Integrand size = 14, antiderivative size = 83 \[ \int x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=b c x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )-\frac {1}{2} c^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{2} x^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{2} b^2 c^2 \log \left (1-\frac {c^2}{x^2}\right )+b^2 c^2 \log (x) \]
b*c*x*(a+b*arccoth(x/c))-1/2*c^2*(a+b*arccoth(x/c))^2+1/2*x^2*(a+b*arccoth (x/c))^2+1/2*b^2*c^2*ln(1-c^2/x^2)+b^2*c^2*ln(x)
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.11 \[ \int x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=\frac {1}{2} \left (2 a b c x+a^2 x^2+2 b x (b c+a x) \text {arctanh}\left (\frac {c}{x}\right )+b^2 \left (-c^2+x^2\right ) \text {arctanh}\left (\frac {c}{x}\right )^2+b (a+b) c^2 \log (-c+x)-a b c^2 \log (c+x)+b^2 c^2 \log (c+x)\right ) \]
(2*a*b*c*x + a^2*x^2 + 2*b*x*(b*c + a*x)*ArcTanh[c/x] + b^2*(-c^2 + x^2)*A rcTanh[c/x]^2 + b*(a + b)*c^2*Log[-c + x] - a*b*c^2*Log[c + x] + b^2*c^2*L og[c + x])/2
Time = 0.60 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6454, 6452, 6544, 6452, 243, 47, 14, 16, 6510}
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.
\(\displaystyle \int x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx\) |
\(\Big \downarrow \) 6454 |
\(\displaystyle -\int x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2d\frac {1}{x}\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-b c \int \frac {x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}\) |
\(\Big \downarrow \) 6544 |
\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-b c \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )d\frac {1}{x}\right )\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-b c \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+b c \int \frac {x}{1-\frac {c^2}{x^2}}d\frac {1}{x}-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-b c \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\frac {1}{2} b c \int \frac {x}{1-\frac {c^2}{x^2}}d\frac {1}{x^2}-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\right )\) |
\(\Big \downarrow \) 47 |
\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-b c \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-\frac {c^2}{x^2}}d\frac {1}{x^2}+\int xd\frac {1}{x^2}\right )-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\right )\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-b c \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-\frac {c^2}{x^2}}d\frac {1}{x^2}+\log \left (\frac {1}{x^2}\right )\right )-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-b c \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (1-\frac {c^2}{x^2}\right )\right )\right )\) |
\(\Big \downarrow \) 6510 |
\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-b c \left (\frac {c \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{2 b}-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (1-\frac {c^2}{x^2}\right )\right )\right )\) |
(x^2*(a + b*ArcTanh[c/x])^2)/2 - b*c*(-(x*(a + b*ArcTanh[c/x])) + (c*(a + b*ArcTanh[c/x])^2)/(2*b) + (b*c*(-Log[1 - c^2/x^2] + Log[x^(-2)]))/2)
3.2.45.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcTanh[c*x])^p, x ], x, x^n], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simpl ify[(m + 1)/n]]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b , c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x ], x] - Simp[e/(d*f^2) 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] && LtQ[m, -1]
Time = 2.84 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.42
method | result | size |
parallelrisch | \(\frac {x^{2} \operatorname {arctanh}\left (\frac {c}{x}\right )^{2} b^{2}}{2}-\frac {\operatorname {arctanh}\left (\frac {c}{x}\right )^{2} b^{2} c^{2}}{2}+b^{2} c^{2} \ln \left (x -c \right )+x^{2} \operatorname {arctanh}\left (\frac {c}{x}\right ) a b +x \,\operatorname {arctanh}\left (\frac {c}{x}\right ) b^{2} c -\operatorname {arctanh}\left (\frac {c}{x}\right ) a b \,c^{2}+\operatorname {arctanh}\left (\frac {c}{x}\right ) b^{2} c^{2}+\frac {a^{2} x^{2}}{2}+a b c x +\frac {a^{2} c^{2}}{2}\) | \(118\) |
