3.201 \(\int \frac{(a+b \tanh ^{-1}(c \sqrt{x}))^2}{x^3} \, dx\)

Optimal. Leaf size=133 \[ \frac{1}{2} c^4 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2-\frac{b c^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{\sqrt{x}}-\frac{b c \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{3 x^{3/2}}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{2 x^2}-\frac{b^2 c^2}{6 x}+\frac{2}{3} b^2 c^4 \log (x)-\frac{2}{3} b^2 c^4 \log \left (1-c^2 x\right ) \]

[Out]

-(b^2*c^2)/(6*x) - (b*c*(a + b*ArcTanh[c*Sqrt[x]]))/(3*x^(3/2)) - (b*c^3*(a + b*ArcTanh[c*Sqrt[x]]))/Sqrt[x] +
(c^4*(a + b*ArcTanh[c*Sqrt[x]])^2)/2 - (a + b*ArcTanh[c*Sqrt[x]])^2/(2*x^2) + (2*b^2*c^4*Log[x])/3 - (2*b^2*c
^4*Log[1 - c^2*x])/3

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Rubi [F]  time = 0.0240127, 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 = {} \[ \int \frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcTanh[c*Sqrt[x]])^2/x^3,x]

[Out]

Defer[Int][(a + b*ArcTanh[c*Sqrt[x]])^2/x^3, x]

Rubi steps

\begin{align*} \int \frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{x^3} \, dx &=\int \frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{x^3} \, dx\\ \end{align*}

Mathematica [A]  time = 0.127832, size = 178, normalized size = 1.34 \[ -\frac{3 a^2+6 a b c^3 x^{3/2}+b c^4 x^2 (3 a+4 b) \log \left (1-c \sqrt{x}\right )-3 a b c^4 x^2 \log \left (c \sqrt{x}+1\right )+2 b \tanh ^{-1}\left (c \sqrt{x}\right ) \left (3 a+b c \sqrt{x} \left (3 c^2 x+1\right )\right )+2 a b c \sqrt{x}+4 b^2 c^4 x^2 \log \left (c \sqrt{x}+1\right )-4 b^2 c^4 x^2 \log (x)-3 b^2 \left (c^4 x^2-1\right ) \tanh ^{-1}\left (c \sqrt{x}\right )^2+b^2 c^2 x}{6 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*Sqrt[x]])^2/x^3,x]

[Out]

-(3*a^2 + 2*a*b*c*Sqrt[x] + b^2*c^2*x + 6*a*b*c^3*x^(3/2) + 2*b*(3*a + b*c*Sqrt[x]*(1 + 3*c^2*x))*ArcTanh[c*Sq
rt[x]] - 3*b^2*(-1 + c^4*x^2)*ArcTanh[c*Sqrt[x]]^2 + b*(3*a + 4*b)*c^4*x^2*Log[1 - c*Sqrt[x]] - 3*a*b*c^4*x^2*
Log[1 + c*Sqrt[x]] + 4*b^2*c^4*x^2*Log[1 + c*Sqrt[x]] - 4*b^2*c^4*x^2*Log[x])/(6*x^2)

