3.200 \(\int \frac {(a+b \tanh ^{-1}(c \sqrt {x}))^2}{x^2} \, dx\)

Optimal. Leaf size=85 \[ c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2-\frac {2 b c \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{\sqrt {x}}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{x}+b^2 c^2 \log (x)-b^2 c^2 \log \left (1-c^2 x\right ) \]

[Out]

c^2*(a+b*arctanh(c*x^(1/2)))^2-(a+b*arctanh(c*x^(1/2)))^2/x+b^2*c^2*ln(x)-b^2*c^2*ln(-c^2*x+1)-2*b*c*(a+b*arct
anh(c*x^(1/2)))/x^(1/2)

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Rubi [F]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcTanh[c*Sqrt[x]])^2/x^2,x]

[Out]

Defer[Int][(a + b*ArcTanh[c*Sqrt[x]])^2/x^2, x]

Rubi steps

\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx &=\int \frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 129, normalized size = 1.52 \[ -\frac {a^2-a b c^2 x \log \left (c \sqrt {x}+1\right )+b c^2 x (a+b) \log \left (1-c \sqrt {x}\right )+2 a b c \sqrt {x}+2 b \tanh ^{-1}\left (c \sqrt {x}\right ) \left (a+b c \sqrt {x}\right )+b^2 c^2 x \log \left (c \sqrt {x}+1\right )-b^2 c^2 x \log (x)-b^2 \left (c^2 x-1\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^2}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*Sqrt[x]])^2/x^2,x]

[Out]

-((a^2 + 2*a*b*c*Sqrt[x] + 2*b*(a + b*c*Sqrt[x])*ArcTanh[c*Sqrt[x]] - b^2*(-1 + c^2*x)*ArcTanh[c*Sqrt[x]]^2 +
b*(a + b)*c^2*x*Log[1 - c*Sqrt[x]] - a*b*c^2*x*Log[1 + c*Sqrt[x]] + b^2*c^2*x*Log[1 + c*Sqrt[x]] - b^2*c^2*x*L
og[x])/x)

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fricas [B]  time = 0.72, size = 157, normalized size = 1.85 \[ \frac {8 \, b^{2} c^{2} x \log \left (\sqrt {x}\right ) + 4 \, {\left (a b - b^{2}\right )} c^{2} x \log \left (c \sqrt {x} + 1\right ) - 4 \, {\left (a b + b^{2}\right )} c^{2} x \log \left (c \sqrt {x} - 1\right ) - 8 \, a b c \sqrt {x} + {\left (b^{2} c^{2} x - b^{2}\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right )^{2} - 4 \, a^{2} - 4 \, {\left (b^{2} c \sqrt {x} + a b\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right )}{4 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(1/2)))^2/x^2,x, algorithm="fricas")

[Out]

1/4*(8*b^2*c^2*x*log(sqrt(x)) + 4*(a*b - b^2)*c^2*x*log(c*sqrt(x) + 1) - 4*(a*b + b^2)*c^2*x*log(c*sqrt(x) - 1
) - 8*a*b*c*sqrt(x) + (b^2*c^2*x - b^2)*log(-(c^2*x + 2*c*sqrt(x) + 1)/(c^2*x - 1))^2 - 4*a^2 - 4*(b^2*c*sqrt(
x) + a*b)*log(-(c^2*x + 2*c*sqrt(x) + 1)/(c^2*x - 1)))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{2}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(1/2)))^2/x^2,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*sqrt(x)) + a)^2/x^2, x)

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maple [B]  time = 0.07, size = 292, normalized size = 3.44 \[ -\frac {a^{2}}{x}-\frac {b^{2} \arctanh \left (c \sqrt {x}\right )^{2}}{x}-\frac {2 c \,b^{2} \arctanh \left (c \sqrt {x}\right )}{\sqrt {x}}-c^{2} b^{2} \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )+c^{2} b^{2} \arctanh \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )-\frac {c^{2} b^{2} \ln \left (c \sqrt {x}-1\right )^{2}}{4}+\frac {c^{2} b^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {1}{2}+\frac {c \sqrt {x}}{2}\right )}{2}+2 c^{2} b^{2} \ln \left (c \sqrt {x}\right )-c^{2} b^{2} \ln \left (c \sqrt {x}-1\right )-c^{2} b^{2} \ln \left (1+c \sqrt {x}\right )-\frac {c^{2} b^{2} \ln \left (1+c \sqrt {x}\right )^{2}}{4}+\frac {c^{2} b^{2} \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {c^{2} b^{2} \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c \sqrt {x}}{2}\right )}{2}-\frac {2 a b \arctanh \left (c \sqrt {x}\right )}{x}-\frac {2 c a b}{\sqrt {x}}-c^{2} a b \ln \left (c \sqrt {x}-1\right )+c^{2} a b \ln \left (1+c \sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^(1/2)))^2/x^2,x)

[Out]

