\frac {b \,e^{2} x}{2 c^{3}}-\frac {b \,d^{2} \arctan \left (c x \right )}{2}+\frac {b \,e^{2} \arctan \left (c x \right )}{2 c^{4}}-\frac {b \,d^{2}}{2 c x}+\frac {i b d e \ln \left (c x \right ) \ln \left (i c x +1\right )}{c^{2}}-\frac {i b d e \ln \left (c x \right ) \ln \left (-i c x +1\right )}{c^{2}}+\frac {i b d e \dilog \left (i c x +1\right )}{c^{2}}-\frac {i b d e \dilog \left (-i c x +1\right )}{c^{2}}\right )\) | \(211\) |
default | \(c^{2} \left (\frac {a \,e^{2} x^{2}}{2 c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}+\frac {b \arctan \left (c x \right ) e^{2} x^{2}}{2 c^{2}}-\frac {b \arctan \left (c x \right ) d^{2}}{2 c^{2} x^{2}}+\frac {2 b \arctan \left (c x \right ) d e \ln \left (c x \right )}{c^{2}}-\frac {b \,e^{2} x}{2 c^{3}}-\frac {b \,d^{2} \arctan \left (c x \right )}{2}+\frac {b \,e^{2} \arctan \left (c x \right )}{2 c^{4}}-\frac {b \,d^{2}}{2 c x}+\frac {i b d e \ln \left (c x \right ) \ln \left (i c x +1\right )}{c^{2}}-\frac {i b d e \ln \left (c x \right ) \ln \left (-i c x +1\right )}{c^{2}}+\frac {i b d e \dilog \left (i c x +1\right )}{c^{2}}-\frac {i b d e \dilog \left (-i c x +1\right )}{c^{2}}\right )\) | \(211\) |
risch | \(\frac {i b \ln \left (-i c x +1\right ) e^{2} x^{2}}{4}+\frac {b \,e^{2} \arctan \left (c x \right )}{2 c^{2}}+\frac {i b \,d^{2} \ln \left (i c x +1\right )}{4 x^{2}}+i b d e \dilog \left (i c x +1\right )-\frac {b \,c^{2} d^{2} \arctan \left (c x \right )}{2}-\frac {b \,e^{2} x}{2 c}-i b d e \dilog \left (-i c x +1\right )-\frac {i b \,c^{2} d^{2} \ln \left (i c x \right )}{4}-\frac {b c \,d^{2}}{2 x}-\frac {i b \,d^{2} \ln \left (-i c x +1\right )}{4 x^{2}}+\frac {i c^{2} b \,d^{2} \ln \left (-i c x \right )}{4}-\frac {i b \,e^{2} \ln \left (i c x +1\right ) x^{2}}{4}+\frac {a \,x^{2} e^{2}}{2}+\frac {a \,e^{2}}{2 c^{2}}-\frac {a \,d^{2}}{2 x^{2}}+2 a d e \ln \left (-i c x \right )\) | \(218\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.60, size = 154, normalized size = 1.20 \begin {gather*} -\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} + \frac {1}{2} \, a x^{2} e^{2} + 2 \, a d e \log \left (x\right ) - \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 \arctan \left (c x\right ) e \log \left (c x\right ) + 2 i \, b c^{2} d {\rm Li}_2\left (i \, c x + 1\right ) e - 2 i \, b c^{2} d {\rm Li}_2\left (-i \, c x + 1\right ) e + b c x e^{2} - {\left (b c^{2} x^{2} e^{2} + b e^{2}\right )} \arctan \left (c x\right )}{2 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.68, size = 157, normalized size = 1.23 \begin {gather*} \left \{\begin {array}{cl} \frac {a\,\left (e^2\,x^4-d^2+4\,d\,e\,x^2\,\ln \left (x\right )\right )}{2\,x^2} & \text {\ if\ \ }c=0\\ \frac {a\,\left (e^2\,x^4-d^2+4\,d\,e\,x^2\,\ln \left (x\right )\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 . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________