3.12.55 \(\int \frac {x^2 (a+b \text {ArcTan}(c x))}{d+e x^2} \, dx\) [1155]

Optimal. Leaf size=555 \[ \frac {a x}{e}+\frac {b x \text {ArcTan}(c x)}{e}-\frac {a \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2}}-\frac {i b \sqrt {-d} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 e^{3/2}}+\frac {i b \sqrt {-d} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 e^{3/2}}-\frac {i b \sqrt {-d} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 e^{3/2}}+\frac {i b \sqrt {-d} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 e^{3/2}}-\frac {b \log \left (1+c^2 x^2\right )}{2 c e}+\frac {i b \sqrt {-d} \text {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 e^{3/2}}-\frac {i b \sqrt {-d} \text {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 e^{3/2}}-\frac {i b \sqrt {-d} \text {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 e^{3/2}}+\frac {i b \sqrt {-d} \text {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 e^{3/2}} \]

[Out]

a*x/e+b*x*arctan(c*x)/e-1/2*b*ln(c^2*x^2+1)/c/e-1/4*I*b*ln(1+I*c*x)*ln(c*((-d)^(1/2)-x*e^(1/2))/(c*(-d)^(1/2)-
I*e^(1/2)))*(-d)^(1/2)/e^(3/2)+1/4*I*b*ln(1-I*c*x)*ln(c*((-d)^(1/2)-x*e^(1/2))/(c*(-d)^(1/2)+I*e^(1/2)))*(-d)^
(1/2)/e^(3/2)-1/4*I*b*ln(1-I*c*x)*ln(c*((-d)^(1/2)+x*e^(1/2))/(c*(-d)^(1/2)-I*e^(1/2)))*(-d)^(1/2)/e^(3/2)+1/4
*I*b*ln(1+I*c*x)*ln(c*((-d)^(1/2)+x*e^(1/2))/(c*(-d)^(1/2)+I*e^(1/2)))*(-d)^(1/2)/e^(3/2)+1/4*I*b*polylog(2,(I
-c*x)*e^(1/2)/(c*(-d)^(1/2)+I*e^(1/2)))*(-d)^(1/2)/e^(3/2)+1/4*I*b*polylog(2,(c*x+I)*e^(1/2)/(c*(-d)^(1/2)+I*e
^(1/2)))*(-d)^(1/2)/e^(3/2)-1/4*I*b*polylog(2,(1-I*c*x)*e^(1/2)/(I*c*(-d)^(1/2)+e^(1/2)))*(-d)^(1/2)/e^(3/2)-1
/4*I*b*polylog(2,(1+I*c*x)*e^(1/2)/(I*c*(-d)^(1/2)+e^(1/2)))*(-d)^(1/2)/e^(3/2)-a*arctan(x*e^(1/2)/d^(1/2))*d^
(1/2)/e^(3/2)

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Rubi [A]
time = 0.49, antiderivative size = 555, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5036, 4930, 266, 5030, 211, 5028, 2456, 2441, 2440, 2438} \begin {gather*} -\frac {a \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2}}+\frac {a x}{e}+\frac {b x \text {ArcTan}(c x)}{e}-\frac {b \log \left (c^2 x^2+1\right )}{2 c e}+\frac {i b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} (i-c x)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 e^{3/2}}-\frac {i b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} (1-i c x)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 e^{3/2}}-\frac {i b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} (i c x+1)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 e^{3/2}}+\frac {i b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} (c x+i)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 e^{3/2}}-\frac {i b \sqrt {-d} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 e^{3/2}}+\frac {i b \sqrt {-d} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 e^{3/2}}-\frac {i b \sqrt {-d} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 e^{3/2}}+\frac {i b \sqrt {-d} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 e^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(a + b*ArcTan[c*x]))/(d + e*x^2),x]

[Out]

