3.195 \(\int x^3 (a+b \tanh ^{-1}(c \sqrt{x}))^2 \, dx\)

Optimal. Leaf size=211 \[ \frac{b x^{5/2} \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{10 c^3}+\frac{b x^{3/2} \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{6 c^5}+\frac{a b \sqrt{x}}{2 c^7}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{4 c^8}+\frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2+\frac{b x^{7/2} \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{14 c}+\frac{b^2 x^3}{84 c^2}+\frac{3 b^2 x^2}{70 c^4}+\frac{71 b^2 x}{420 c^6}+\frac{44 b^2 \log \left (1-c^2 x\right )}{105 c^8}+\frac{b^2 \sqrt{x} \tanh ^{-1}\left (c \sqrt{x}\right )}{2 c^7} \]

[Out]

(a*b*Sqrt[x])/(2*c^7) + (71*b^2*x)/(420*c^6) + (3*b^2*x^2)/(70*c^4) + (b^2*x^3)/(84*c^2) + (b^2*Sqrt[x]*ArcTan
h[c*Sqrt[x]])/(2*c^7) + (b*x^(3/2)*(a + b*ArcTanh[c*Sqrt[x]]))/(6*c^5) + (b*x^(5/2)*(a + b*ArcTanh[c*Sqrt[x]])
)/(10*c^3) + (b*x^(7/2)*(a + b*ArcTanh[c*Sqrt[x]]))/(14*c) - (a + b*ArcTanh[c*Sqrt[x]])^2/(4*c^8) + (x^4*(a +
b*ArcTanh[c*Sqrt[x]])^2)/4 + (44*b^2*Log[1 - c^2*x])/(105*c^8)

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Rubi [F]  time = 0.0251889, 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 = {} \[ \int x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^3*(a + b*ArcTanh[c*Sqrt[x]])^2,x]

[Out]

Defer[Int][x^3*(a + b*ArcTanh[c*Sqrt[x]])^2, x]

Rubi steps

\begin{align*} \int x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2 \, dx &=\int x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2 \, dx\\ \end{align*}

Mathematica [A]  time = 0.112294, size = 224, normalized size = 1.06 \[ \frac{105 a^2 c^8 x^4+30 a b c^7 x^{7/2}+42 a b c^5 x^{5/2}+70 a b c^3 x^{3/2}+2 b c \sqrt{x} \tanh ^{-1}\left (c \sqrt{x}\right ) \left (105 a c^7 x^{7/2}+b \left (15 c^6 x^3+21 c^4 x^2+35 c^2 x+105\right )\right )+210 a b c \sqrt{x}+b (105 a+176 b) \log \left (1-c \sqrt{x}\right )-105 a b \log \left (c \sqrt{x}+1\right )+5 b^2 c^6 x^3+18 b^2 c^4 x^2+105 b^2 \left (c^8 x^4-1\right ) \tanh ^{-1}\left (c \sqrt{x}\right )^2+71 b^2 c^2 x+176 b^2 \log \left (c \sqrt{x}+1\right )}{420 c^8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*ArcTanh[c*Sqrt[x]])^2,x]

[Out]

(210*a*b*c*Sqrt[x] + 71*b^2*c^2*x + 70*a*b*c^3*x^(3/2) + 18*b^2*c^4*x^2 + 42*a*b*c^5*x^(5/2) + 5*b^2*c^6*x^3 +
30*a*b*c^7*x^(7/2) + 105*a^2*c^8*x^4 + 2*b*c*Sqrt[x]*(105*a*c^7*x^(7/2) + b*(105 + 35*c^2*x + 21*c^4*x^2 + 15
*c^6*x^3))*ArcTanh[c*Sqrt[x]] + 105*b^2*(-1 + c^8*x^4)*ArcTanh[c*Sqrt[x]]^2 + b*(105*a + 176*b)*Log[1 - c*Sqrt
[x]] - 105*a*b*Log[1 + c*Sqrt[x]] + 176*b^2*Log[1 + c*Sqrt[x]])/(420*c^8)

