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

Optimal. Leaf size=173 \[ \frac{2 b x^{3/2} \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{9 c^3}+\frac{2 a b \sqrt{x}}{3 c^5}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{3 c^6}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2+\frac{2 b x^{5/2} \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{15 c}+\frac{b^2 x^2}{30 c^2}+\frac{8 b^2 x}{45 c^4}+\frac{23 b^2 \log \left (1-c^2 x\right )}{45 c^6}+\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left (c \sqrt{x}\right )}{3 c^5} \]

[Out]

(2*a*b*Sqrt[x])/(3*c^5) + (8*b^2*x)/(45*c^4) + (b^2*x^2)/(30*c^2) + (2*b^2*Sqrt[x]*ArcTanh[c*Sqrt[x]])/(3*c^5)
+ (2*b*x^(3/2)*(a + b*ArcTanh[c*Sqrt[x]]))/(9*c^3) + (2*b*x^(5/2)*(a + b*ArcTanh[c*Sqrt[x]]))/(15*c) - (a + b
*ArcTanh[c*Sqrt[x]])^2/(3*c^6) + (x^3*(a + b*ArcTanh[c*Sqrt[x]])^2)/3 + (23*b^2*Log[1 - c^2*x])/(45*c^6)

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Rubi [F]  time = 0.023916, 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 = {} \[ \int x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.103839, size = 194, normalized size = 1.12 \[ \frac{30 a^2 c^6 x^3+12 a b c^5 x^{5/2}+20 a b c^3 x^{3/2}+4 b c \sqrt{x} \tanh ^{-1}\left (c \sqrt{x}\right ) \left (15 a c^5 x^{5/2}+b \left (3 c^4 x^2+5 c^2 x+15\right )\right )+60 a b c \sqrt{x}+2 b (15 a+23 b) \log \left (1-c \sqrt{x}\right )-30 a b \log \left (c \sqrt{x}+1\right )+3 b^2 c^4 x^2+30 b^2 \left (c^6 x^3-1\right ) \tanh ^{-1}\left (c \sqrt{x}\right )^2+16 b^2 c^2 x+46 b^2 \log \left (c \sqrt{x}+1\right )}{90 c^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(60*a*b*c*Sqrt[x] + 16*b^2*c^2*x + 20*a*b*c^3*x^(3/2) + 3*b^2*c^4*x^2 + 12*a*b*c^5*x^(5/2) + 30*a^2*c^6*x^3 +
4*b*c*Sqrt[x]*(15*a*c^5*x^(5/2) + b*(15 + 5*c^2*x + 3*c^4*x^2))*ArcTanh[c*Sqrt[x]] + 30*b^2*(-1 + c^6*x^3)*Arc
Tanh[c*Sqrt[x]]^2 + 2*b*(15*a + 23*b)*Log[1 - c*Sqrt[x]] - 30*a*b*Log[1 + c*Sqrt[x]] + 46*b^2*Log[1 + c*Sqrt[x
]])/(90*c^6)