(a*x)/e + (b*x*ArcTan[c*x])/e - (a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e^(3/2) - ((I/4)*b*Sqrt[-d]*Log[1 + I*
c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/e^(3/2) + ((I/4)*b*Sqrt[-d]*Log[1 - I*c*x]*Log[
(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/e^(3/2) - ((I/4)*b*Sqrt[-d]*Log[1 - I*c*x]*Log[(c*(Sqrt[
-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/e^(3/2) + ((I/4)*b*Sqrt[-d]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqr
t[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/e^(3/2) - (b*Log[1 + c^2*x^2])/(2*c*e) + ((I/4)*b*Sqrt[-d]*PolyLog[2, (Sqr
t[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/e^(3/2) - ((I/4)*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*S
qrt[-d] + Sqrt[e])])/e^(3/2) - ((I/4)*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/e
^(3/2) + ((I/4)*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/e^(3/2)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2456

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Dist[b*c
*n*p, Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0
] && (EqQ[n, 1] || EqQ[p, 1])

Rule 5028

Int[ArcTan[(c_.)*(x_)]/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Dist[I/2, Int[Log[1 - I*c*x]/(d + e*x^2), x], x] -
Dist[I/2, Int[Log[1 + I*c*x]/(d + e*x^2), x], x] /; FreeQ[{c, d, e}, x]

Rule 5030

Int[(ArcTan[(c_.)*(x_)]*(b_.) + (a_))/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Dist[a, Int[1/(d + e*x^2), x], x] +
Dist[b, Int[ArcTan[c*x]/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]

Rule 5036

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[d*(f^2/e), Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/
(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