-a^2/x-b^2/x*arctanh(c*x^(1/2))^2-2*c*b^2*arctanh(c*x^(1/2))/x^(1/2)-c^2*b^2*arctanh(c*x^(1/2))*ln(c*x^(1/2)-1
)+c^2*b^2*arctanh(c*x^(1/2))*ln(1+c*x^(1/2))-1/4*c^2*b^2*ln(c*x^(1/2)-1)^2+1/2*c^2*b^2*ln(c*x^(1/2)-1)*ln(1/2+
1/2*c*x^(1/2))+2*c^2*b^2*ln(c*x^(1/2))-c^2*b^2*ln(c*x^(1/2)-1)-c^2*b^2*ln(1+c*x^(1/2))-1/4*c^2*b^2*ln(1+c*x^(1
/2))^2+1/2*c^2*b^2*ln(-1/2*c*x^(1/2)+1/2)*ln(1+c*x^(1/2))-1/2*c^2*b^2*ln(-1/2*c*x^(1/2)+1/2)*ln(1/2+1/2*c*x^(1
/2))-2*a*b/x*arctanh(c*x^(1/2))-2*c*a*b/x^(1/2)-c^2*a*b*ln(c*x^(1/2)-1)+c^2*a*b*ln(1+c*x^(1/2))

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maxima [B]  time = 0.33, size = 174, normalized size = 2.05 \[ {\left ({\left (c \log \left (c \sqrt {x} + 1\right ) - c \log \left (c \sqrt {x} - 1\right ) - \frac {2}{\sqrt {x}}\right )} c - \frac {2 \, \operatorname {artanh}\left (c \sqrt {x}\right )}{x}\right )} a b + \frac {1}{4} \, {\left ({\left (2 \, {\left (\log \left (c \sqrt {x} - 1\right ) - 2\right )} \log \left (c \sqrt {x} + 1\right ) - \log \left (c \sqrt {x} + 1\right )^{2} - \log \left (c \sqrt {x} - 1\right )^{2} - 4 \, \log \left (c \sqrt {x} - 1\right ) + 4 \, \log \relax (x)\right )} c^{2} + 4 \, {\left (c \log \left (c \sqrt {x} + 1\right ) - c \log \left (c \sqrt {x} - 1\right ) - \frac {2}{\sqrt {x}}\right )} c \operatorname {artanh}\left (c \sqrt {x}\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (c \sqrt {x}\right )^{2}}{x} - \frac {a^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(1/2)))^2/x^2,x, algorithm="maxima")

[Out]

((c*log(c*sqrt(x) + 1) - c*log(c*sqrt(x) - 1) - 2/sqrt(x))*c - 2*arctanh(c*sqrt(x))/x)*a*b + 1/4*((2*(log(c*sq
rt(x) - 1) - 2)*log(c*sqrt(x) + 1) - log(c*sqrt(x) + 1)^2 - log(c*sqrt(x) - 1)^2 - 4*log(c*sqrt(x) - 1) + 4*lo
g(x))*c^2 + 4*(c*log(c*sqrt(x) + 1) - c*log(c*sqrt(x) - 1) - 2/sqrt(x))*c*arctanh(c*sqrt(x)))*b^2 - b^2*arctan
h(c*sqrt(x))^2/x - a^2/x

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mupad [B]  time = 1.79, size = 278, normalized size = 3.27 \[ 2\,b^2\,c^2\,\ln \left (\sqrt {x}\right )-\frac {a^2}{x}-b^2\,c^2\,\ln \left (c\,\sqrt {x}-1\right )-b^2\,c^2\,\ln \left (c\,\sqrt {x}+1\right )+\frac {b^2\,c^2\,{\ln \left (c\,\sqrt {x}+1\right )}^2}{4}+\frac {b^2\,c^2\,{\ln \left (1-c\,\sqrt {x}\right )}^2}{4}-\frac {b^2\,{\ln \left (c\,\sqrt {x}+1\right )}^2}{4\,x}-\frac {b^2\,{\ln \left (1-c\,\sqrt {x}\right )}^2}{4\,x}-a\,b\,c^2\,\ln \left (c\,\sqrt {x}-1\right )+a\,b\,c^2\,\ln \left (c\,\sqrt {x}+1\right )-\frac {2\,a\,b\,c}{\sqrt {x}}-\frac {a\,b\,\ln \left (c\,\sqrt {x}+1\right )}{x}+\frac {a\,b\,\ln \left (1-c\,\sqrt {x}\right )}{x}-\frac {b^2\,c^2\,\ln \left (c\,\sqrt {x}+1\right )\,\ln \left (1-c\,\sqrt {x}\right )}{2}-\frac {b^2\,c\,\ln \left (c\,\sqrt {x}+1\right )}{\sqrt {x}}+\frac {b^2\,c\,\ln \left (1-c\,\sqrt {x}\right )}{\sqrt {x}}+\frac {b^2\,\ln \left (c\,\sqrt {x}+1\right )\,\ln \left (1-c\,\sqrt {x}\right )}{2\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^(1/2)))^2/x^2,x)

[Out]

