Optimal. Leaf size=343 \[ -\frac {i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac {2 b e^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}-i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 b c d^2 \log \left (2-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+2 d e x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{c}+\frac {4 b d e \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c}+\frac {1}{3} e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac {i b^2 e^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{3 c^3}-\frac {b^2 e^2 \tan ^{-1}(c x)}{3 c^3}+\frac {b^2 e^2 x}{3 c^2}-i b^2 c d^2 \text {Li}_2\left (\frac {2}{1-i c x}-1\right )+\frac {2 i b^2 d e \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.58, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {4980, 4846, 4920, 4854, 2402, 2315, 4852, 4924, 4868, 2447, 4916, 321, 203} \[ -\frac {i b^2 e^2 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3}-i b^2 c d^2 \text {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )+\frac {2 i b^2 d e \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}-\frac {i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac {2 b e^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}-i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 b c d^2 \log \left (2-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+2 d e x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{c}+\frac {4 b d e \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c}+\frac {1}{3} e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {b^2 e^2 x}{3 c^2}-\frac {b^2 e^2 \tan ^{-1}(c x)}{3 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 321
Rule 2315
Rule 2402
Rule 2447
Rule 4846
Rule 4852
Rule 4854
Rule 4868
Rule 4916
Rule 4920
Rule 4924
Rule 4980
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx &=\int \left (2 d e \left (a+b \tan ^{-1}(c x)\right )^2+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x^2}+e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx+(2 d e) \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+e^2 \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 d e x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\left (2 b c d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx-(4 b c d e) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac {1}{3} \left (2 b c e^2\right ) \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 d e x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\left (2 i b c d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{x (i+c x)} \, dx+(4 b d e) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx-\frac {\left (2 b e^2\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac {\left (2 b e^2\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}\\ &=-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 d e x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {4 b d e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}+2 b c d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )-\left (2 b^2 c^2 d^2\right ) \int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx-\left (4 b^2 d e\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\frac {1}{3} \left (b^2 e^2\right ) \int \frac {x^2}{1+c^2 x^2} \, dx-\frac {\left (2 b e^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c^2}\\ &=\frac {b^2 e^2 x}{3 c^2}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 d e x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {4 b d e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {2 b e^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+2 b c d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )-i b^2 c d^2 \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )+\frac {\left (4 i b^2 d e\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c}-\frac {\left (b^2 e^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 c^2}+\frac {\left (2 b^2 e^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2}\\ &=\frac {b^2 e^2 x}{3 c^2}-\frac {b^2 e^2 \tan ^{-1}(c x)}{3 c^3}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 d e x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {4 b d e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {2 b e^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+2 b c d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )-i b^2 c d^2 \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )+\frac {2 i b^2 d e \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c}-\frac {\left (2 i b^2 e^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{3 c^3}\\ &=\frac {b^2 e^2 x}{3 c^2}-\frac {b^2 e^2 \tan ^{-1}(c x)}{3 c^3}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 d e x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {4 b d e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {2 b e^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+2 b c d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )-i b^2 c d^2 \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )+\frac {2 i b^2 d e \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c}-\frac {i b^2 e^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{3 c^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.83, size = 349, normalized size = 1.02 \[ \frac {1}{3} \left (-\frac {3 a^2 d^2}{x}+6 a^2 d e x+a^2 e^2 x^3-\frac {3 a b d^2 \left (c x \left (\log \left (c^2 x^2+1\right )-2 \log (c x)\right )+2 \tan ^{-1}(c x)\right )}{x}+\frac {6 a b d e \left (2 c x \tan ^{-1}(c x)-\log \left (c^2 x^2+1\right )\right )}{c}+\frac {a b e^2 \left (2 c^3 x^3 \tan ^{-1}(c x)-c^2 x^2+\log \left (c^2 x^2+1\right )\right )}{c^3}+\frac {b^2 e^2 \left (\left (c^3 x^3+i\right ) \tan ^{-1}(c x)^2-\tan ^{-1}(c x) \left (c^2 x^2+2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+1\right )+i \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+c x\right )}{c^3}+3 b^2 c d^2 \left (\tan ^{-1}(c x) \left (\left (-\frac {1}{c x}-i\right ) \tan ^{-1}(c x)+2 \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )\right )-i \text {Li}_2\left (e^{2 i \tan ^{-1}(c x)}\right )\right )+\frac {6 b^2 d e \left (\tan ^{-1}(c x) \left ((c x-i) \tan ^{-1}(c x)+2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-i \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )\right )}{c}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} e^{2} x^{4} + 2 \, a^{2} d e x^{2} + a^{2} d^{2} + {\left (b^{2} e^{2} x^{4} + 2 \, b^{2} d e x^{2} + b^{2} d^{2}\right )} \arctan \left (c x\right )^{2} + 2 \, {\left (a b e^{2} x^{4} + 2 \, a b d e x^{2} + a b d^{2}\right )} \arctan \left (c x\right )}{x^{2}}, x\right ) \]
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 [B] time = 0.14, size = 997, normalized size = 2.91 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [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]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^2}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )^{2}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________