Optimal. Leaf size=154 \[ -\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{2 x^2}-\frac {1}{2} a c^2 e \log \left (c^2 x^2+1\right )+a c^2 e \log (x)-\frac {b c \left (e \log \left (c^2 x^2+1\right )+d\right )}{2 x}-\frac {1}{2} b c^2 \tan ^{-1}(c x) \left (e \log \left (c^2 x^2+1\right )+d\right )+\frac {1}{2} i b c^2 e \text {Li}_2(-i c x)-\frac {1}{2} i b c^2 e \text {Li}_2(i c x)+b c^2 e \tan ^{-1}(c x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {4852, 325, 203, 5021, 801, 635, 260, 4848, 2391} \[ \frac {1}{2} i b c^2 e \text {PolyLog}(2,-i c x)-\frac {1}{2} i b c^2 e \text {PolyLog}(2,i c x)-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{2 x^2}-\frac {1}{2} a c^2 e \log \left (c^2 x^2+1\right )+a c^2 e \log (x)-\frac {b c \left (e \log \left (c^2 x^2+1\right )+d\right )}{2 x}-\frac {1}{2} b c^2 \tan ^{-1}(c x) \left (e \log \left (c^2 x^2+1\right )+d\right )+b c^2 e \tan ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 260
Rule 325
Rule 635
Rule 801
Rule 2391
Rule 4848
Rule 4852
Rule 5021
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^3} \, dx &=-\frac {b c \left (d+e \log \left (1+c^2 x^2\right )\right )}{2 x}-\frac {1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{2 x^2}-\left (2 c^2 e\right ) \int \left (\frac {-a-b c x}{2 x \left (1+c^2 x^2\right )}-\frac {b \tan ^{-1}(c x)}{2 x}\right ) \, dx\\ &=-\frac {b c \left (d+e \log \left (1+c^2 x^2\right )\right )}{2 x}-\frac {1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{2 x^2}-\left (c^2 e\right ) \int \frac {-a-b c x}{x \left (1+c^2 x^2\right )} \, dx+\left (b c^2 e\right ) \int \frac {\tan ^{-1}(c x)}{x} \, dx\\ &=-\frac {b c \left (d+e \log \left (1+c^2 x^2\right )\right )}{2 x}-\frac {1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{2 x^2}-\left (c^2 e\right ) \int \left (-\frac {a}{x}+\frac {c (-b+a c x)}{1+c^2 x^2}\right ) \, dx+\frac {1}{2} \left (i b c^2 e\right ) \int \frac {\log (1-i c x)}{x} \, dx-\frac {1}{2} \left (i b c^2 e\right ) \int \frac {\log (1+i c x)}{x} \, dx\\ &=a c^2 e \log (x)-\frac {b c \left (d+e \log \left (1+c^2 x^2\right )\right )}{2 x}-\frac {1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{2 x^2}+\frac {1}{2} i b c^2 e \text {Li}_2(-i c x)-\frac {1}{2} i b c^2 e \text {Li}_2(i c x)-\left (c^3 e\right ) \int \frac {-b+a c x}{1+c^2 x^2} \, dx\\ &=a c^2 e \log (x)-\frac {b c \left (d+e \log \left (1+c^2 x^2\right )\right )}{2 x}-\frac {1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{2 x^2}+\frac {1}{2} i b c^2 e \text {Li}_2(-i c x)-\frac {1}{2} i b c^2 e \text {Li}_2(i c x)+\left (b c^3 e\right ) \int \frac {1}{1+c^2 x^2} \, dx-\left (a c^4 e\right ) \int \frac {x}{1+c^2 x^2} \, dx\\ &=b c^2 e \tan ^{-1}(c x)+a c^2 e \log (x)-\frac {1}{2} a c^2 e \log \left (1+c^2 x^2\right )-\frac {b c \left (d+e \log \left (1+c^2 x^2\right )\right )}{2 x}-\frac {1}{2} b c^2 \tan ^{-1}(c x) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{2 x^2}+\frac {1}{2} i b c^2 e \text {Li}_2(-i c x)-\frac {1}{2} i b c^2 e \text {Li}_2(i c x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 189, normalized size = 1.23 \[ -\frac {-2 a c^2 e x^2 \log (x)+a c^2 e x^2 \log \left (c^2 x^2+1\right )+a e \log \left (c^2 x^2+1\right )+a d+b c^2 d x^2 \tan ^{-1}(c x)-i b c^2 e x^2 \text {Li}_2(-i c x)+i b c^2 e x^2 \text {Li}_2(i c x)+b c e x \log \left (c^2 x^2+1\right )-2 b c^2 e x^2 \tan ^{-1}(c x)+b c^2 e x^2 \log \left (c^2 x^2+1\right ) \tan ^{-1}(c x)+b e \log \left (c^2 x^2+1\right ) \tan ^{-1}(c x)+b c d x+b d \tan ^{-1}(c x)}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b d \arctan \left (c x\right ) + a d + {\left (b e \arctan \left (c x\right ) + a e\right )} \log \left (c^{2} x^{2} + 1\right )}{x^{3}}, x\right ) \]
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 [F] time = 38.69, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arctan \left (c x \right )\right ) \left (d +e \ln \left (c^{2} x^{2}+1\right )\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b d - \frac {1}{2} \, {\left (c^{2} {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {\log \left (c^{2} x^{2} + 1\right )}{x^{2}}\right )} a e + \frac {{\left (2 \, c^{4} x^{2} \int \frac {x \arctan \left (c x\right )}{c^{2} x^{2} + 1}\,{d x} + 2 \, c^{2} x^{2} \arctan \left (c x\right ) + 2 \, c^{2} x^{2} \int \frac {\arctan \left (c x\right )}{c^{2} x^{3} + x}\,{d x} - {\left (c x + {\left (c^{2} x^{2} + 1\right )} \arctan \left (c x\right )\right )} \log \left (c^{2} x^{2} + 1\right )\right )} b e}{2 \, x^{2}} - \frac {a d}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (c^2\,x^2+1\right )\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e \log {\left (c^{2} x^{2} + 1 \right )}\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________