Integrand size = 16, antiderivative size = 40 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2} \, dx=-\frac {b c}{\sqrt {x}}+b c^2 \text {arctanh}\left (c \sqrt {x}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x} \] Output:
-b*c/x^(1/2)+b*c^2*arctanh(c*x^(1/2))-(a+b*arctanh(c*x^(1/2)))/x
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.68 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2} \, dx=-\frac {a}{x}-\frac {b c}{\sqrt {x}}-\frac {b \text {arctanh}\left (c \sqrt {x}\right )}{x}-\frac {1}{2} b c^2 \log \left (1-c \sqrt {x}\right )+\frac {1}{2} b c^2 \log \left (1+c \sqrt {x}\right ) \] Input:
Integrate[(a + b*ArcTanh[c*Sqrt[x]])/x^2,x]
Output:
-(a/x) - (b*c)/Sqrt[x] - (b*ArcTanh[c*Sqrt[x]])/x - (b*c^2*Log[1 - c*Sqrt[ x]])/2 + (b*c^2*Log[1 + c*Sqrt[x]])/2
Time = 0.22 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6452, 61, 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.
\(\displaystyle \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{2} b c \int \frac {1}{x^{3/2} \left (1-c^2 x\right )}dx-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {1}{2} b c \left (c^2 \int \frac {1}{\sqrt {x} \left (1-c^2 x\right )}dx-\frac {2}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} b c \left (2 c^2 \int \frac {1}{1-c^2 x}d\sqrt {x}-\frac {2}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} b c \left (2 c \text {arctanh}\left (c \sqrt {x}\right )-\frac {2}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x}\) |
Input:
Int[(a + b*ArcTanh[c*Sqrt[x]])/x^2,x]
Output:
-((a + b*ArcTanh[c*Sqrt[x]])/x) + (b*c*(-2/Sqrt[x] + 2*c*ArcTanh[c*Sqrt[x] ]))/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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 = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.42
method | result | size |
parts | \(-\frac {a}{x}+2 b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 c^{2} x}-\frac {\ln \left (c \sqrt {x}-1\right )}{4}-\frac {1}{2 c \sqrt {x}}+\frac {\ln \left (1+c \sqrt {x}\right )}{4}\right )\) | \(57\) |
derivativedivides | \(2 c^{2} \left (-\frac {a}{2 c^{2} x}+b \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 c^{2} x}-\frac {\ln \left (c \sqrt {x}-1\right )}{4}-\frac {1}{2 c \sqrt {x}}+\frac {\ln \left (1+c \sqrt {x}\right )}{4}\right )\right )\) | \(61\) |
default | \(2 c^{2} \left (-\frac {a}{2 c^{2} x}+b \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 c^{2} x}-\frac {\ln \left (c \sqrt {x}-1\right )}{4}-\frac {1}{2 c \sqrt {x}}+\frac {\ln \left (1+c \sqrt {x}\right )}{4}\right )\right )\) | \(61\) |
Input:
int((a+b*arctanh(c*x^(1/2)))/x^2,x,method=_RETURNVERBOSE)
Output:
-a/x+2*b*c^2*(-1/2/c^2/x*arctanh(c*x^(1/2))-1/4*ln(c*x^(1/2)-1)-1/2/c/x^(1 /2)+1/4*ln(1+c*x^(1/2)))
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.32 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2} \, dx=-\frac {2 \, b c \sqrt {x} - {\left (b c^{2} x - b\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right ) + 2 \, a}{2 \, x} \] Input:
integrate((a+b*arctanh(c*x^(1/2)))/x^2,x, algorithm="fricas")
Output:
-1/2*(2*b*c*sqrt(x) - (b*c^2*x - b)*log(-(c^2*x + 2*c*sqrt(x) + 1)/(c^2*x - 1)) + 2*a)/x
Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (34) = 68\).
