Integrand size = 26, antiderivative size = 160 \[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x} \, dx=-\frac {3 b \sqrt {x}}{2 c^5}-\frac {b x^{3/2}}{6 c^3}+\frac {3 b \text {arctanh}\left (c \sqrt {x}\right )}{2 c^6}-\frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c^4}-\frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{2 c^2}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{b c^6}+\frac {2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c^6}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{c^6} \] Output:
-3/2*b*x^(1/2)/c^5-1/6*b*x^(3/2)/c^3+3/2*b*arctanh(c*x^(1/2))/c^6-x*(a+b*a rctanh(c*x^(1/2)))/c^4-1/2*x^2*(a+b*arctanh(c*x^(1/2)))/c^2-(a+b*arctanh(c *x^(1/2)))^2/b/c^6+2*(a+b*arctanh(c*x^(1/2)))*ln(2/(1-c*x^(1/2)))/c^6+b*po lylog(2,1-2/(1-c*x^(1/2)))/c^6
Time = 0.35 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.81 \[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x} \, dx=-\frac {9 b c \sqrt {x}+6 a c^2 x+b c^3 x^{3/2}+3 a c^4 x^2-6 b \text {arctanh}\left (c \sqrt {x}\right )^2+3 b \text {arctanh}\left (c \sqrt {x}\right ) \left (-3+2 c^2 x+c^4 x^2-4 \log \left (1+e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )+6 a \log \left (1-c^2 x\right )+6 b \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )}{6 c^6} \] Input:
Integrate[(x^2*(a + b*ArcTanh[c*Sqrt[x]]))/(1 - c^2*x),x]
Output:
-1/6*(9*b*c*Sqrt[x] + 6*a*c^2*x + b*c^3*x^(3/2) + 3*a*c^4*x^2 - 6*b*ArcTan h[c*Sqrt[x]]^2 + 3*b*ArcTanh[c*Sqrt[x]]*(-3 + 2*c^2*x + c^4*x^2 - 4*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + 6*a*Log[1 - c^2*x] + 6*b*PolyLog[2, -E^(-2 *ArcTanh[c*Sqrt[x]])])/c^6
Time = 1.39 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.36, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {7267, 6542, 6452, 254, 2009, 6542, 6452, 262, 219, 6546, 6470, 2849, 2752}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 2 \int \frac {x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}\) |
\(\Big \downarrow \) 6542 |
\(\displaystyle 2 \left (\frac {\int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{c^2}\right )\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle 2 \left (\frac {\int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \int \frac {x^2}{1-c^2 x}d\sqrt {x}}{c^2}\right )\) |
\(\Big \downarrow \) 254 |
\(\displaystyle 2 \left (\frac {\int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \int \left (-\frac {x}{c^2}+\frac {1}{c^4 \left (1-c^2 x\right )}-\frac {1}{c^4}\right )d\sqrt {x}}{c^2}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {\int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )\) |
\(\Big \downarrow \) 6542 |
\(\displaystyle 2 \left (\frac {\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int \sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle 2 \left (\frac {\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \int \frac {x}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle 2 \left (\frac {\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\int \frac {1}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 \left (\frac {\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )\) |
\(\Big \downarrow \) 6546 |
\(\displaystyle 2 \left (\frac {\frac {\frac {\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c \sqrt {x}}d\sqrt {x}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )\) |
\(\Big \downarrow \) 6470 |
\(\displaystyle 2 \left (\frac {\frac {\frac {\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}-b \int \frac {\log \left (\frac {2}{1-c \sqrt {x}}\right )}{1-c^2 x}d\sqrt {x}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )\) |
