Integrand size = 14, antiderivative size = 62 \[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \, dx=\frac {b \sqrt {x}}{2 c^3}+\frac {b x^{3/2}}{6 c}-\frac {b \text {arctanh}\left (c \sqrt {x}\right )}{2 c^4}+\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \]
1/6*b*x^(3/2)/c-1/2*b*arctanh(c*x^(1/2))/c^4+1/2*x^2*(a+b*arctanh(c*x^(1/2 )))+1/2*b*x^(1/2)/c^3
Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.42 \[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \, dx=\frac {b \sqrt {x}}{2 c^3}+\frac {b x^{3/2}}{6 c}+\frac {a x^2}{2}+\frac {1}{2} b x^2 \text {arctanh}\left (c \sqrt {x}\right )+\frac {b \log \left (1-c \sqrt {x}\right )}{4 c^4}-\frac {b \log \left (1+c \sqrt {x}\right )}{4 c^4} \]
(b*Sqrt[x])/(2*c^3) + (b*x^(3/2))/(6*c) + (a*x^2)/2 + (b*x^2*ArcTanh[c*Sqr t[x]])/2 + (b*Log[1 - c*Sqrt[x]])/(4*c^4) - (b*Log[1 + c*Sqrt[x]])/(4*c^4)
Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6452, 60, 60, 73, 219}
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.
\(\displaystyle \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \, dx\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \int \frac {x^{3/2}}{1-c^2 x}dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\int \frac {\sqrt {x}}{1-c^2 x}dx}{c^2}-\frac {2 x^{3/2}}{3 c^2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\frac {\int \frac {1}{\sqrt {x} \left (1-c^2 x\right )}dx}{c^2}-\frac {2 \sqrt {x}}{c^2}}{c^2}-\frac {2 x^{3/2}}{3 c^2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\frac {2 \int \frac {1}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {2 \sqrt {x}}{c^2}}{c^2}-\frac {2 x^{3/2}}{3 c^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\frac {2 \text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {2 \sqrt {x}}{c^2}}{c^2}-\frac {2 x^{3/2}}{3 c^2}\right )\) |
(x^2*(a + b*ArcTanh[c*Sqrt[x]]))/2 - (b*c*((-2*x^(3/2))/(3*c^2) + ((-2*Sqr t[x])/c^2 + (2*ArcTanh[c*Sqrt[x]])/c^3)/c^2))/4
3.2.89.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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[((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]
Time = 0.70 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.02
method | result | size |
parts | \(\frac {a \,x^{2}}{2}+\frac {2 b \left (\frac {c^{4} x^{2} \operatorname {arctanh}\left (c \sqrt {x}\right )}{4}+\frac {c^{3} x^{\frac {3}{2}}}{12}+\frac {c \sqrt {x}}{4}+\frac {\ln \left (c \sqrt {x}-1\right )}{8}-\frac {\ln \left (1+c \sqrt {x}\right )}{8}\right )}{c^{4}}\) | \(63\) |
derivativedivides | \(\frac {\frac {a \,c^{4} x^{2}}{2}+2 b \left (\frac {c^{4} x^{2} \operatorname {arctanh}\left (c \sqrt {x}\right )}{4}+\frac {c^{3} x^{\frac {3}{2}}}{12}+\frac {c \sqrt {x}}{4}+\frac {\ln \left (c \sqrt {x}-1\right )}{8}-\frac {\ln \left (1+c \sqrt {x}\right )}{8}\right )}{c^{4}}\) | \(67\) |
default | \(\frac {\frac {a \,c^{4} x^{2}}{2}+2 b \left (\frac {c^{4} x^{2} \operatorname {arctanh}\left (c \sqrt {x}\right )}{4}+\frac {c^{3} x^{\frac {3}{2}}}{12}+\frac {c \sqrt {x}}{4}+\frac {\ln \left (c \sqrt {x}-1\right )}{8}-\frac {\ln \left (1+c \sqrt {x}\right )}{8}\right )}{c^{4}}\) | \(67\) |