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Maple [B]  time = 0.056, size = 396, normalized size = 1.9 \begin{align*}{\frac{{b}^{2}}{2\,{c}^{7}}{\it Artanh} \left ( c\sqrt{x} \right ) \sqrt{x}}+{\frac{ab}{10\,{c}^{3}}{x}^{{\frac{5}{2}}}}+{\frac{ab}{14\,c}{x}^{{\frac{7}{2}}}}+{\frac{{b}^{2}}{14\,c}{\it Artanh} \left ( c\sqrt{x} \right ){x}^{{\frac{7}{2}}}}+{\frac{ab{x}^{4}}{2}{\it Artanh} \left ( c\sqrt{x} \right ) }-{\frac{{b}^{2}}{4\,{c}^{8}}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( 1+c\sqrt{x} \right ) }+{\frac{{b}^{2}}{10\,{c}^{3}}{\it Artanh} \left ( c\sqrt{x} \right ){x}^{{\frac{5}{2}}}}+{\frac{{b}^{2}}{6\,{c}^{5}}{\it Artanh} \left ( c\sqrt{x} \right ){x}^{{\frac{3}{2}}}}+{\frac{{b}^{2}}{4\,{c}^{8}}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( c\sqrt{x}-1 \right ) }+{\frac{ab}{6\,{c}^{5}}{x}^{{\frac{3}{2}}}}-{\frac{{b}^{2}}{8\,{c}^{8}}\ln \left ( c\sqrt{x}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }+{\frac{{b}^{2}}{8\,{c}^{8}}\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}}{8\,{c}^{8}}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ( 1+c\sqrt{x} \right ) }+{\frac{ab}{4\,{c}^{8}}\ln \left ( c\sqrt{x}-1 \right ) }-{\frac{ab}{4\,{c}^{8}}\ln \left ( 1+c\sqrt{x} \right ) }+{\frac{71\,{b}^{2}x}{420\,{c}^{6}}}+{\frac{{x}^{3}{b}^{2}}{84\,{c}^{2}}}+{\frac{{b}^{2}}{16\,{c}^{8}} \left ( \ln \left ( 1+c\sqrt{x} \right ) \right ) ^{2}}+{\frac{44\,{b}^{2}}{105\,{c}^{8}}\ln \left ( c\sqrt{x}-1 \right ) }+{\frac{44\,{b}^{2}}{105\,{c}^{8}}\ln \left ( 1+c\sqrt{x} \right ) }+{\frac{{b}^{2}}{16\,{c}^{8}} \left ( \ln \left ( c\sqrt{x}-1 \right ) \right ) ^{2}}+{\frac{3\,{b}^{2}{x}^{2}}{70\,{c}^{4}}}+{\frac{{x}^{4}{b}^{2}}{4} \left ({\it Artanh} \left ( c\sqrt{x} \right ) \right ) ^{2}}+{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{ab}{2\,{c}^{7}}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arctanh(c*x^(1/2)))^2,x)

[Out]

1/14/c*b^2*arctanh(c*x^(1/2))*x^(7/2)+1/2*a*b*x^4*arctanh(c*x^(1/2))-1/4/c^8*b^2*arctanh(c*x^(1/2))*ln(1+c*x^(
1/2))+1/10/c^3*a*b*x^(5/2)+1/14/c*x^(7/2)*a*b+1/10/c^3*b^2*arctanh(c*x^(1/2))*x^(5/2)+1/6/c^5*b^2*arctanh(c*x^
(1/2))*x^(3/2)+1/6/c^5*a*b*x^(3/2)-1/8/c^8*b^2*ln(c*x^(1/2)-1)*ln(1/2+1/2*c*x^(1/2))+1/8/c^8*b^2*ln(-1/2*c*x^(
1/2)+1/2)*ln(1/2+1/2*c*x^(1/2))+1/4/c^8*b^2*arctanh(c*x^(1/2))*ln(c*x^(1/2)-1)-1/8/c^8*b^2*ln(-1/2*c*x^(1/2)+1
/2)*ln(1+c*x^(1/2))+1/4/c^8*a*b*ln(c*x^(1/2)-1)-1/4/c^8*a*b*ln(1+c*x^(1/2))+71/420*b^2*x/c^6+1/84*b^2*x^3/c^2+
1/4*b^2*x^4*arctanh(c*x^(1/2))^2+1/16/c^8*b^2*ln(1+c*x^(1/2))^2+44/105/c^8*b^2*ln(c*x^(1/2)-1)+44/105/c^8*b^2*
ln(1+c*x^(1/2))+1/16/c^8*b^2*ln(c*x^(1/2)-1)^2+3/70*b^2*x^2/c^4+1/4*a^2*x^4+1/2*a*b*x^(1/2)/c^7+1/2*b^2*arctan
h(c*x^(1/2))*x^(1/2)/c^7

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Maxima [A]  time = 0.987937, size = 358, normalized size = 1.7 \begin{align*} \frac{1}{4} \, b^{2} x^{4} \operatorname{artanh}\left (c \sqrt{x}\right )^{2} + \frac{1}{4} \, a^{2} x^{4} + \frac{1}{420} \,{\left (210 \, x^{4} \operatorname{artanh}\left (c \sqrt{x}\right ) + c{\left (\frac{2 \,{\left (15 \, c^{6} x^{\frac{7}{2}} + 21 \, c^{4} x^{\frac{5}{2}} + 35 \, c^{2} x^{\frac{3}{2}} + 105 \, \sqrt{x}\right )}}{c^{8}} - \frac{105 \, \log \left (c \sqrt{x} + 1\right )}{c^{9}} + \frac{105 \, \log \left (c \sqrt{x} - 1\right )}{c^{9}}\right )}\right )} a b + \frac{1}{1680} \,{\left (4 \, c{\left (\frac{2 \,{\left (15 \, c^{6} x^{\frac{7}{2}} + 21 \, c^{4} x^{\frac{5}{2}} + 35 \, c^{2} x^{\frac{3}{2}} + 105 \, \sqrt{x}\right )}}{c^{8}} - \frac{105 \, \log \left (c \sqrt{x} + 1\right )}{c^{9}} + \frac{105 \, \log \left (c \sqrt{x} - 1\right )}{c^{9}}\right )} \operatorname{artanh}\left (c \sqrt{x}\right ) + \frac{20 \, c^{6} x^{3} + 72 \, c^{4} x^{2} + 284 \, c^{2} x - 2 \,{\left (105 \, \log \left (c \sqrt{x} - 1\right ) - 352\right )} \log \left (c \sqrt{x} + 1\right ) + 105 \, \log \left (c \sqrt{x} + 1\right )^{2} + 105 \, \log \left (c \sqrt{x} - 1\right )^{2} + 704 \, \log \left (c \sqrt{x} - 1\right )}{c^{8}}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arctanh(c*x^(1/2)))^2,x, algorithm="maxima")