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Maple [B]  time = 0.062, size = 332, normalized size = 2.5 \begin{align*} -{\frac{{a}^{2}}{2\,{x}^{2}}}-{\frac{{b}^{2}}{2\,{x}^{2}} \left ({\it Artanh} \left ( c\sqrt{x} \right ) \right ) ^{2}}-{\frac{{c}^{4}{b}^{2}}{2}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( c\sqrt{x}-1 \right ) }-{\frac{{b}^{2}c}{3}{\it Artanh} \left ( c\sqrt{x} \right ){x}^{-{\frac{3}{2}}}}-{{c}^{3}{b}^{2}{\it Artanh} \left ( c\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}+{\frac{{c}^{4}{b}^{2}}{2}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( 1+c\sqrt{x} \right ) }-{\frac{{c}^{4}{b}^{2}}{8} \left ( \ln \left ( c\sqrt{x}-1 \right ) \right ) ^{2}}+{\frac{{c}^{4}{b}^{2}}{4}\ln \left ( c\sqrt{x}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }-{\frac{{c}^{4}{b}^{2}}{4}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }+{\frac{{c}^{4}{b}^{2}}{4}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ( 1+c\sqrt{x} \right ) }-{\frac{{c}^{4}{b}^{2}}{8} \left ( \ln \left ( 1+c\sqrt{x} \right ) \right ) ^{2}}-{\frac{2\,{c}^{4}{b}^{2}}{3}\ln \left ( c\sqrt{x}-1 \right ) }-{\frac{{b}^{2}{c}^{2}}{6\,x}}+{\frac{4\,{c}^{4}{b}^{2}}{3}\ln \left ( c\sqrt{x} \right ) }-{\frac{2\,{c}^{4}{b}^{2}}{3}\ln \left ( 1+c\sqrt{x} \right ) }-{\frac{ab}{{x}^{2}}{\it Artanh} \left ( c\sqrt{x} \right ) }-{\frac{{c}^{4}ab}{2}\ln \left ( c\sqrt{x}-1 \right ) }-{\frac{cab}{3}{x}^{-{\frac{3}{2}}}}-{{c}^{3}ab{\frac{1}{\sqrt{x}}}}+{\frac{{c}^{4}ab}{2}\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^3,x)

[Out]

-1/2*a^2/x^2-1/2*b^2/x^2*arctanh(c*x^(1/2))^2-1/2*c^4*b^2*arctanh(c*x^(1/2))*ln(c*x^(1/2)-1)-1/3*c*b^2/x^(3/2)
*arctanh(c*x^(1/2))-c^3*b^2/x^(1/2)*arctanh(c*x^(1/2))+1/2*c^4*b^2*arctanh(c*x^(1/2))*ln(1+c*x^(1/2))-1/8*c^4*
b^2*ln(c*x^(1/2)-1)^2+1/4*c^4*b^2*ln(c*x^(1/2)-1)*ln(1/2+1/2*c*x^(1/2))-1/4*c^4*b^2*ln(-1/2*c*x^(1/2)+1/2)*ln(
1/2+1/2*c*x^(1/2))+1/4*c^4*b^2*ln(-1/2*c*x^(1/2)+1/2)*ln(1+c*x^(1/2))-1/8*c^4*b^2*ln(1+c*x^(1/2))^2-2/3*c^4*b^
2*ln(c*x^(1/2)-1)-1/6*b^2*c^2/x+4/3*c^4*b^2*ln(c*x^(1/2))-2/3*c^4*b^2*ln(1+c*x^(1/2))-a*b/x^2*arctanh(c*x^(1/2
))-1/2*c^4*a*b*ln(c*x^(1/2)-1)-1/3*c*a*b/x^(3/2)-c^3*a*b/x^(1/2)+1/2*c^4*a*b*ln(1+c*x^(1/2))

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Maxima [B]  time = 0.984568, size = 316, normalized size = 2.38 \begin{align*} \frac{1}{6} \,{\left ({\left (3 \, c^{3} \log \left (c \sqrt{x} + 1\right ) - 3 \, c^{3} \log \left (c \sqrt{x} - 1\right ) - \frac{2 \,{\left (3 \, c^{2} x + 1\right )}}{x^{\frac{3}{2}}}\right )} c - \frac{6 \, \operatorname{artanh}\left (c \sqrt{x}\right )}{x^{2}}\right )} a b + \frac{1}{24} \,{\left ({\left (16 \, c^{2} \log \left (x\right ) - \frac{3 \, c^{2} x \log \left (c \sqrt{x} + 1\right )^{2} + 3 \, c^{2} x \log \left (c \sqrt{x} - 1\right )^{2} + 16 \, c^{2} x \log \left (c \sqrt{x} - 1\right ) - 2 \,{\left (3 \, c^{2} x \log \left (c \sqrt{x} - 1\right ) - 8 \, c^{2} x\right )} \log \left (c \sqrt{x} + 1\right ) + 4}{x}\right )} c^{2} + 4 \,{\left (3 \, c^{3} \log \left (c \sqrt{x} + 1\right ) - 3 \, c^{3} \log \left (c \sqrt{x} - 1\right ) - \frac{2 \,{\left (3 \, c^{2} x + 1\right )}}{x^{\frac{3}{2}}}\right )} c \operatorname{artanh}\left (c \sqrt{x}\right )\right )} b^{2} - \frac{b^{2} \operatorname{artanh}\left (c \sqrt{x}\right )^{2}}{2 \, x^{2}} - \frac{a^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(1/2)))^2/x^3,x, algorithm="maxima")

