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

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

[Out]

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

________________________________________________________________________________________

Rubi [F]  time = 0.01, 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 = {} \[ \int x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2 \, dx \]

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 = 0.09, 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)

________________________________________________________________________________________

fricas [A]  time = 0.92, size = 207, normalized size = 1.60 \[ \frac {12 \, a^{2} c^{4} x^{2} + 4 \, b^{2} c^{2} x + 3 \, {\left (b^{2} c^{4} x^{2} - b^{2}\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right )^{2} + 4 \, {\left (3 \, a b c^{4} - 3 \, a b + 4 \, b^{2}\right )} \log \left (c \sqrt {x} + 1\right ) - 4 \, {\left (3 \, a b c^{4} - 3 \, a b - 4 \, b^{2}\right )} \log \left (c \sqrt {x} - 1\right ) + 4 \, {\left (3 \, a b c^{4} x^{2} - 3 \, a b c^{4} + {\left (b^{2} c^{3} x + 3 \, b^{2} c\right )} \sqrt {x}\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right ) + 8 \, {\left (a b c^{3} x + 3 \, a b c\right )} \sqrt {x}}{24 \, c^{4}} \]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maple [B]  time = 0.05, size = 317, normalized size = 2.46 \[ \frac {a^{2} x^{2}}{2}+\frac {b^{2} x^{2} \arctanh \left (c \sqrt {x}\right )^{2}}{2}+\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) x^{\frac {3}{2}}}{3 c}+\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) \sqrt {x}}{c^{3}}+\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2 c^{4}}-\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2 c^{4}}+\frac {b^{2} \ln \left (c \sqrt {x}-1\right )^{2}}{8 c^{4}}-\frac {b^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {1}{2}+\frac {c \sqrt {x}}{2}\right )}{4 c^{4}}+\frac {b^{2} \ln \left (1+c \sqrt {x}\right )^{2}}{8 c^{4}}-\frac {b^{2} \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (1+c \sqrt {x}\right )}{4 c^{4}}+\frac {b^{2} \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c \sqrt {x}}{2}\right )}{4 c^{4}}+\frac {b^{2} x}{6 c^{2}}+\frac {2 b^{2} \ln \left (c \sqrt {x}-1\right )}{3 c^{4}}+\frac {2 b^{2} \ln \left (1+c \sqrt {x}\right )}{3 c^{4}}+x^{2} a b \arctanh \left (c \sqrt {x}\right )+\frac {a b \,x^{\frac {3}{2}}}{3 c}+\frac {a b \sqrt {x}}{c^{3}}+\frac {a b \ln \left (c \sqrt {x}-1\right )}{2 c^{4}}-\frac {a b \ln \left (1+c \sqrt {x}\right )}{2 c^{4}} \]

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))+1/8/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))+1/8/c^4*b^2*ln(1+c*x^(1/2))^2-1/4/c^4*b
^2*ln(-1/2*c*x^(1/2)+1/2)*ln(1+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))+1/6*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))+x^2*a*b*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 = 0.33, size = 215, normalized size = 1.67 \[ \frac {1}{2} \, b^{2} x^{2} \operatorname {artanh}\left (c \sqrt {x}\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + \frac {1}{6} \, {\left (6 \, x^{2} \operatorname {artanh}\left (c \sqrt {x}\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{\frac {3}{2}} + 3 \, \sqrt {x}\right )}}{c^{4}} - \frac {3 \, \log \left (c \sqrt {x} + 1\right )}{c^{5}} + \frac {3 \, \log \left (c \sqrt {x} - 1\right )}{c^{5}}\right )}\right )} a b + \frac {1}{24} \, {\left (4 \, c {\left (\frac {2 \, {\left (c^{2} x^{\frac {3}{2}} + 3 \, \sqrt {x}\right )}}{c^{4}} - \frac {3 \, \log \left (c \sqrt {x} + 1\right )}{c^{5}} + \frac {3 \, \log \left (c \sqrt {x} - 1\right )}{c^{5}}\right )} \operatorname {artanh}\left (c \sqrt {x}\right ) + \frac {4 \, c^{2} x - 2 \, {\left (3 \, \log \left (c \sqrt {x} - 1\right ) - 8\right )} \log \left (c \sqrt {x} + 1\right ) + 3 \, \log \left (c \sqrt {x} + 1\right )^{2} + 3 \, \log \left (c \sqrt {x} - 1\right )^{2} + 16 \, \log \left (c \sqrt {x} - 1\right )}{c^{4}}\right )} b^{2} \]

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

________________________________________________________________________________________

mupad [B]  time = 1.28, size = 143, normalized size = 1.11 \[ \frac {a^2\,x^2}{2}-\frac {b^2\,{\mathrm {atanh}\left (c\,\sqrt {x}\right )}^2}{2\,c^4}+\frac {2\,b^2\,\ln \left (c^2\,x-1\right )}{3\,c^4}+\frac {b^2\,x^2\,{\mathrm {atanh}\left (c\,\sqrt {x}\right )}^2}{2}+\frac {b^2\,x}{6\,c^2}+\frac {b^2\,x^{3/2}\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{3\,c}+\frac {b^2\,\sqrt {x}\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{c^3}+\frac {a\,b\,x^{3/2}}{3\,c}+\frac {a\,b\,\sqrt {x}}{c^3}-\frac {a\,b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{c^4}+a\,b\,x^2\,\mathrm {atanh}\left (c\,\sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*atanh(c*x^(1/2)))^2,x)

[Out]

(a^2*x^2)/2 - (b^2*atanh(c*x^(1/2))^2)/(2*c^4) + (2*b^2*log(c^2*x - 1))/(3*c^4) + (b^2*x^2*atanh(c*x^(1/2))^2)
/2 + (b^2*x)/(6*c^2) + (b^2*x^(3/2)*atanh(c*x^(1/2)))/(3*c) + (b^2*x^(1/2)*atanh(c*x^(1/2)))/c^3 + (a*b*x^(3/2
))/(3*c) + (a*b*x^(1/2))/c^3 - (a*b*atanh(c*x^(1/2)))/c^4 + a*b*x^2*atanh(c*x^(1/2))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )^{2}\, dx \]

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)

________________________________________________________________________________________