parts | \(\frac {a^{2} x^{2}}{2}-b^{2} c^{2} \left (-\frac {x^{2} \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{2 c^{2}}-\frac {x \,\operatorname {arctanh}\left (\frac {c}{x}\right )}{c}+\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{2}-\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )}{2}-\frac {\ln \left (\frac {c}{x}-1\right )^{2}}{8}+\frac {\ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{4}-\frac {\ln \left (1+\frac {c}{x}\right )}{2}-\frac {\ln \left (\frac {c}{x}-1\right )}{2}+\ln \left (\frac {c}{x}\right )+\frac {\left (\ln \left (1+\frac {c}{x}\right )-\ln \left (\frac {c}{2 x}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right )}{4}-\frac {\ln \left (1+\frac {c}{x}\right )^{2}}{8}\right )-2 a b \,c^{2} \left (-\frac {x^{2} \operatorname {arctanh}\left (\frac {c}{x}\right )}{2 c^{2}}-\frac {x}{2 c}+\frac {\ln \left (1+\frac {c}{x}\right )}{4}-\frac {\ln \left (\frac {c}{x}-1\right )}{4}\right )\) | \(227\) |
derivativedivides | \(-c^{2} \left (-\frac {a^{2} x^{2}}{2 c^{2}}+b^{2} \left (-\frac {x^{2} \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{2 c^{2}}-\frac {x \,\operatorname {arctanh}\left (\frac {c}{x}\right )}{c}+\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{2}-\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )}{2}-\frac {\ln \left (\frac {c}{x}-1\right )^{2}}{8}+\frac {\ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{4}-\frac {\ln \left (1+\frac {c}{x}\right )}{2}-\frac {\ln \left (\frac {c}{x}-1\right )}{2}+\ln \left (\frac {c}{x}\right )+\frac {\left (\ln \left (1+\frac {c}{x}\right )-\ln \left (\frac {c}{2 x}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right )}{4}-\frac {\ln \left (1+\frac {c}{x}\right )^{2}}{8}\right )+2 a b \left (-\frac {x^{2} \operatorname {arctanh}\left (\frac {c}{x}\right )}{2 c^{2}}-\frac {x}{2 c}+\frac {\ln \left (1+\frac {c}{x}\right )}{4}-\frac {\ln \left (\frac {c}{x}-1\right )}{4}\right )\right )\) | \(228\) |
default | \(-c^{2} \left (-\frac {a^{2} x^{2}}{2 c^{2}}+b^{2} \left (-\frac {x^{2} \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{2 c^{2}}-\frac {x \,\operatorname {arctanh}\left (\frac {c}{x}\right )}{c}+\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{2}-\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )}{2}-\frac {\ln \left (\frac {c}{x}-1\right )^{2}}{8}+\frac {\ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{4}-\frac {\ln \left (1+\frac {c}{x}\right )}{2}-\frac {\ln \left (\frac {c}{x}-1\right )}{2}+\ln \left (\frac {c}{x}\right )+\frac {\left (\ln \left (1+\frac {c}{x}\right )-\ln \left (\frac {c}{2 x}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right )}{4}-\frac {\ln \left (1+\frac {c}{x}\right )^{2}}{8}\right )+2 a b \left (-\frac {x^{2} \operatorname {arctanh}\left (\frac {c}{x}\right )}{2 c^{2}}-\frac {x}{2 c}+\frac {\ln \left (1+\frac {c}{x}\right )}{4}-\frac {\ln \left (\frac {c}{x}-1\right )}{4}\right )\right )\) | \(228\) |
risch | \(\text {Expression too large to display}\) | \(14545\) |
1/2*x^2*arctanh(c/x)^2*b^2-1/2*arctanh(c/x)^2*b^2*c^2+b^2*c^2*ln(x-c)+x^2* arctanh(c/x)*a*b+x*arctanh(c/x)*b^2*c-arctanh(c/x)*a*b*c^2+arctanh(c/x)*b^ 2*c^2+1/2*a^2*x^2+a*b*c*x+1/2*a^2*c^2
Time = 0.26 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.34 \[ \int x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=a b c x + \frac {1}{2} \, a^{2} x^{2} - \frac {1}{2} \, {\left (a b - b^{2}\right )} c^{2} \log \left (c + x\right ) + \frac {1}{2} \, {\left (a b + b^{2}\right )} c^{2} \log \left (-c + x\right ) - \frac {1}{8} \, {\left (b^{2} c^{2} - b^{2} x^{2}\right )} \log \left (-\frac {c + x}{c - x}\right )^{2} + \frac {1}{2} \, {\left (b^{2} c x + a b x^{2}\right )} \log \left (-\frac {c + x}{c - x}\right ) \]
a*b*c*x + 1/2*a^2*x^2 - 1/2*(a*b - b^2)*c^2*log(c + x) + 1/2*(a*b + b^2)*c ^2*log(-c + x) - 1/8*(b^2*c^2 - b^2*x^2)*log(-(c + x)/(c - x))^2 + 1/2*(b^ 2*c*x + a*b*x^2)*log(-(c + x)/(c - x))