Rubi steps

\begin {align*} \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{d+e x^2} \, dx &=\frac {\int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{e}-\frac {d \int \frac {a+b \tan ^{-1}(c x)}{d+e x^2} \, dx}{e}\\ &=\frac {a x}{e}+\frac {b \int \tan ^{-1}(c x) \, dx}{e}-\frac {(a d) \int \frac {1}{d+e x^2} \, dx}{e}-\frac {(b d) \int \frac {\tan ^{-1}(c x)}{d+e x^2} \, dx}{e}\\ &=\frac {a x}{e}+\frac {b x \tan ^{-1}(c x)}{e}-\frac {a \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2}}-\frac {(b c) \int \frac {x}{1+c^2 x^2} \, dx}{e}-\frac {(i b d) \int \frac {\log (1-i c x)}{d+e x^2} \, dx}{2 e}+\frac {(i b d) \int \frac {\log (1+i c x)}{d+e x^2} \, dx}{2 e}\\ &=\frac {a x}{e}+\frac {b x \tan ^{-1}(c x)}{e}-\frac {a \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2}}-\frac {b \log \left (1+c^2 x^2\right )}{2 c e}-\frac {(i b d) \int \left (\frac {\sqrt {-d} \log (1-i c x)}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \log (1-i c x)}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{2 e}+\frac {(i b d) \int \left (\frac {\sqrt {-d} \log (1+i c x)}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \log (1+i c x)}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{2 e}\\ &=\frac {a x}{e}+\frac {b x \tan ^{-1}(c x)}{e}-\frac {a \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2}}-\frac {b \log \left (1+c^2 x^2\right )}{2 c e}-\frac {\left (i b \sqrt {-d}\right ) \int \frac {\log (1-i c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 e}-\frac {\left (i b \sqrt {-d}\right ) \int \frac {\log (1-i c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 e}+\frac {\left (i b \sqrt {-d}\right ) \int \frac {\log (1+i c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 e}+\frac {\left (i b \sqrt {-d}\right ) \int \frac {\log (1+i c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 e}\\ &=\frac {a x}{e}+\frac {b x \tan ^{-1}(c x)}{e}-\frac {a \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2}}-\frac {i b \sqrt {-d} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 e^{3/2}}+\frac {i b \sqrt {-d} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 e^{3/2}}-\frac {i b \sqrt {-d} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 e^{3/2}}+\frac {i b \sqrt {-d} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 e^{3/2}}-\frac {b \log \left (1+c^2 x^2\right )}{2 c e}-\frac {\left (b c \sqrt {-d}\right ) \int \frac {\log \left (-\frac {i c \left (\sqrt {-d}-\sqrt {e} x\right )}{-i c \sqrt {-d}+\sqrt {e}}\right )}{1-i c x} \, dx}{4 e^{3/2}}-\frac {\left (b c \sqrt {-d}\right ) \int \frac {\log \left (\frac {i c \left (\sqrt {-d}-\sqrt {e} x\right )}{i c \sqrt {-d}+\sqrt {e}}\right )}{1+i c x} \, dx}{4 e^{3/2}}+\frac {\left (b c \sqrt {-d}\right ) \int \frac {\log \left (-\frac {i c \left (\sqrt {-d}+\sqrt {e} x\right )}{-i c \sqrt {-d}-\sqrt {e}}\right )}{1-i c x} \, dx}{4 e^{3/2}}+\frac {\left (b c \sqrt {-d}\right ) \int \frac {\log \left (\frac {i c \left (\sqrt {-d}+\sqrt {e} x\right )}{i c \sqrt {-d}-\sqrt {e}}\right )}{1+i c x} \, dx}{4 e^{3/2}}\\ &=\frac {a x}{e}+\frac {b x \tan ^{-1}(c x)}{e}-\frac {a \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2}}-\frac {i b \sqrt {-d} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 e^{3/2}}+\frac {i b \sqrt {-d} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 e^{3/2}}-\frac {i b \sqrt {-d} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 e^{3/2}}+\frac {i b \sqrt {-d} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 e^{3/2}}-\frac {b \log \left (1+c^2 x^2\right )}{2 c e}+\frac {\left (i b \sqrt {-d}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{-i c \sqrt {-d}-\sqrt {e}}\right )}{x} \, dx,x,1-i c x\right )}{4 e^{3/2}}-\frac {\left (i b \sqrt {-d}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{i c \sqrt {-d}-\sqrt {e}}\right )}{x} \, dx,x,1+i c x\right )}{4 e^{3/2}}-\frac {\left (i b \sqrt {-d}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{-i c \sqrt {-d}+\sqrt {e}}\right )}{x} \, dx,x,1-i c x\right )}{4 e^{3/2}}+\frac {\left (i b \sqrt {-d}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{i c \sqrt {-d}+\sqrt {e}}\right )}{x} \, dx,x,1+i c x\right )}{4 e^{3/2}}\\ &=\frac {a x}{e}+\frac {b x \tan ^{-1}(c x)}{e}-\frac {a \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2}}-\frac {i b \sqrt {-d} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 e^{3/2}}+\frac {i b \sqrt {-d} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 e^{3/2}}-\frac {i b \sqrt {-d} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 e^{3/2}}+\frac {i b \sqrt {-d} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 e^{3/2}}-\frac {b \log \left (1+c^2 x^2\right )}{2 c e}+\frac {i b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 e^{3/2}}-\frac {i b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 e^{3/2}}-\frac {i b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 e^{3/2}}+\frac {i b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 e^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 2.35, size = 766, normalized size = 1.38 \begin {gather*} \frac {a x}{e}-\frac {a \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2}}+\frac {b \left (4 c x \text {ArcTan}(c x)-2 \log \left (1+c^2 x^2\right )+\frac {c^2 d \left (-4 \text {ArcTan}(c x) \tanh ^{-1}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )-2 \text {ArcCos}\left (\frac {c^2 d+e}{-c^2 d+e}\right ) \tanh ^{-1}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )-\left (\text {ArcCos}\left (\frac {c^2 d+e}{-c^2 d+e}\right )-2 i \tanh ^{-1}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 c d \left (i e+\sqrt {-c^2 d e}\right ) (-i+c x)}{\left (c^2 d-e\right ) \left (-c d+\sqrt {-c^2 d e} x\right )}\right )-\left (\text {ArcCos}\left (\frac {c^2 d+e}{-c^2 d+e}\right )+2 i \tanh ^{-1}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 c d \left (-i e+\sqrt {-c^2 d e}\right ) (i+c x)}{\left (c^2 d-e\right ) \left (-c d+\sqrt {-c^2 d e} x\right )}\right )+\left (\text {ArcCos}\left (\frac {c^2 d+e}{-c^2 d+e}\right )+2 i \tanh ^{-1}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+2 i \tanh ^{-1}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{-i \text {ArcTan}(c x)}}{\sqrt {-c^2 d+e} \sqrt {-c^2 d-e+\left (-c^2 d+e\right ) \cos (2 \text {ArcTan}(c x))}}\right )+\left (\text {ArcCos}\left (\frac {c^2 d+e}{-c^2 d+e}\right )-2 i \tanh ^{-1}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )-2 i \tanh ^{-1}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{i \text {ArcTan}(c x)}}{\sqrt {-c^2 d+e} \sqrt {-c^2 d-e+\left (-c^2 d+e\right ) \cos (2 \text {ArcTan}(c x))}}\right )+i \left (-\text {PolyLog}\left (2,\frac {\left (c^2 d+e-2 i \sqrt {-c^2 d e}\right ) \left (c d+\sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (c d-\sqrt {-c^2 d e} x\right )}\right )+\text {PolyLog}\left (2,\frac {\left (c^2 d+e+2 i \sqrt {-c^2 d e}\right ) \left (c d+\sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (c d-\sqrt {-c^2 d e} x\right )}\right )\right )\right )}{\sqrt {-c^2 d e}}\right )}{4 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(a + b*ArcTan[c*x]))/(d + e*x^2),x]