2*b^2*c^2*log(x^(1/2)) - a^2/x - b^2*c^2*log(c*x^(1/2) - 1) - b^2*c^2*log(c*x^(1/2) + 1) + (b^2*c^2*log(c*x^(1
/2) + 1)^2)/4 + (b^2*c^2*log(1 - c*x^(1/2))^2)/4 - (b^2*log(c*x^(1/2) + 1)^2)/(4*x) - (b^2*log(1 - c*x^(1/2))^
2)/(4*x) - a*b*c^2*log(c*x^(1/2) - 1) + a*b*c^2*log(c*x^(1/2) + 1) - (2*a*b*c)/x^(1/2) - (a*b*log(c*x^(1/2) +
1))/x + (a*b*log(1 - c*x^(1/2)))/x - (b^2*c^2*log(c*x^(1/2) + 1)*log(1 - c*x^(1/2)))/2 - (b^2*c*log(c*x^(1/2)
+ 1))/x^(1/2) + (b^2*c*log(1 - c*x^(1/2)))/x^(1/2) + (b^2*log(c*x^(1/2) + 1)*log(1 - c*x^(1/2)))/(2*x)

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sympy [A]  time = 22.59, size = 680, normalized size = 8.00 \[ \begin {cases} - \frac {a^{2}}{x} + \frac {2 a b \operatorname {atanh}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{x} - \frac {b^{2} \operatorname {atanh}^{2}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{x} & \text {for}\: c = - \sqrt {\frac {1}{x}} \\- \frac {a^{2}}{x} - \frac {2 a b \operatorname {atanh}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{x} - \frac {b^{2} \operatorname {atanh}^{2}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{x} & \text {for}\: c = \sqrt {\frac {1}{x}} \\- \frac {a^{2}}{x} & \text {for}\: c = 0 \\- \frac {a^{2} c^{2} x^{\frac {3}{2}}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {a^{2} \sqrt {x}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 a 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 {2 a b c^{3} x^{2}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {4 a 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 {2 a b c x}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 a b \sqrt {x} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {b^{2} c^{4} x^{\frac {5}{2}} \log {\relax (x )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 b^{2} c^{4} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \frac {1}{c} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {b^{2} c^{4} x^{\frac {5}{2}} \operatorname {atanh}^{2}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 b^{2} c^{4} x^{\frac {5}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 b^{2} c^{3} x^{2} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {b^{2} c^{2} x^{\frac {3}{2}} \log {\relax (x )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 b^{2} c^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} - \frac {1}{c} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 b^{2} c^{2} x^{\frac {3}{2}} \operatorname {atanh}^{2}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 b^{2} c^{2} x^{\frac {3}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 b^{2} c x \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {b^{2} \sqrt {x} \operatorname {atanh}^{2}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x**(1/2)))**2/x**2,x)

[Out]

Piecewise((-a**2/x + 2*a*b*atanh(sqrt(x)*sqrt(1/x))/x - b**2*atanh(sqrt(x)*sqrt(1/x))**2/x, Eq(c, -sqrt(1/x)))
, (-a**2/x - 2*a*b*atanh(sqrt(x)*sqrt(1/x))/x - b**2*atanh(sqrt(x)*sqrt(1/x))**2/x, Eq(c, sqrt(1/x))), (-a**2/
x, Eq(c, 0)), (-a**2*c**2*x**(3/2)/(c**2*x**(5/2) - x**(3/2)) + a**2*sqrt(x)/(c**2*x**(5/2) - x**(3/2)) + 2*a*
b*c**4*x**(5/2)*atanh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) - 2*a*b*c**3*x**2/(c**2*x**(5/2) - x**(3/2)) - 4*a
*b*c**2*x**(3/2)*atanh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) + 2*a*b*c*x/(c**2*x**(5/2) - x**(3/2)) + 2*a*b*sq
rt(x)*atanh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) + b**2*c**4*x**(5/2)*log(x)/(c**2*x**(5/2) - x**(3/2)) - 2*b
**2*c**4*x**(5/2)*log(sqrt(x) - 1/c)/(c**2*x**(5/2) - x**(3/2)) + b**2*c**4*x**(5/2)*atanh(c*sqrt(x))**2/(c**2
*x**(5/2) - x**(3/2)) - 2*b**2*c**4*x**(5/2)*atanh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) - 2*b**2*c**3*x**2*at
anh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) - b**2*c**2*x**(3/2)*log(x)/(c**2*x**(5/2) - x**(3/2)) + 2*b**2*c**2
*x**(3/2)*log(sqrt(x) - 1/c)/(c**2*x**(5/2) - x**(3/2)) - 2*b**2*c**2*x**(3/2)*atanh(c*sqrt(x))**2/(c**2*x**(5
/2) - x**(3/2)) + 2*b**2*c**2*x**(3/2)*atanh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) + 2*b**2*c*x*atanh(c*sqrt(x
))/(c**2*x**(5/2) - x**(3/2)) + b**2*sqrt(x)*atanh(c*sqrt(x))**2/(c**2*x**(5/2) - x**(3/2)), True))

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