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]

-(a+b*arctanh(c*x^(1/2)))^2/c^2+x*(a+b*arctanh(c*x^(1/2)))^2+b^2*ln(-c^2*x+1)/c^2+2*a*b*x^(1/2)/c+2*b^2*arctan
h(c*x^(1/2))*x^(1/2)/c

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

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*}

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Mathematica [A]  time = 0.06, 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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fricas [B]  time = 1.13, size = 165, normalized size = 1.94 \[ \frac {4 \, a^{2} c^{2} x + 8 \, a b c \sqrt {x} + {\left (b^{2} c^{2} x - b^{2}\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right )^{2} + 4 \, {\left (a b c^{2} - a b + b^{2}\right )} \log \left (c \sqrt {x} + 1\right ) - 4 \, {\left (a b c^{2} - a b - b^{2}\right )} \log \left (c \sqrt {x} - 1\right ) + 4 \, {\left (a b c^{2} x - a b c^{2} + b^{2} c \sqrt {x}\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right )}{4 \, c^{2}} \]

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

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)

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maple [B]  time = 0.05, size = 272, normalized size = 3.20 \[ a^{2} x +x \,b^{2} \arctanh \left (c \sqrt {x}\right )^{2}+\frac {2 b^{2} \arctanh \left (c \sqrt {x}\right ) \sqrt {x}}{c}+\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{c^{2}}-\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{c^{2}}+\frac {b^{2} \ln \left (c \sqrt {x}-1\right )^{2}}{4 c^{2}}-\frac {b^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {1}{2}+\frac {c \sqrt {x}}{2}\right )}{2 c^{2}}+\frac {b^{2} \ln \left (c \sqrt {x}-1\right )}{c^{2}}+\frac {b^{2} \ln \left (1+c \sqrt {x}\right )}{c^{2}}+\frac {b^{2} \ln \left (1+c \sqrt {x}\right )^{2}}{4 c^{2}}+\frac {b^{2} \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c \sqrt {x}}{2}\right )}{2 c^{2}}-\frac {b^{2} \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (1+c \sqrt {x}\right )}{2 c^{2}}+2 a b x \arctanh \left (c \sqrt {x}\right )+\frac {2 a b \sqrt {x}}{c}+\frac {a b \ln \left (c \sqrt {x}-1\right )}{c^{2}}-\frac {a b \ln \left (1+c \sqrt {x}\right )}{c^{2}} \]

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+x*b^2*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))+1/4/c^2*b^2*ln(c*x^(1/2)-1)^2-1/2/c^2*b^2*ln(c*x^(1/2)-1)*ln(1
/2+1/2*c*x^(1/2))+1/c^2*b^2*ln(c*x^(1/2)-1)+1/c^2*b^2*ln(1+c*x^(1/2))+1/4/c^2*b^2*ln(1+c*x^(1/2))^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))-1/2/c^2*b^2*ln(-1/2*c*x^(1/2)+1/2)*ln(1+c*x^(1/2))+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 = 0.34, size = 175, normalized size = 2.06 \[ {\left (c {\left (\frac {2 \, \sqrt {x}}{c^{2}} - \frac {\log \left (c \sqrt {x} + 1\right )}{c^{3}} + \frac {\log \left (c \sqrt {x} - 1\right )}{c^{3}}\right )} + 2 \, x \operatorname {artanh}\left (c \sqrt {x}\right )\right )} a b + \frac {1}{4} \, {\left (4 \, c {\left (\frac {2 \, \sqrt {x}}{c^{2}} - \frac {\log \left (c \sqrt {x} + 1\right )}{c^{3}} + \frac {\log \left (c \sqrt {x} - 1\right )}{c^{3}}\right )} \operatorname {artanh}\left (c \sqrt {x}\right ) + 4 \, x \operatorname {artanh}\left (c \sqrt {x}\right )^{2} - \frac {2 \, {\left (\log \left (c \sqrt {x} - 1\right ) - 2\right )} \log \left (c \sqrt {x} + 1\right ) - \log \left (c \sqrt {x} + 1\right )^{2} - \log \left (c \sqrt {x} - 1\right )^{2} - 4 \, \log \left (c \sqrt {x} - 1\right )}{c^{2}}\right )} b^{2} + a^{2} x \]

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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mupad [B]  time = 1.06, size = 94, normalized size = 1.11 \[ a^2\,x+\frac {c\,\left (2\,b^2\,\sqrt {x}\,\mathrm {atanh}\left (c\,\sqrt {x}\right )+2\,a\,b\,\sqrt {x}\right )-b^2\,{\mathrm {atanh}\left (c\,\sqrt {x}\right )}^2+b^2\,\ln \left (c^2\,x-1\right )-2\,a\,b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{c^2}+b^2\,x\,{\mathrm {atanh}\left (c\,\sqrt {x}\right )}^2+2\,a\,b\,x\,\mathrm {atanh}\left (c\,\sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^(1/2)))^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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