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Maple [B]  time = 0.049, size = 358, normalized size = 2.1 \begin{align*}{\frac{{x}^{3}{a}^{2}}{3}}+{\frac{{x}^{3}{b}^{2}}{3} \left ({\it Artanh} \left ( c\sqrt{x} \right ) \right ) ^{2}}+{\frac{2\,{b}^{2}}{15\,c}{\it Artanh} \left ( c\sqrt{x} \right ){x}^{{\frac{5}{2}}}}+{\frac{2\,{b}^{2}}{9\,{c}^{3}}{\it Artanh} \left ( c\sqrt{x} \right ){x}^{{\frac{3}{2}}}}+{\frac{2\,{b}^{2}}{3\,{c}^{5}}{\it Artanh} \left ( c\sqrt{x} \right ) \sqrt{x}}+{\frac{{b}^{2}}{3\,{c}^{6}}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( c\sqrt{x}-1 \right ) }-{\frac{{b}^{2}}{3\,{c}^{6}}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( 1+c\sqrt{x} \right ) }-{\frac{{b}^{2}}{6\,{c}^{6}}\ln \left ( c\sqrt{x}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }+{\frac{{b}^{2}}{12\,{c}^{6}} \left ( \ln \left ( c\sqrt{x}-1 \right ) \right ) ^{2}}+{\frac{{b}^{2}}{6\,{c}^{6}}\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}}{6\,{c}^{6}}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ( 1+c\sqrt{x} \right ) }+{\frac{{b}^{2}}{12\,{c}^{6}} \left ( \ln \left ( 1+c\sqrt{x} \right ) \right ) ^{2}}+{\frac{{b}^{2}{x}^{2}}{30\,{c}^{2}}}+{\frac{8\,{b}^{2}x}{45\,{c}^{4}}}+{\frac{23\,{b}^{2}}{45\,{c}^{6}}\ln \left ( c\sqrt{x}-1 \right ) }+{\frac{23\,{b}^{2}}{45\,{c}^{6}}\ln \left ( 1+c\sqrt{x} \right ) }+{\frac{2\,ab{x}^{3}}{3}{\it Artanh} \left ( c\sqrt{x} \right ) }+{\frac{2\,ab}{15\,c}{x}^{{\frac{5}{2}}}}+{\frac{2\,ab}{9\,{c}^{3}}{x}^{{\frac{3}{2}}}}+{\frac{2\,ab}{3\,{c}^{5}}\sqrt{x}}+{\frac{ab}{3\,{c}^{6}}\ln \left ( c\sqrt{x}-1 \right ) }-{\frac{ab}{3\,{c}^{6}}\ln \left ( 1+c\sqrt{x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*a^2+1/3*b^2*x^3*arctanh(c*x^(1/2))^2+2/15/c*b^2*arctanh(c*x^(1/2))*x^(5/2)+2/9/c^3*b^2*arctanh(c*x^(1/
2))*x^(3/2)+2/3*b^2*arctanh(c*x^(1/2))*x^(1/2)/c^5+1/3/c^6*b^2*arctanh(c*x^(1/2))*ln(c*x^(1/2)-1)-1/3/c^6*b^2*
arctanh(c*x^(1/2))*ln(1+c*x^(1/2))-1/6/c^6*b^2*ln(c*x^(1/2)-1)*ln(1/2+1/2*c*x^(1/2))+1/12/c^6*b^2*ln(c*x^(1/2)
-1)^2+1/6/c^6*b^2*ln(-1/2*c*x^(1/2)+1/2)*ln(1/2+1/2*c*x^(1/2))-1/6/c^6*b^2*ln(-1/2*c*x^(1/2)+1/2)*ln(1+c*x^(1/
2))+1/12/c^6*b^2*ln(1+c*x^(1/2))^2+1/30*b^2*x^2/c^2+8/45*b^2*x/c^4+23/45/c^6*b^2*ln(c*x^(1/2)-1)+23/45/c^6*b^2
*ln(1+c*x^(1/2))+2/3*a*b*x^3*arctanh(c*x^(1/2))+2/15/c*a*b*x^(5/2)+2/9/c^3*a*b*x^(3/2)+2/3*a*b*x^(1/2)/c^5+1/3
/c^6*a*b*ln(c*x^(1/2)-1)-1/3/c^6*a*b*ln(1+c*x^(1/2))