1/2*a*x^2+2*b/c^4*(1/4*c^4*x^2*arctanh(c*x^(1/2))+1/12*c^3*x^(3/2)+1/4*c*x ^(1/2)+1/8*ln(c*x^(1/2)-1)-1/8*ln(1+c*x^(1/2)))
Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.13 \[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \, dx=\frac {6 \, a c^{4} x^{2} + 3 \, {\left (b c^{4} x^{2} - b\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right ) + 2 \, {\left (b c^{3} x + 3 \, b c\right )} \sqrt {x}}{12 \, c^{4}} \]
1/12*(6*a*c^4*x^2 + 3*(b*c^4*x^2 - b)*log(-(c^2*x + 2*c*sqrt(x) + 1)/(c^2* x - 1)) + 2*(b*c^3*x + 3*b*c)*sqrt(x))/c^4
\[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \, dx=\int x \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )\, dx \]
Time = 0.18 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.11 \[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \, dx=\frac {1}{2} \, a x^{2} + \frac {1}{12} \, {\left (6 \, x^{2} \operatorname {artanh}\left (c \sqrt {x}\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{\frac {3}{2}} + 3 \, \sqrt {x}\right )}}{c^{4}} - \frac {3 \, \log \left (c \sqrt {x} + 1\right )}{c^{5}} + \frac {3 \, \log \left (c \sqrt {x} - 1\right )}{c^{5}}\right )}\right )} b \]
1/2*a*x^2 + 1/12*(6*x^2*arctanh(c*sqrt(x)) + c*(2*(c^2*x^(3/2) + 3*sqrt(x) )/c^4 - 3*log(c*sqrt(x) + 1)/c^5 + 3*log(c*sqrt(x) - 1)/c^5))*b
Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (46) = 92\).
Time = 0.28 (sec) , antiderivative size = 239, normalized size of antiderivative = 3.85 \[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \, dx=\frac {1}{2} \, a x^{2} + \frac {2}{3} \, b c {\left (\frac {\frac {3 \, {\left (c \sqrt {x} + 1\right )}^{2}}{{\left (c \sqrt {x} - 1\right )}^{2}} - \frac {3 \, {\left (c \sqrt {x} + 1\right )}}{c \sqrt {x} - 1} + 2}{c^{5} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}^{3}} + \frac {3 \, {\left (\frac {{\left (c \sqrt {x} + 1\right )}^{3}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {c \sqrt {x} + 1}{c \sqrt {x} - 1}\right )} \log \left (-\frac {\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} + 1}{\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} - 1}\right )}{c^{5} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}^{4}}\right )} \]
1/2*a*x^2 + 2/3*b*c*((3*(c*sqrt(x) + 1)^2/(c*sqrt(x) - 1)^2 - 3*(c*sqrt(x) + 1)/(c*sqrt(x) - 1) + 2)/(c^5*((c*sqrt(x) + 1)/(c*sqrt(x) - 1) - 1)^3) + 3*((c*sqrt(x) + 1)^3/(c*sqrt(x) - 1)^3 + (c*sqrt(x) + 1)/(c*sqrt(x) - 1)) *log(-(c*((c*sqrt(x) + 1)/(c*sqrt(x) - 1) + 1)/((c*sqrt(x) + 1)*c/(c*sqrt( x) - 1) - c) + 1)/(c*((c*sqrt(x) + 1)/(c*sqrt(x) - 1) + 1)/((c*sqrt(x) + 1 )*c/(c*sqrt(x) - 1) - c) - 1))/(c^5*((c*sqrt(x) + 1)/(c*sqrt(x) - 1) - 1)^ 4))
Time = 3.59 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.79 \[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \, dx=\frac {\frac {b\,c^3\,x^{3/2}}{6}-\frac {b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{2}+\frac {b\,c\,\sqrt {x}}{2}}{c^4}+\frac {a\,x^2}{2}+\frac {b\,x^2\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{2} \]