Optimal. Leaf size=70 \[ \frac {1}{2} i a^2 c \text {Li}_2(-i a x)-\frac {1}{2} i a^2 c \text {Li}_2(i a x)-\frac {1}{2} a^2 c \tan ^{-1}(a x)-\frac {c \tan ^{-1}(a x)}{2 x^2}-\frac {a c}{2 x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4950, 4852, 325, 203, 4848, 2391} \[ \frac {1}{2} i a^2 c \text {PolyLog}(2,-i a x)-\frac {1}{2} i a^2 c \text {PolyLog}(2,i a x)-\frac {1}{2} a^2 c \tan ^{-1}(a x)-\frac {c \tan ^{-1}(a x)}{2 x^2}-\frac {a c}{2 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 325
Rule 2391
Rule 4848
Rule 4852
Rule 4950
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)}{x^3} \, dx &=c \int \frac {\tan ^{-1}(a x)}{x^3} \, dx+\left (a^2 c\right ) \int \frac {\tan ^{-1}(a x)}{x} \, dx\\ &=-\frac {c \tan ^{-1}(a x)}{2 x^2}+\frac {1}{2} (a c) \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx+\frac {1}{2} \left (i a^2 c\right ) \int \frac {\log (1-i a x)}{x} \, dx-\frac {1}{2} \left (i a^2 c\right ) \int \frac {\log (1+i a x)}{x} \, dx\\ &=-\frac {a c}{2 x}-\frac {c \tan ^{-1}(a x)}{2 x^2}+\frac {1}{2} i a^2 c \text {Li}_2(-i a x)-\frac {1}{2} i a^2 c \text {Li}_2(i a x)-\frac {1}{2} \left (a^3 c\right ) \int \frac {1}{1+a^2 x^2} \, dx\\ &=-\frac {a c}{2 x}-\frac {1}{2} a^2 c \tan ^{-1}(a x)-\frac {c \tan ^{-1}(a x)}{2 x^2}+\frac {1}{2} i a^2 c \text {Li}_2(-i a x)-\frac {1}{2} i a^2 c \text {Li}_2(i a x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 74, normalized size = 1.06 \[ -\frac {a c \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-a^2 x^2\right )}{2 x}+\frac {1}{2} i a^2 c \text {Li}_2(-i a x)-\frac {1}{2} i a^2 c \text {Li}_2(i a x)-\frac {c \tan ^{-1}(a x)}{2 x^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 110, normalized size = 1.57 \[ a^{2} c \arctan \left (a x \right ) \ln \left (a x \right )-\frac {c \arctan \left (a x \right )}{2 x^{2}}-\frac {a c}{2 x}-\frac {a^{2} c \arctan \left (a x \right )}{2}+\frac {i a^{2} c \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}-\frac {i a^{2} c \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}+\frac {i a^{2} c \dilog \left (i a x +1\right )}{2}-\frac {i a^{2} c \dilog \left (-i a x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.50, size = 95, normalized size = 1.36 \[ -\frac {\pi a^{2} c x^{2} \log \left (a^{2} x^{2} + 1\right ) - 4 \, a^{2} c x^{2} \arctan \left (a x\right ) \log \left (a x\right ) + 2 i \, a^{2} c x^{2} {\rm Li}_2\left (i \, a x + 1\right ) - 2 i \, a^{2} c x^{2} {\rm Li}_2\left (-i \, a x + 1\right ) + 2 \, a c x + 2 \, {\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.56, size = 71, normalized size = 1.01 \[ \left \{\begin {array}{cl} 0 & \text {\ if\ \ }a=0\\ -\frac {c\,\mathrm {atan}\left (a\,x\right )}{2\,x^2}-\frac {c\,\left (a^3\,\mathrm {atan}\left (a\,x\right )+\frac {a^2}{x}\right )}{2\,a}-\frac {a^2\,c\,{\mathrm {Li}}_{\mathrm {2}}\left (1-a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {a^2\,c\,{\mathrm {Li}}_{\mathrm {2}}\left (1+a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ c \left (\int \frac {\operatorname {atan}{\left (a x \right )}}{x^{3}}\, dx + \int \frac {a^{2} \operatorname {atan}{\left (a x \right )}}{x}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________