[Out]

1/6*((3*c^3*log(c*sqrt(x) + 1) - 3*c^3*log(c*sqrt(x) - 1) - 2*(3*c^2*x + 1)/x^(3/2))*c - 6*arctanh(c*sqrt(x))/
x^2)*a*b + 1/24*((16*c^2*log(x) - (3*c^2*x*log(c*sqrt(x) + 1)^2 + 3*c^2*x*log(c*sqrt(x) - 1)^2 + 16*c^2*x*log(
c*sqrt(x) - 1) - 2*(3*c^2*x*log(c*sqrt(x) - 1) - 8*c^2*x)*log(c*sqrt(x) + 1) + 4)/x)*c^2 + 4*(3*c^3*log(c*sqrt
(x) + 1) - 3*c^3*log(c*sqrt(x) - 1) - 2*(3*c^2*x + 1)/x^(3/2))*c*arctanh(c*sqrt(x)))*b^2 - 1/2*b^2*arctanh(c*s
qrt(x))^2/x^2 - 1/2*a^2/x^2

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Fricas [A]  time = 1.92422, size = 471, normalized size = 3.54 \begin{align*} \frac{32 \, b^{2} c^{4} x^{2} \log \left (\sqrt{x}\right ) + 4 \,{\left (3 \, a b - 4 \, b^{2}\right )} c^{4} x^{2} \log \left (c \sqrt{x} + 1\right ) - 4 \,{\left (3 \, a b + 4 \, b^{2}\right )} c^{4} x^{2} \log \left (c \sqrt{x} - 1\right ) - 4 \, b^{2} c^{2} x + 3 \,{\left (b^{2} c^{4} x^{2} - b^{2}\right )} \log \left (-\frac{c^{2} x + 2 \, c \sqrt{x} + 1}{c^{2} x - 1}\right )^{2} - 12 \, a^{2} - 4 \,{\left (3 \, a b +{\left (3 \, b^{2} c^{3} x + b^{2} c\right )} \sqrt{x}\right )} \log \left (-\frac{c^{2} x + 2 \, c \sqrt{x} + 1}{c^{2} x - 1}\right ) - 8 \,{\left (3 \, a b c^{3} x + a b c\right )} \sqrt{x}}{24 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(1/2)))^2/x^3,x, algorithm="fricas")

[Out]

1/24*(32*b^2*c^4*x^2*log(sqrt(x)) + 4*(3*a*b - 4*b^2)*c^4*x^2*log(c*sqrt(x) + 1) - 4*(3*a*b + 4*b^2)*c^4*x^2*l
og(c*sqrt(x) - 1) - 4*b^2*c^2*x + 3*(b^2*c^4*x^2 - b^2)*log(-(c^2*x + 2*c*sqrt(x) + 1)/(c^2*x - 1))^2 - 12*a^2
- 4*(3*a*b + (3*b^2*c^3*x + b^2*c)*sqrt(x))*log(-(c^2*x + 2*c*sqrt(x) + 1)/(c^2*x - 1)) - 8*(3*a*b*c^3*x + a*
b*c)*sqrt(x))/x^2

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Sympy [A]  time = 179.052, size = 972, normalized size = 7.31 \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**3,x)