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Maxima [A]  time = 1.00531, size = 325, normalized size = 1.88 \begin{align*} \frac{1}{3} \, b^{2} x^{3} \operatorname{artanh}\left (c \sqrt{x}\right )^{2} + \frac{1}{3} \, a^{2} x^{3} + \frac{1}{45} \,{\left (30 \, x^{3} \operatorname{artanh}\left (c \sqrt{x}\right ) + c{\left (\frac{2 \,{\left (3 \, c^{4} x^{\frac{5}{2}} + 5 \, c^{2} x^{\frac{3}{2}} + 15 \, \sqrt{x}\right )}}{c^{6}} - \frac{15 \, \log \left (c \sqrt{x} + 1\right )}{c^{7}} + \frac{15 \, \log \left (c \sqrt{x} - 1\right )}{c^{7}}\right )}\right )} a b + \frac{1}{180} \,{\left (4 \, c{\left (\frac{2 \,{\left (3 \, c^{4} x^{\frac{5}{2}} + 5 \, c^{2} x^{\frac{3}{2}} + 15 \, \sqrt{x}\right )}}{c^{6}} - \frac{15 \, \log \left (c \sqrt{x} + 1\right )}{c^{7}} + \frac{15 \, \log \left (c \sqrt{x} - 1\right )}{c^{7}}\right )} \operatorname{artanh}\left (c \sqrt{x}\right ) + \frac{6 \, c^{4} x^{2} + 32 \, c^{2} x - 2 \,{\left (15 \, \log \left (c \sqrt{x} - 1\right ) - 46\right )} \log \left (c \sqrt{x} + 1\right ) + 15 \, \log \left (c \sqrt{x} + 1\right )^{2} + 15 \, \log \left (c \sqrt{x} - 1\right )^{2} + 92 \, \log \left (c \sqrt{x} - 1\right )}{c^{6}}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2*x^3*arctanh(c*sqrt(x))^2 + 1/3*a^2*x^3 + 1/45*(30*x^3*arctanh(c*sqrt(x)) + c*(2*(3*c^4*x^(5/2) + 5*c^2
*x^(3/2) + 15*sqrt(x))/c^6 - 15*log(c*sqrt(x) + 1)/c^7 + 15*log(c*sqrt(x) - 1)/c^7))*a*b + 1/180*(4*c*(2*(3*c^
4*x^(5/2) + 5*c^2*x^(3/2) + 15*sqrt(x))/c^6 - 15*log(c*sqrt(x) + 1)/c^7 + 15*log(c*sqrt(x) - 1)/c^7)*arctanh(c
*sqrt(x)) + (6*c^4*x^2 + 32*c^2*x - 2*(15*log(c*sqrt(x) - 1) - 46)*log(c*sqrt(x) + 1) + 15*log(c*sqrt(x) + 1)^
2 + 15*log(c*sqrt(x) - 1)^2 + 92*log(c*sqrt(x) - 1))/c^6)*b^2

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Fricas [A]  time = 2.18744, size = 567, normalized size = 3.28 \begin{align*} \frac{60 \, a^{2} c^{6} x^{3} + 6 \, b^{2} c^{4} x^{2} + 32 \, b^{2} c^{2} x + 15 \,{\left (b^{2} c^{6} x^{3} - b^{2}\right )} \log \left (-\frac{c^{2} x + 2 \, c \sqrt{x} + 1}{c^{2} x - 1}\right )^{2} + 4 \,{\left (15 \, a b c^{6} - 15 \, a b + 23 \, b^{2}\right )} \log \left (c \sqrt{x} + 1\right ) - 4 \,{\left (15 \, a b c^{6} - 15 \, a b - 23 \, b^{2}\right )} \log \left (c \sqrt{x} - 1\right ) + 4 \,{\left (15 \, a b c^{6} x^{3} - 15 \, a b c^{6} +{\left (3 \, b^{2} c^{5} x^{2} + 5 \, b^{2} c^{3} x + 15 \, 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^{5} x^{2} + 5 \, a b c^{3} x + 15 \, a b c\right )} \sqrt{x}}{180 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/180*(60*a^2*c^6*x^3 + 6*b^2*c^4*x^2 + 32*b^2*c^2*x + 15*(b^2*c^6*x^3 - b^2)*log(-(c^2*x + 2*c*sqrt(x) + 1)/(
c^2*x - 1))^2 + 4*(15*a*b*c^6 - 15*a*b + 23*b^2)*log(c*sqrt(x) + 1) - 4*(15*a*b*c^6 - 15*a*b - 23*b^2)*log(c*s
qrt(x) - 1) + 4*(15*a*b*c^6*x^3 - 15*a*b*c^6 + (3*b^2*c^5*x^2 + 5*b^2*c^3*x + 15*b^2*c)*sqrt(x))*log(-(c^2*x +
2*c*sqrt(x) + 1)/(c^2*x - 1)) + 8*(3*a*b*c^5*x^2 + 5*a*b*c^3*x + 15*a*b*c)*sqrt(x))/c^6

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

[Out]

Integral(x**2*(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} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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