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

Optimal. Leaf size=129 \[ \frac{a b \sqrt{x}}{c^3}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{2 c^4}+\frac{b x^{3/2} \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{3 c}+\frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2+\frac{b^2 x}{6 c^2}+\frac{2 b^2 \log \left (1-c^2 x\right )}{3 c^4}+\frac{b^2 \sqrt{x} \tanh ^{-1}\left (c \sqrt{x}\right )}{c^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [F]  time = 0.013963, 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 = {} \[ \int x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*(a + b*ArcTanh[c*Sqrt[x]])^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0859147, size = 160, normalized size = 1.24 \[ \frac{3 a^2 c^4 x^2+2 a b c^3 x^{3/2}+2 b c \sqrt{x} \tanh ^{-1}\left (c \sqrt{x}\right ) \left (3 a c^3 x^{3/2}+b \left (c^2 x+3\right )\right )+6 a b c \sqrt{x}+b (3 a+4 b) \log \left (1-c \sqrt{x}\right )-3 a b \log \left (c \sqrt{x}+1\right )+3 b^2 \left (c^4 x^2-1\right ) \tanh ^{-1}\left (c \sqrt{x}\right )^2+b^2 c^2 x+4 b^2 \log \left (c \sqrt{x}+1\right )}{6 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.051, size = 317, normalized size = 2.5 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{{b}^{2}{x}^{2}}{2} \left ({\it Artanh} \left ( c\sqrt{x} \right ) \right ) ^{2}}+{\frac{{b}^{2}}{3\,c}{\it Artanh} \left ( c\sqrt{x} \right ){x}^{{\frac{3}{2}}}}+{\frac{{b}^{2}}{{c}^{3}}{\it Artanh} \left ( c\sqrt{x} \right ) \sqrt{x}}+{\frac{{b}^{2}}{2\,{c}^{4}}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( c\sqrt{x}-1 \right ) }-{\frac{{b}^{2}}{2\,{c}^{4}}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( 1+c\sqrt{x} \right ) }-{\frac{{b}^{2}}{4\,{c}^{4}}\ln \left ( c\sqrt{x}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }+{\frac{{b}^{2}}{8\,{c}^{4}} \left ( \ln \left ( c\sqrt{x}-1 \right ) \right ) ^{2}}+{\frac{{b}^{2}}{4\,{c}^{4}}\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}^{4}}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ( 1+c\sqrt{x} \right ) }+{\frac{{b}^{2}}{8\,{c}^{4}} \left ( \ln \left ( 1+c\sqrt{x} \right ) \right ) ^{2}}+{\frac{{b}^{2}x}{6\,{c}^{2}}}+{\frac{2\,{b}^{2}}{3\,{c}^{4}}\ln \left ( c\sqrt{x}-1 \right ) }+{\frac{2\,{b}^{2}}{3\,{c}^{4}}\ln \left ( 1+c\sqrt{x} \right ) }+ab{x}^{2}{\it Artanh} \left ( c\sqrt{x} \right ) +{\frac{ab}{3\,c}{x}^{{\frac{3}{2}}}}+{\frac{ab}{{c}^{3}}\sqrt{x}}+{\frac{ab}{2\,{c}^{4}}\ln \left ( c\sqrt{x}-1 \right ) }-{\frac{ab}{2\,{c}^{4}}\ln \left ( 1+c\sqrt{x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.02672, size = 290, normalized size = 2.25 \begin{align*} \frac{1}{2} \, b^{2} x^{2} \operatorname{artanh}\left (c \sqrt{x}\right )^{2} + \frac{1}{2} \, a^{2} x^{2} + \frac{1}{6} \,{\left (6 \, x^{2} \operatorname{artanh}\left (c \sqrt{x}\right ) + c{\left (\frac{2 \,{\left (c^{2} x^{\frac{3}{2}} + 3 \, \sqrt{x}\right )}}{c^{4}} - \frac{3 \, \log \left (c \sqrt{x} + 1\right )}{c^{5}} + \frac{3 \, \log \left (c \sqrt{x} - 1\right )}{c^{5}}\right )}\right )} a b + \frac{1}{24} \,{\left (4 \, c{\left (\frac{2 \,{\left (c^{2} x^{\frac{3}{2}} + 3 \, \sqrt{x}\right )}}{c^{4}} - \frac{3 \, \log \left (c \sqrt{x} + 1\right )}{c^{5}} + \frac{3 \, \log \left (c \sqrt{x} - 1\right )}{c^{5}}\right )} \operatorname{artanh}\left (c \sqrt{x}\right ) + \frac{4 \, c^{2} x - 2 \,{\left (3 \, \log \left (c \sqrt{x} - 1\right ) - 8\right )} \log \left (c \sqrt{x} + 1\right ) + 3 \, \log \left (c \sqrt{x} + 1\right )^{2} + 3 \, \log \left (c \sqrt{x} - 1\right )^{2} + 16 \, \log \left (c \sqrt{x} - 1\right )}{c^{4}}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^2*x^2*arctanh(c*sqrt(x))^2 + 1/2*a^2*x^2 + 1/6*(6*x^2*arctanh(c*sqrt(x)) + c*(2*(c^2*x^(3/2) + 3*sqrt(x)
)/c^4 - 3*log(c*sqrt(x) + 1)/c^5 + 3*log(c*sqrt(x) - 1)/c^5))*a*b + 1/24*(4*c*(2*(c^2*x^(3/2) + 3*sqrt(x))/c^4
- 3*log(c*sqrt(x) + 1)/c^5 + 3*log(c*sqrt(x) - 1)/c^5)*arctanh(c*sqrt(x)) + (4*c^2*x - 2*(3*log(c*sqrt(x) - 1
) - 8)*log(c*sqrt(x) + 1) + 3*log(c*sqrt(x) + 1)^2 + 3*log(c*sqrt(x) - 1)^2 + 16*log(c*sqrt(x) - 1))/c^4)*b^2

________________________________________________________________________________________

Fricas [A]  time = 2.20403, size = 479, normalized size = 3.71 \begin{align*} \frac{12 \, a^{2} c^{4} x^{2} + 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} + 4 \,{\left (3 \, a b c^{4} - 3 \, a b + 4 \, b^{2}\right )} \log \left (c \sqrt{x} + 1\right ) - 4 \,{\left (3 \, a b c^{4} - 3 \, a b - 4 \, b^{2}\right )} \log \left (c \sqrt{x} - 1\right ) + 4 \,{\left (3 \, a b c^{4} x^{2} - 3 \, a b c^{4} +{\left (b^{2} c^{3} x + 3 \, 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 (a b c^{3} x + 3 \, a b c\right )} \sqrt{x}}{24 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \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(x*(a+b*atanh(c*x**(1/2)))**2,x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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