[Out]

Piecewise((-a**2/(2*x**2) + a*b*atanh(sqrt(x)*sqrt(1/x))/x**2 - b**2*atanh(sqrt(x)*sqrt(1/x))**2/(2*x**2), Eq(
c, -sqrt(1/x))), (-a**2/(2*x**2) - a*b*atanh(sqrt(x)*sqrt(1/x))/x**2 - b**2*atanh(sqrt(x)*sqrt(1/x))**2/(2*x**
2), Eq(c, sqrt(1/x))), (-a**2/(2*x**2), Eq(c, 0)), (-3*a**2*c**2*x**(3/2)/(6*c**2*x**(7/2) - 6*x**(5/2)) + 3*a
**2*sqrt(x)/(6*c**2*x**(7/2) - 6*x**(5/2)) + 6*a*b*c**6*x**(7/2)*atanh(c*sqrt(x))/(6*c**2*x**(7/2) - 6*x**(5/2
)) - 6*a*b*c**5*x**3/(6*c**2*x**(7/2) - 6*x**(5/2)) - 6*a*b*c**4*x**(5/2)*atanh(c*sqrt(x))/(6*c**2*x**(7/2) -
6*x**(5/2)) + 4*a*b*c**3*x**2/(6*c**2*x**(7/2) - 6*x**(5/2)) - 6*a*b*c**2*x**(3/2)*atanh(c*sqrt(x))/(6*c**2*x*
*(7/2) - 6*x**(5/2)) + 2*a*b*c*x/(6*c**2*x**(7/2) - 6*x**(5/2)) + 6*a*b*sqrt(x)*atanh(c*sqrt(x))/(6*c**2*x**(7
/2) - 6*x**(5/2)) + 4*b**2*c**6*x**(7/2)*log(x)/(6*c**2*x**(7/2) - 6*x**(5/2)) - 8*b**2*c**6*x**(7/2)*log(sqrt
(x) - 1/c)/(6*c**2*x**(7/2) - 6*x**(5/2)) + 3*b**2*c**6*x**(7/2)*atanh(c*sqrt(x))**2/(6*c**2*x**(7/2) - 6*x**(
5/2)) - 8*b**2*c**6*x**(7/2)*atanh(c*sqrt(x))/(6*c**2*x**(7/2) - 6*x**(5/2)) - b**2*c**6*x**(7/2)/(6*c**2*x**(
7/2) - 6*x**(5/2)) - 6*b**2*c**5*x**3*atanh(c*sqrt(x))/(6*c**2*x**(7/2) - 6*x**(5/2)) - 4*b**2*c**4*x**(5/2)*l
og(x)/(6*c**2*x**(7/2) - 6*x**(5/2)) + 8*b**2*c**4*x**(5/2)*log(sqrt(x) - 1/c)/(6*c**2*x**(7/2) - 6*x**(5/2))
- 3*b**2*c**4*x**(5/2)*atanh(c*sqrt(x))**2/(6*c**2*x**(7/2) - 6*x**(5/2)) + 8*b**2*c**4*x**(5/2)*atanh(c*sqrt(
x))/(6*c**2*x**(7/2) - 6*x**(5/2)) + 4*b**2*c**3*x**2*atanh(c*sqrt(x))/(6*c**2*x**(7/2) - 6*x**(5/2)) - 3*b**2
*c**2*x**(3/2)*atanh(c*sqrt(x))**2/(6*c**2*x**(7/2) - 6*x**(5/2)) + b**2*c**2*x**(3/2)/(6*c**2*x**(7/2) - 6*x*
*(5/2)) + 2*b**2*c*x*atanh(c*sqrt(x))/(6*c**2*x**(7/2) - 6*x**(5/2)) + 3*b**2*sqrt(x)*atanh(c*sqrt(x))**2/(6*c
**2*x**(7/2) - 6*x**(5/2)), 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^{3}}\,{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^3,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*sqrt(x)) + a)^2/x^3, x)