\(\Big \downarrow \) 2849 |
\(\displaystyle 2 \left (\frac {\frac {\frac {\frac {b \int \frac {\log \left (\frac {2}{1-c \sqrt {x}}\right )}{1-\frac {2}{1-c \sqrt {x}}}d\frac {1}{1-c \sqrt {x}}}{c}+\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )\) |
\(\Big \downarrow \) 2752 |
\(\displaystyle 2 \left (\frac {\frac {\frac {\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{2 c}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )\) |
Input:
Int[(x^2*(a + b*ArcTanh[c*Sqrt[x]]))/(1 - c^2*x),x]
Output:
2*(-(((x^2*(a + b*ArcTanh[c*Sqrt[x]]))/4 - (b*c*(-(Sqrt[x]/c^4) - x^(3/2)/ (3*c^2) + ArcTanh[c*Sqrt[x]]/c^5))/4)/c^2) + (-(((x*(a + b*ArcTanh[c*Sqrt[ x]]))/2 - (b*c*(-(Sqrt[x]/c^2) + ArcTanh[c*Sqrt[x]]/c^3))/2)/c^2) + (-1/2* (a + b*ArcTanh[c*Sqrt[x]])^2/(b*c^2) + (((a + b*ArcTanh[c*Sqrt[x]])*Log[2/ (1 - c*Sqrt[x])])/c + (b*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/(2*c))/c)/c^2) /c^2)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol ] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c *(p/e) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 , 0]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTanh[c* x])^p, x], x] - Simp[d*(f^2/e) Int[(f*x)^(m - 2)*((a + b*ArcTanh[c*x])^p/ (d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ (c*d) Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.07 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.35
method | result | size |
parts | \(-\frac {a \,x^{2}}{2 c^{2}}-\frac {a x}{c^{4}}-\frac {a \ln \left (c^{2} x -1\right )}{c^{6}}-\frac {2 b \left (\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{4} x^{2}}{4}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{2} x}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{8}-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{8}+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {c^{3} x^{\frac {3}{2}}}{12}+\frac {3 c \sqrt {x}}{4}+\frac {3 \ln \left (c \sqrt {x}-1\right )}{8}-\frac {3 \ln \left (1+c \sqrt {x}\right )}{8}\right )}{c^{6}}\) | \(216\) |
derivativedivides | \(-\frac {2 \left (a \left (\frac {c^{4} x^{2}}{4}+\frac {c^{2} x}{2}+\frac {\ln \left (c \sqrt {x}-1\right )}{2}+\frac {\ln \left (1+c \sqrt {x}\right )}{2}\right )+b \left (\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{4} x^{2}}{4}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{2} x}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{8}-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{8}+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {c^{3} x^{\frac {3}{2}}}{12}+\frac {3 c \sqrt {x}}{4}+\frac {3 \ln \left (c \sqrt {x}-1\right )}{8}-\frac {3 \ln \left (1+c \sqrt {x}\right )}{8}\right )\right )}{c^{6}}\) | \(224\) |
default | \(-\frac {2 \left (a \left (\frac {c^{4} x^{2}}{4}+\frac {c^{2} x}{2}+\frac {\ln \left (c \sqrt {x}-1\right )}{2}+\frac {\ln \left (1+c \sqrt {x}\right )}{2}\right )+b \left (\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{4} x^{2}}{4}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{2} x}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{8}-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{8}+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {c^{3} x^{\frac {3}{2}}}{12}+\frac {3 c \sqrt {x}}{4}+\frac {3 \ln \left (c \sqrt {x}-1\right )}{8}-\frac {3 \ln \left (1+c \sqrt {x}\right )}{8}\right )\right )}{c^{6}}\) | \(224\) |
Input:
int(x^2*(a+b*arctanh(c*x^(1/2)))/(-c^2*x+1),x,method=_RETURNVERBOSE)
Output:
