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]

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

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Rubi [F]  time = 0.0238263, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 0.110467, 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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Maple [B]  time = 0.054, size = 292, normalized size = 3.4 \begin{align*} -{\frac{{a}^{2}}{x}}-{\frac{{b}^{2}}{x} \left ({\it Artanh} \left ( c\sqrt{x} \right ) \right ) ^{2}}-2\,{\frac{{b}^{2}c{\it Artanh} \left ( c\sqrt{x} \right ) }{\sqrt{x}}}-{c}^{2}{b}^{2}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( c\sqrt{x}-1 \right ) +{c}^{2}{b}^{2}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( 1+c\sqrt{x} \right ) +{\frac{{b}^{2}{c}^{2}}{2}\ln \left ( c\sqrt{x}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }-{\frac{{b}^{2}{c}^{2}}{4} \left ( \ln \left ( c\sqrt{x}-1 \right ) \right ) ^{2}}-{c}^{2}{b}^{2}\ln \left ( c\sqrt{x}-1 \right ) +2\,{c}^{2}{b}^{2}\ln \left ( c\sqrt{x} \right ) -{c}^{2}{b}^{2}\ln \left ( 1+c\sqrt{x} \right ) +{\frac{{b}^{2}{c}^{2}}{2}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ( 1+c\sqrt{x} \right ) }-{\frac{{b}^{2}{c}^{2}}{2}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }-{\frac{{b}^{2}{c}^{2}}{4} \left ( \ln \left ( 1+c\sqrt{x} \right ) \right ) ^{2}}-2\,{\frac{ab{\it Artanh} \left ( c\sqrt{x} \right ) }{x}}-2\,{\frac{cab}{\sqrt{x}}}-{c}^{2}ab\ln \left ( c\sqrt{x}-1 \right ) +{c}^{2}ab\ln \left ( 1+c\sqrt{x} \right ) \end{align*}

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/x^(1/2)*arctanh(c*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))+1/2*c^2*b^2*ln(c*x^(1/2)-1)*ln(1/2+1/2*c*x^(1/2))-1/4*c^2*b^2*ln(
c*x^(1/2)-1)^2-c^2*b^2*ln(c*x^(1/2)-1)+2*c^2*b^2*ln(c*x^(1/2))-c^2*b^2*ln(1+c*x^(1/2))+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))-1/4*c^2*b^2*ln(1+c*x^(1/2
))^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 = 1.01967, size = 235, normalized size = 2.76 \begin{align*}{\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 \left (x\right )\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} \end{align*}

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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Fricas [B]  time = 1.83377, size = 375, normalized size = 4.41 \begin{align*} \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} \end{align*}

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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Sympy [A]  time = 42.9305, size = 680, normalized size = 8. \begin{align*} \text{result too large to display} \end{align*}

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**4*x**(5/2)/(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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (c \sqrt{x}\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}

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)