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

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

[Out]

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

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Rubi [F]  time = 0.0063505, 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 = {} \[ \int \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcTanh[c*Sqrt[x]])^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0614956, size = 115, normalized size = 1.35 \[ \frac{a^2 c^2 x+2 a b c \sqrt{x}+b (a+b) \log \left (1-c \sqrt{x}\right )-a b \log \left (c \sqrt{x}+1\right )+2 b c \sqrt{x} \tanh ^{-1}\left (c \sqrt{x}\right ) \left (a c \sqrt{x}+b\right )+b^2 \left (c^2 x-1\right ) \tanh ^{-1}\left (c \sqrt{x}\right )^2+b^2 \log \left (c \sqrt{x}+1\right )}{c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.05, size = 272, normalized size = 3.2 \begin{align*}{a}^{2}x+{b}^{2}x \left ({\it Artanh} \left ( c\sqrt{x} \right ) \right ) ^{2}+2\,{\frac{{b}^{2}{\it Artanh} \left ( c\sqrt{x} \right ) \sqrt{x}}{c}}+{\frac{{b}^{2}}{{c}^{2}}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( c\sqrt{x}-1 \right ) }-{\frac{{b}^{2}}{{c}^{2}}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( 1+c\sqrt{x} \right ) }-{\frac{{b}^{2}}{2\,{c}^{2}}\ln \left ( c\sqrt{x}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }+{\frac{{b}^{2}}{4\,{c}^{2}} \left ( \ln \left ( c\sqrt{x}-1 \right ) \right ) ^{2}}+{\frac{{b}^{2}}{{c}^{2}}\ln \left ( c\sqrt{x}-1 \right ) }+{\frac{{b}^{2}}{{c}^{2}}\ln \left ( 1+c\sqrt{x} \right ) }-{\frac{{b}^{2}}{2\,{c}^{2}}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ( 1+c\sqrt{x} \right ) }+{\frac{{b}^{2}}{2\,{c}^{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}}{4\,{c}^{2}} \left ( \ln \left ( 1+c\sqrt{x} \right ) \right ) ^{2}}+2\,abx{\it Artanh} \left ( c\sqrt{x} \right ) +2\,{\frac{ab\sqrt{x}}{c}}+{\frac{ab}{{c}^{2}}\ln \left ( c\sqrt{x}-1 \right ) }-{\frac{ab}{{c}^{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)

[Out]

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

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Maxima [B]  time = 1.01408, size = 236, normalized size = 2.78 \begin{align*}{\left (c{\left (\frac{2 \, \sqrt{x}}{c^{2}} - \frac{\log \left (c \sqrt{x} + 1\right )}{c^{3}} + \frac{\log \left (c \sqrt{x} - 1\right )}{c^{3}}\right )} + 2 \, x \operatorname{artanh}\left (c \sqrt{x}\right )\right )} a b + \frac{1}{4} \,{\left (4 \, c{\left (\frac{2 \, \sqrt{x}}{c^{2}} - \frac{\log \left (c \sqrt{x} + 1\right )}{c^{3}} + \frac{\log \left (c \sqrt{x} - 1\right )}{c^{3}}\right )} \operatorname{artanh}\left (c \sqrt{x}\right ) + 4 \, x \operatorname{artanh}\left (c \sqrt{x}\right )^{2} - \frac{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 )}{c^{2}}\right )} b^{2} + 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, algorithm="maxima")

[Out]

(c*(2*sqrt(x)/c^2 - log(c*sqrt(x) + 1)/c^3 + log(c*sqrt(x) - 1)/c^3) + 2*x*arctanh(c*sqrt(x)))*a*b + 1/4*(4*c*
(2*sqrt(x)/c^2 - log(c*sqrt(x) + 1)/c^3 + log(c*sqrt(x) - 1)/c^3)*arctanh(c*sqrt(x)) + 4*x*arctanh(c*sqrt(x))^
2 - (2*(log(c*sqrt(x) - 1) - 2)*log(c*sqrt(x) + 1) - log(c*sqrt(x) + 1)^2 - log(c*sqrt(x) - 1)^2 - 4*log(c*sqr
t(x) - 1))/c^2)*b^2 + a^2*x

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Fricas [B]  time = 1.99055, size = 382, normalized size = 4.49 \begin{align*} \frac{4 \, a^{2} c^{2} x + 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 \,{\left (a b c^{2} - a b + b^{2}\right )} \log \left (c \sqrt{x} + 1\right ) - 4 \,{\left (a b c^{2} - a b - b^{2}\right )} \log \left (c \sqrt{x} - 1\right ) + 4 \,{\left (a b c^{2} x - a b c^{2} + b^{2} c \sqrt{x}\right )} \log \left (-\frac{c^{2} x + 2 \, c \sqrt{x} + 1}{c^{2} x - 1}\right )}{4 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{atanh}{\left (c \sqrt{x} \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c*sqrt(x)))**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{artanh}\left (c \sqrt{x}\right ) + a\right )}^{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, algorithm="giac")

[Out]

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