Time = 2.56 (sec) , antiderivative size = 231, normalized size of antiderivative = 5.78 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2} \, dx=\begin {cases} - \frac {a}{x} + \frac {b \operatorname {atanh}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{x} & \text {for}\: c = - \sqrt {\frac {1}{x}} \\- \frac {a}{x} - \frac {b \operatorname {atanh}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{x} & \text {for}\: c = \sqrt {\frac {1}{x}} \\- \frac {a c^{2} x^{\frac {3}{2}}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {a \sqrt {x}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {b c^{4} x^{\frac {5}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {b c^{3} x^{2}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 b c^{2} x^{\frac {3}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {b c x}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {b \sqrt {x} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((a+b*atanh(c*x**(1/2)))/x**2,x)
Output:
Piecewise((-a/x + b*atanh(sqrt(x)*sqrt(1/x))/x, Eq(c, -sqrt(1/x))), (-a/x - b*atanh(sqrt(x)*sqrt(1/x))/x, Eq(c, sqrt(1/x))), (-a*c**2*x**(3/2)/(c**2 *x**(5/2) - x**(3/2)) + a*sqrt(x)/(c**2*x**(5/2) - x**(3/2)) + b*c**4*x**( 5/2)*atanh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) - b*c**3*x**2/(c**2*x**(5 /2) - x**(3/2)) - 2*b*c**2*x**(3/2)*atanh(c*sqrt(x))/(c**2*x**(5/2) - x**( 3/2)) + b*c*x/(c**2*x**(5/2) - x**(3/2)) + b*sqrt(x)*atanh(c*sqrt(x))/(c** 2*x**(5/2) - x**(3/2)), True))
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.28 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2} \, dx=\frac {1}{2} \, {\left ({\left (c \log \left (c \sqrt {x} + 1\right ) - c \log \left (c \sqrt {x} - 1\right ) - \frac {2}{\sqrt {x}}\right )} c - \frac {2 \, \operatorname {artanh}\left (c \sqrt {x}\right )}{x}\right )} b - \frac {a}{x} \] Input:
integrate((a+b*arctanh(c*x^(1/2)))/x^2,x, algorithm="maxima")
Output:
1/2*((c*log(c*sqrt(x) + 1) - c*log(c*sqrt(x) - 1) - 2/sqrt(x))*c - 2*arcta nh(c*sqrt(x))/x)*b - a/x
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (34) = 68\).
Time = 0.13 (sec) , antiderivative size = 168, normalized size of antiderivative = 4.20 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2} \, dx=2 \, {\left (\frac {{\left (c \sqrt {x} + 1\right )} b c \log \left (-\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1}\right )}{{\left (c \sqrt {x} - 1\right )} {\left (\frac {{\left (c \sqrt {x} + 1\right )}^{2}}{{\left (c \sqrt {x} - 1\right )}^{2}} + \frac {2 \, {\left (c \sqrt {x} + 1\right )}}{c \sqrt {x} - 1} + 1\right )}} + \frac {\frac {2 \, {\left (c \sqrt {x} + 1\right )} a c}{c \sqrt {x} - 1} + \frac {{\left (c \sqrt {x} + 1\right )} b c}{c \sqrt {x} - 1} + b c}{\frac {{\left (c \sqrt {x} + 1\right )}^{2}}{{\left (c \sqrt {x} - 1\right )}^{2}} + \frac {2 \, {\left (c \sqrt {x} + 1\right )}}{c \sqrt {x} - 1} + 1}\right )} c \] Input:
integrate((a+b*arctanh(c*x^(1/2)))/x^2,x, algorithm="giac")
Output:
2*((c*sqrt(x) + 1)*b*c*log(-(c*sqrt(x) + 1)/(c*sqrt(x) - 1))/((c*sqrt(x) - 1)*((c*sqrt(x) + 1)^2/(c*sqrt(x) - 1)^2 + 2*(c*sqrt(x) + 1)/(c*sqrt(x) - 1) + 1)) + (2*(c*sqrt(x) + 1)*a*c/(c*sqrt(x) - 1) + (c*sqrt(x) + 1)*b*c/(c *sqrt(x) - 1) + b*c)/((c*sqrt(x) + 1)^2/(c*sqrt(x) - 1)^2 + 2*(c*sqrt(x) + 1)/(c*sqrt(x) - 1) + 1))*c
Time = 4.35 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.30 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2} \, dx=b\,c\,\mathrm {atan}\left (\frac {c^2\,\sqrt {x}}{\sqrt {-c^2}}\right )\,\sqrt {-c^2}-\frac {a}{x}-\frac {b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )+b\,c\,\sqrt {x}}{x} \] Input:
int((a + b*atanh(c*x^(1/2)))/x^2,x)
Output:
b*c*atan((c^2*x^(1/2))/(-c^2)^(1/2))*(-c^2)^(1/2) - a/x - (b*atanh(c*x^(1/ 2)) + b*c*x^(1/2))/x
Time = 0.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2} \, dx=\frac {\mathit {atanh} \left (\sqrt {x}\, c \right ) b \,c^{2} x -\mathit {atanh} \left (\sqrt {x}\, c \right ) b -\sqrt {x}\, b c -a}{x} \] Input:
int((a+b*atanh(c*x^(1/2)))/x^2,x)
Output:
(atanh(sqrt(x)*c)*b*c**2*x - atanh(sqrt(x)*c)*b - sqrt(x)*b*c - a)/x