[Out]

1/4*b^2*x^4*arctanh(c*sqrt(x))^2 + 1/4*a^2*x^4 + 1/420*(210*x^4*arctanh(c*sqrt(x)) + c*(2*(15*c^6*x^(7/2) + 21
*c^4*x^(5/2) + 35*c^2*x^(3/2) + 105*sqrt(x))/c^8 - 105*log(c*sqrt(x) + 1)/c^9 + 105*log(c*sqrt(x) - 1)/c^9))*a
*b + 1/1680*(4*c*(2*(15*c^6*x^(7/2) + 21*c^4*x^(5/2) + 35*c^2*x^(3/2) + 105*sqrt(x))/c^8 - 105*log(c*sqrt(x) +
1)/c^9 + 105*log(c*sqrt(x) - 1)/c^9)*arctanh(c*sqrt(x)) + (20*c^6*x^3 + 72*c^4*x^2 + 284*c^2*x - 2*(105*log(c
*sqrt(x) - 1) - 352)*log(c*sqrt(x) + 1) + 105*log(c*sqrt(x) + 1)^2 + 105*log(c*sqrt(x) - 1)^2 + 704*log(c*sqrt
(x) - 1))/c^8)*b^2

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Fricas [A]  time = 2.12589, size = 662, normalized size = 3.14 \begin{align*} \frac{420 \, a^{2} c^{8} x^{4} + 20 \, b^{2} c^{6} x^{3} + 72 \, b^{2} c^{4} x^{2} + 284 \, b^{2} c^{2} x + 105 \,{\left (b^{2} c^{8} x^{4} - b^{2}\right )} \log \left (-\frac{c^{2} x + 2 \, c \sqrt{x} + 1}{c^{2} x - 1}\right )^{2} + 4 \,{\left (105 \, a b c^{8} - 105 \, a b + 176 \, b^{2}\right )} \log \left (c \sqrt{x} + 1\right ) - 4 \,{\left (105 \, a b c^{8} - 105 \, a b - 176 \, b^{2}\right )} \log \left (c \sqrt{x} - 1\right ) + 4 \,{\left (105 \, a b c^{8} x^{4} - 105 \, a b c^{8} +{\left (15 \, b^{2} c^{7} x^{3} + 21 \, b^{2} c^{5} x^{2} + 35 \, b^{2} c^{3} x + 105 \, 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 (15 \, a b c^{7} x^{3} + 21 \, a b c^{5} x^{2} + 35 \, a b c^{3} x + 105 \, a b c\right )} \sqrt{x}}{1680 \, c^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arctanh(c*x^(1/2)))^2,x, algorithm="fricas")

[Out]

1/1680*(420*a^2*c^8*x^4 + 20*b^2*c^6*x^3 + 72*b^2*c^4*x^2 + 284*b^2*c^2*x + 105*(b^2*c^8*x^4 - b^2)*log(-(c^2*
x + 2*c*sqrt(x) + 1)/(c^2*x - 1))^2 + 4*(105*a*b*c^8 - 105*a*b + 176*b^2)*log(c*sqrt(x) + 1) - 4*(105*a*b*c^8
- 105*a*b - 176*b^2)*log(c*sqrt(x) - 1) + 4*(105*a*b*c^8*x^4 - 105*a*b*c^8 + (15*b^2*c^7*x^3 + 21*b^2*c^5*x^2
+ 35*b^2*c^3*x + 105*b^2*c)*sqrt(x))*log(-(c^2*x + 2*c*sqrt(x) + 1)/(c^2*x - 1)) + 8*(15*a*b*c^7*x^3 + 21*a*b*
c^5*x^2 + 35*a*b*c^3*x + 105*a*b*c)*sqrt(x))/c^8

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \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**3*(a+b*atanh(c*x**(1/2)))**2,x)

[Out]

Integral(x**3*(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^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arctanh(c*x^(1/2)))^2,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*sqrt(x)) + a)^2*x^3, x)