[Out]

(a*x)/e - (a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e^(3/2) + (b*(4*c*x*ArcTan[c*x] - 2*Log[1 + c^2*x^2] + (c^2*
d*(-4*ArcTan[c*x]*ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] - 2*ArcCos[(c^2*d + e)/(-(c^2*d) + e)]*ArcTanh[(c*e*x)/S
qrt[-(c^2*d*e)]] - (ArcCos[(c^2*d + e)/(-(c^2*d) + e)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c*d*(
I*e + Sqrt[-(c^2*d*e)])*(-I + c*x))/((c^2*d - e)*(-(c*d) + Sqrt[-(c^2*d*e)]*x))] - (ArcCos[(c^2*d + e)/(-(c^2*
d) + e)] + (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c*d*((-I)*e + Sqrt[-(c^2*d*e)])*(I + c*x))/((c^2*d
- e)*(-(c*d) + Sqrt[-(c^2*d*e)]*x))] + (ArcCos[(c^2*d + e)/(-(c^2*d) + e)] + (2*I)*ArcTanh[(c*d)/(Sqrt[-(c^2*d
*e)]*x)] + (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[-(c^2*d) + e]*E^(I*Ar
cTan[c*x])*Sqrt[-(c^2*d) - e + (-(c^2*d) + e)*Cos[2*ArcTan[c*x]]])] + (ArcCos[(c^2*d + e)/(-(c^2*d) + e)] - (2
*I)*ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(Sqrt[2]*Sqrt[-(c^2*d*e
)]*E^(I*ArcTan[c*x]))/(Sqrt[-(c^2*d) + e]*Sqrt[-(c^2*d) - e + (-(c^2*d) + e)*Cos[2*ArcTan[c*x]]])] + I*(-PolyL
og[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(c*d + Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c*d - Sqrt[-(c^2*d*e)]*x
))] + PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(c*d + Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c*d - Sqrt[-(
c^2*d*e)]*x))])))/Sqrt[-(c^2*d*e)]))/(4*c*e)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.87, size = 2437, normalized size = 4.39

method	result	size
risch	\(\frac {i a}{c e}-\frac {b \ln \left (c^{2} x^{2}+1\right )}{2 c e}-\frac {i a d \arctanh \left (\frac {2 \left (-i c x +1\right ) e -2 e}{2 c \sqrt {d e}}\right )}{e \sqrt {d e}}+\frac {i b \ln \left (-i c x +1\right ) x}{2 e}+\frac {b}{c e}-\frac {b d \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {d e}-\left (-i c x +1\right ) e +e}{c \sqrt {d e}+e}\right )}{4 e \sqrt {d e}}+\frac {b d \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {d e}+\left (-i c x +1\right ) e -e}{c \sqrt {d e}-e}\right )}{4 e \sqrt {d e}}-\frac {b d \dilog \left (\frac {c \sqrt {d e}-\left (-i c x +1\right ) e +e}{c \sqrt {d e}+e}\right )}{4 e \sqrt {d e}}+\frac {b d \dilog \left (\frac {c \sqrt {d e}+\left (-i c x +1\right ) e -e}{c \sqrt {d e}-e}\right )}{4 e \sqrt {d e}}+\frac {a x}{e}-\frac {i b \ln \left (i c x +1\right ) x}{2 e}-\frac {b d \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {d e}-\left (i c x +1\right ) e +e}{c \sqrt {d e}+e}\right )}{4 e \sqrt {d e}}+\frac {b d \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {d e}+\left (i c x +1\right ) e -e}{c \sqrt {d e}-e}\right )}{4 e \sqrt {d e}}-\frac {b d \dilog \left (\frac {c \sqrt {d e}-\left (i c x +1\right ) e +e}{c \sqrt {d e}+e}\right )}{4 e \sqrt {d e}}+\frac {b d \dilog \left (\frac {c \sqrt {d e}+\left (i c x +1\right ) e -e}{c \sqrt {d e}-e}\right )}{4 e \sqrt {d e}}\)	\(511\)
derivativedivides	\(\text {Expression too large to display}\)	\(2437\)
default	\(\text {Expression too large to display}\)	\(2437\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arctan(c*x))/(e*x^2+d),x,method=_RETURNVERBOSE)

