Optimal. Leaf size=349 \[ \frac {2 x}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 \text {ArcTan}(a x)}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \text {ArcTan}(a x)^2}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.25, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5084, 5010,
5008, 4266, 2611, 2320, 6724, 5018, 197} \begin {gather*} -\frac {x \text {ArcTan}(a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}+\frac {2 x}{a^2 c \sqrt {a^2 c x^2+c}}+\frac {2 i \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (-i e^{i \text {ArcTan}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (i e^{i \text {ArcTan}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \text {ArcTan}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}+\frac {2 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \text {ArcTan}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {2 \text {ArcTan}(a x)}{a^3 c \sqrt {a^2 c x^2+c}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 197
Rule 2320
Rule 2611
Rule 4266
Rule 5008
Rule 5010
Rule 5018
Rule 5084
Rule 6724
Rubi steps
\begin {align*} \int \frac {x^2 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac {\int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2}+\frac {\int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a^2 c}\\ &=-\frac {2 \tan ^{-1}(a x)}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^2}{a^2 c \sqrt {c+a^2 c x^2}}+\frac {2 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2}+\frac {\sqrt {1+a^2 x^2} \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{a^2 c \sqrt {c+a^2 c x^2}}\\ &=\frac {2 x}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 \tan ^{-1}(a x)}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^2}{a^2 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c \sqrt {c+a^2 c x^2}}\\ &=\frac {2 x}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 \tan ^{-1}(a x)}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^2}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c \sqrt {c+a^2 c x^2}}\\ &=\frac {2 x}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 \tan ^{-1}(a x)}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^2}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {\left (2 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\left (2 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c \sqrt {c+a^2 c x^2}}\\ &=\frac {2 x}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 \tan ^{-1}(a x)}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^2}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}\\ &=\frac {2 x}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 \tan ^{-1}(a x)}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^2}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {2 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.25, size = 228, normalized size = 0.65 \begin {gather*} -\frac {\sqrt {1+a^2 x^2} \left (-\frac {2 a x}{\sqrt {1+a^2 x^2}}+\frac {2 \text {ArcTan}(a x)}{\sqrt {1+a^2 x^2}}+\frac {a x \text {ArcTan}(a x)^2}{\sqrt {1+a^2 x^2}}-\text {ArcTan}(a x)^2 \log \left (1-i e^{i \text {ArcTan}(a x)}\right )+\text {ArcTan}(a x)^2 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-2 i \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )+2 i \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )+2 \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )-2 \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )\right )}{a^3 c \sqrt {c \left (1+a^2 x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 1.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \arctan \left (a x \right )^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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 {x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,{\mathrm {atan}\left (a\,x\right )}^2}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________