Time = 0.18 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.25 \[ \int x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=\frac {a^{2} x^{2}}{2} - a b c^{2} \operatorname {atanh}{\left (\frac {c}{x} \right )} + a b c x + a b x^{2} \operatorname {atanh}{\left (\frac {c}{x} \right )} + b^{2} c^{2} \log {\left (- c + x \right )} - \frac {b^{2} c^{2} \operatorname {atanh}^{2}{\left (\frac {c}{x} \right )}}{2} + b^{2} c^{2} \operatorname {atanh}{\left (\frac {c}{x} \right )} + b^{2} c x \operatorname {atanh}{\left (\frac {c}{x} \right )} + \frac {b^{2} x^{2} \operatorname {atanh}^{2}{\left (\frac {c}{x} \right )}}{2} \]
a**2*x**2/2 - a*b*c**2*atanh(c/x) + a*b*c*x + a*b*x**2*atanh(c/x) + b**2*c **2*log(-c + x) - b**2*c**2*atanh(c/x)**2/2 + b**2*c**2*atanh(c/x) + b**2* c*x*atanh(c/x) + b**2*x**2*atanh(c/x)**2/2
Time = 0.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.64 \[ \int x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=\frac {1}{2} \, b^{2} x^{2} \operatorname {artanh}\left (\frac {c}{x}\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {artanh}\left (\frac {c}{x}\right ) - {\left (c \log \left (c + x\right ) - c \log \left (-c + x\right ) - 2 \, x\right )} c\right )} a b + \frac {1}{8} \, {\left ({\left (\log \left (c + x\right )^{2} - 2 \, {\left (\log \left (c + x\right ) - 2\right )} \log \left (-c + x\right ) + \log \left (-c + x\right )^{2} + 4 \, \log \left (c + x\right )\right )} c^{2} - 4 \, {\left (c \log \left (c + x\right ) - c \log \left (-c + x\right ) - 2 \, x\right )} c \operatorname {artanh}\left (\frac {c}{x}\right )\right )} b^{2} \]
1/2*b^2*x^2*arctanh(c/x)^2 + 1/2*a^2*x^2 + 1/2*(2*x^2*arctanh(c/x) - (c*lo g(c + x) - c*log(-c + x) - 2*x)*c)*a*b + 1/8*((log(c + x)^2 - 2*(log(c + x ) - 2)*log(-c + x) + log(-c + x)^2 + 4*log(c + x))*c^2 - 4*(c*log(c + x) - c*log(-c + x) - 2*x)*c*arctanh(c/x))*b^2
Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (77) = 154\).
Time = 0.29 (sec) , antiderivative size = 268, normalized size of antiderivative = 3.23 \[ \int x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=-\frac {2 \, b^{2} c^{3} \log \left (-\frac {c + x}{c - x} - 1\right ) - 2 \, b^{2} c^{3} \log \left (-\frac {c + x}{c - x}\right ) + \frac {b^{2} {\left (c + x\right )} c^{3} \log \left (-\frac {c + x}{c - x}\right )^{2}}{{\left (c - x\right )} {\left (\frac {{\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}} + \frac {2 \, {\left (c + x\right )}}{c - x} + 1\right )}} + \frac {2 \, {\left (b^{2} c^{3} + \frac {2 \, a b {\left (c + x\right )} c^{3}}{c - x} + \frac {b^{2} {\left (c + x\right )} c^{3}}{c - x}\right )} \log \left (-\frac {c + x}{c - x}\right )}{\frac {{\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}} + \frac {2 \, {\left (c + x\right )}}{c - x} + 1} + \frac {4 \, {\left (a b c^{3} + \frac {a^{2} {\left (c + x\right )} c^{3}}{c - x} + \frac {a b {\left (c + x\right )} c^{3}}{c - x}\right )}}{\frac {{\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}} + \frac {2 \, {\left (c + x\right )}}{c - x} + 1}}{2 \, c} \]
-1/2*(2*b^2*c^3*log(-(c + x)/(c - x) - 1) - 2*b^2*c^3*log(-(c + x)/(c - x) ) + b^2*(c + x)*c^3*log(-(c + x)/(c - x))^2/((c - x)*((c + x)^2/(c - x)^2 + 2*(c + x)/(c - x) + 1)) + 2*(b^2*c^3 + 2*a*b*(c + x)*c^3/(c - x) + b^2*( c + x)*c^3/(c - x))*log(-(c + x)/(c - x))/((c + x)^2/(c - x)^2 + 2*(c + x) /(c - x) + 1) + 4*(a*b*c^3 + a^2*(c + x)*c^3/(c - x) + a*b*(c + x)*c^3/(c - x))/((c + x)^2/(c - x)^2 + 2*(c + x)/(c - x) + 1))/c
Time = 3.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.22 \[ \int x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=\frac {a^2\,x^2}{2}-\frac {b^2\,c^2\,{\mathrm {atanh}\left (\frac {c}{x}\right )}^2}{2}+\frac {b^2\,x^2\,{\mathrm {atanh}\left (\frac {c}{x}\right )}^2}{2}+\frac {b^2\,c^2\,\ln \left (x^2-c^2\right )}{2}-a\,b\,c^2\,\mathrm {atanh}\left (\frac {c}{x}\right )+a\,b\,x^2\,\mathrm {atanh}\left (\frac {c}{x}\right )+b^2\,c\,x\,\mathrm {atanh}\left (\frac {c}{x}\right )+a\,b\,c\,x \]