-1/2*a/c^2*x^2-a/c^4*x-a/c^6*ln(c^2*x-1)-2*b/c^6*(1/4*arctanh(c*x^(1/2))*c ^4*x^2+1/2*arctanh(c*x^(1/2))*c^2*x+1/2*arctanh(c*x^(1/2))*ln(c*x^(1/2)-1) +1/2*arctanh(c*x^(1/2))*ln(1+c*x^(1/2))-1/2*dilog(1/2*c*x^(1/2)+1/2)-1/4*l n(c*x^(1/2)-1)*ln(1/2*c*x^(1/2)+1/2)+1/8*ln(c*x^(1/2)-1)^2-1/8*ln(1+c*x^(1 /2))^2+1/4*(ln(1+c*x^(1/2))-ln(1/2*c*x^(1/2)+1/2))*ln(-1/2*c*x^(1/2)+1/2)+ 1/12*c^3*x^(3/2)+3/4*c*x^(1/2)+3/8*ln(c*x^(1/2)-1)-3/8*ln(1+c*x^(1/2)))
\[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x} \, dx=\int { -\frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )} x^{2}}{c^{2} x - 1} \,d x } \] Input:
integrate(x^2*(a+b*arctanh(c*x^(1/2)))/(-c^2*x+1),x, algorithm="fricas")
Output:
integral(-(b*x^2*arctanh(c*sqrt(x)) + a*x^2)/(c^2*x - 1), x)
\[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x} \, dx=- \int \frac {a x^{2}}{c^{2} x - 1}\, dx - \int \frac {b x^{2} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x - 1}\, dx \] Input:
integrate(x**2*(a+b*atanh(c*x**(1/2)))/(-c**2*x+1),x)
Output:
-Integral(a*x**2/(c**2*x - 1), x) - Integral(b*x**2*atanh(c*sqrt(x))/(c**2 *x - 1), x)
Time = 0.21 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.30 \[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x} \, dx=-\frac {1}{2} \, a {\left (\frac {c^{2} x^{2} + 2 \, x}{c^{4}} + \frac {2 \, \log \left (c^{2} x - 1\right )}{c^{6}}\right )} - \frac {{\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right )\right )} b}{c^{6}} + \frac {3 \, b \log \left (c \sqrt {x} + 1\right )}{4 \, c^{6}} - \frac {3 \, b \log \left (c \sqrt {x} - 1\right )}{4 \, c^{6}} - \frac {2 \, b c^{3} x^{\frac {3}{2}} + 3 \, b \log \left (c \sqrt {x} + 1\right )^{2} - 3 \, b \log \left (-c \sqrt {x} + 1\right )^{2} + 18 \, b c \sqrt {x} + 3 \, {\left (b c^{4} x^{2} + 2 \, b c^{2} x\right )} \log \left (c \sqrt {x} + 1\right ) - 3 \, {\left (b c^{4} x^{2} + 2 \, b c^{2} x + 2 \, b \log \left (c \sqrt {x} + 1\right )\right )} \log \left (-c \sqrt {x} + 1\right )}{12 \, c^{6}} \] Input:
integrate(x^2*(a+b*arctanh(c*x^(1/2)))/(-c^2*x+1),x, algorithm="maxima")
Output:
-1/2*a*((c^2*x^2 + 2*x)/c^4 + 2*log(c^2*x - 1)/c^6) - (log(c*sqrt(x) + 1)* log(-1/2*c*sqrt(x) + 1/2) + dilog(1/2*c*sqrt(x) + 1/2))*b/c^6 + 3/4*b*log( c*sqrt(x) + 1)/c^6 - 3/4*b*log(c*sqrt(x) - 1)/c^6 - 1/12*(2*b*c^3*x^(3/2) + 3*b*log(c*sqrt(x) + 1)^2 - 3*b*log(-c*sqrt(x) + 1)^2 + 18*b*c*sqrt(x) + 3*(b*c^4*x^2 + 2*b*c^2*x)*log(c*sqrt(x) + 1) - 3*(b*c^4*x^2 + 2*b*c^2*x + 2*b*log(c*sqrt(x) + 1))*log(-c*sqrt(x) + 1))/c^6
\[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x} \, dx=\int { -\frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )} x^{2}}{c^{2} x - 1} \,d x } \] Input:
integrate(x^2*(a+b*arctanh(c*x^(1/2)))/(-c^2*x+1),x, algorithm="giac")
Output:
integrate(-(b*arctanh(c*sqrt(x)) + a)*x^2/(c^2*x - 1), x)
Timed out. \[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x} \, dx=\int -\frac {x^2\,\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}{c^2\,x-1} \,d x \] Input:
int(-(x^2*(a + b*atanh(c*x^(1/2))))/(c^2*x - 1),x)
Output:
int(-(x^2*(a + b*atanh(c*x^(1/2))))/(c^2*x - 1), x)
\[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x} \, dx=\frac {-2 \left (\int \frac {\mathit {atanh} \left (\sqrt {x}\, c \right ) x^{2}}{c^{2} x -1}d x \right ) b \,c^{6}-2 \,\mathrm {log}\left (c^{2} x -1\right ) a -a \,c^{4} x^{2}-2 a \,c^{2} x}{2 c^{6}} \] Input:
int(x^2*(a+b*atanh(c*x^(1/2)))/(-c^2*x+1),x)
Output:
( - 2*int((atanh(sqrt(x)*c)*x**2)/(c**2*x - 1),x)*b*c**6 - 2*log(c**2*x - 1)*a - a*c**4*x**2 - 2*a*c**2*x)/(2*c**6)