[Out]

1/c^3*(-5/8*b*c^4*d/(c^2*d-e)^2*ln(c^2*d*(1+I*c*x)^4/(c^2*x^2+1)^2+2*c^2*d*(1+I*c*x)^2/(c^2*x^2+1)-e*(1+I*c*x)
^4/(c^2*x^2+1)^2+c^2*d+2*e*(1+I*c*x)^2/(c^2*x^2+1)-e)+3/8*b*c^2*e/(c^2*d-e)^2*ln(c^2*d*(1+I*c*x)^4/(c^2*x^2+1)
^2+2*c^2*d*(1+I*c*x)^2/(c^2*x^2+1)-e*(1+I*c*x)^4/(c^2*x^2+1)^2+c^2*d+2*e*(1+I*c*x)^2/(c^2*x^2+1)-e)+3/4*b*c^3*
(d*e)^(1/2)/e^2*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2*e)/c/(d*e)^(1/2))+a*c^3/e*x+2*b*c^2
/(c^2*d-e)*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))+b*c^2/e*ln((1+I*c*x)^2/(c^2*x^2+1)+1)-1/4*b*c^2/e*ln(c^2*d*(1+I*c*x
)^4/(c^2*x^2+1)^2+2*c^2*d*(1+I*c*x)^2/(c^2*x^2+1)-e*(1+I*c*x)^4/(c^2*x^2+1)^2+c^2*d+2*e*(1+I*c*x)^2/(c^2*x^2+1
)-e)+1/8*b*c^2/(c^2*d-e)*ln(c^2*d*(1+I*c*x)^4/(c^2*x^2+1)^2+2*c^2*d*(1+I*c*x)^2/(c^2*x^2+1)-e*(1+I*c*x)^4/(c^2
*x^2+1)^2+c^2*d+2*e*(1+I*c*x)^2/(c^2*x^2+1)-e)-1/4*b*c^3*(d*e)^(1/2)*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2
*x^2+1)+2*c^2*d+2*e)/c/(d*e)^(1/2))/(c^2*d-e)^2+1/4*b*c^4/e^2*d*sum((_R1^2*c^2*d-_R1^2*e+c^2*d+3*e)/(_R1^2*c^2
*d-_R1^2*e+c^2*d+e)*(I*arctan(c*x)*ln((_R1-(1+I*c*x)/(c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-(1+I*c*x)/(c^2*x^2+1)^
(1/2))/_R1)),_R1=RootOf((c^2*d-e)*_Z^4+(2*c^2*d+2*e)*_Z^2+c^2*d-e))-1/4*b*c^4/e^2*d*ln(c^2*d*(1+I*c*x)^4/(c^2*
x^2+1)^2+2*c^2*d*(1+I*c*x)^2/(c^2*x^2+1)-e*(1+I*c*x)^4/(c^2*x^2+1)^2+c^2*d+2*e*(1+I*c*x)^2/(c^2*x^2+1)-e)+I*b*
c^2*arctan(c*x)/e-1/4*b*c^4/e^2*d*sum((_R1^2*c^2*d-_R1^2*e+c^2*d-e)/(_R1^2*c^2*d-_R1^2*e+c^2*d+e)*(I*arctan(c*
x)*ln((_R1-(1+I*c*x)/(c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-(1+I*c*x)/(c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf((c^2*d-e
)*_Z^4+(2*c^2*d+2*e)*_Z^2+c^2*d-e))+b*c^3*arctan(c*x)/e*x-a*c^3*d/e/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+1/8*b*
c^6/e*d^2/(c^2*d-e)^2*ln(c^2*d*(1+I*c*x)^4/(c^2*x^2+1)^2+2*c^2*d*(1+I*c*x)^2/(c^2*x^2+1)-e*(1+I*c*x)^4/(c^2*x^
2+1)^2+c^2*d+2*e*(1+I*c*x)^2/(c^2*x^2+1)-e)-2*b*c^4/e*d/(c^2*d-e)*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))+1/8*b*c^6/e^
2*d^2/(c^2*d-e)*ln(c^2*d*(1+I*c*x)^4/(c^2*x^2+1)^2+2*c^2*d*(1+I*c*x)^2/(c^2*x^2+1)-e*(1+I*c*x)^4/(c^2*x^2+1)^2
+c^2*d+2*e*(1+I*c*x)^2/(c^2*x^2+1)-e)-1/8*b*c^5*(d*e)^(1/2)/e^3*d*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^
2+1)+2*c^2*d+2*e)/c/(d*e)^(1/2))+1/2*b*c^3*(d*e)^(1/2)/e*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^
2*d+2*e)/c/(d*e)^(1/2))/(c^2*d-e)+3/8*b*c*(d*e)^(1/2)/d/e*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c
^2*d+2*e)/c/(d*e)^(1/2))-1/4*b*c^4/e*d/(c^2*d-e)*ln(c^2*d*(1+I*c*x)^4/(c^2*x^2+1)^2+2*c^2*d*(1+I*c*x)^2/(c^2*x
^2+1)-e*(1+I*c*x)^4/(c^2*x^2+1)^2+c^2*d+2*e*(1+I*c*x)^2/(c^2*x^2+1)-e)+1/8*b*c^8/e^2*d^3/(c^2*d-e)^2*ln(c^2*d*
(1+I*c*x)^4/(c^2*x^2+1)^2+2*c^2*d*(1+I*c*x)^2/(c^2*x^2+1)-e*(1+I*c*x)^4/(c^2*x^2+1)^2+c^2*d+2*e*(1+I*c*x)^2/(c
^2*x^2+1)-e)+3/4*b*c*(d*e)^(1/2)/d*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2*e)/c/(d*e)^(1/2)
)/(c^2*d-e)-5/4*b*c^5*(d*e)^(1/2)/e^2*d*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2*e)/c/(d*e)^
(1/2))/(c^2*d-e)-1/2*b*c^5*(d*e)^(1/2)*d/e*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2*e)/c/(d*
e)^(1/2))/(c^2*d-e)^2+1/8*b*c^9*(d*e)^(1/2)/e^3*d^3*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2
*e)/c/(d*e)^(1/2))/(c^2*d-e)^2+3/8*b*c*(d*e)^(1/2)/d*e*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*
d+2*e)/c/(d*e)^(1/2))/(c^2*d-e)^2+1/4*b*c^7*(d*e)^(1/2)/e^2*d^2*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+
1)+2*c^2*d+2*e)/c/(d*e)^(1/2))/(c^2*d-e)^2)

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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]

integrate(x^2*(a+b*arctan(c*x))/(e*x^2+d),x, algorithm="maxima")

[Out]

-(sqrt(d)*arctan(x*e^(1/2)/sqrt(d))*e^(-3/2) - x*e^(-1))*a + 2*b*integrate(1/2*x^2*arctan(c*x)/(x^2*e + d), x)

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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]

integrate(x^2*(a+b*arctan(c*x))/(e*x^2+d),x, algorithm="fricas")

[Out]

integral((b*x^2*arctan(c*x) + a*x^2)/(x^2*e + d), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*atan(c*x))/(e*x**2+d),x)

[Out]

Integral(x**2*(a + b*atan(c*x))/(d + e*x**2), x)

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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]

integrate(x^2*(a+b*arctan(c*x))/(e*x^2+d),x, algorithm="giac")

[Out]

sage0*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{e\,x^2+d} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(a + b*atan(c*x)))/(d + e*x^2),x)

[Out]

int((x^2*(a + b*atan(c*x)))/(d + e*x^2), x)

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