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

Optimal. Leaf size=40 \[ -\frac{a+b \tanh ^{-1}\left (c \sqrt{x}\right )}{x}+b c^2 \tanh ^{-1}\left (c \sqrt{x}\right )-\frac{b c}{\sqrt{x}} \]

[Out]

-((b*c)/Sqrt[x]) + b*c^2*ArcTanh[c*Sqrt[x]] - (a + b*ArcTanh[c*Sqrt[x]])/x

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Rubi [A]  time = 0.0229893, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6097, 51, 63, 206} \[ -\frac{a+b \tanh ^{-1}\left (c \sqrt{x}\right )}{x}+b c^2 \tanh ^{-1}\left (c \sqrt{x}\right )-\frac{b c}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*Sqrt[x]])/x^2,x]

[Out]

-((b*c)/Sqrt[x]) + b*c^2*ArcTanh[c*Sqrt[x]] - (a + b*ArcTanh[c*Sqrt[x]])/x

Rule 6097

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
nh[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 - c^2*x^(2*n)), x], x
] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0241474, size = 67, normalized size = 1.68 \[ -\frac{a}{x}-\frac{1}{2} b c^2 \log \left (1-c \sqrt{x}\right )+\frac{1}{2} b c^2 \log \left (c \sqrt{x}+1\right )-\frac{b c}{\sqrt{x}}-\frac{b \tanh ^{-1}\left (c \sqrt{x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.033, size = 55, normalized size = 1.4 \begin{align*} -{\frac{a}{x}}-{\frac{b}{x}{\it Artanh} \left ( c\sqrt{x} \right ) }-{bc{\frac{1}{\sqrt{x}}}}-{\frac{{c}^{2}b}{2}\ln \left ( c\sqrt{x}-1 \right ) }+{\frac{{c}^{2}b}{2}\ln \left ( 1+c\sqrt{x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.961342, size = 69, normalized size = 1.72 \begin{align*} \frac{1}{2} \,{\left ({\left (c \log \left (c \sqrt{x} + 1\right ) - c \log \left (c \sqrt{x} - 1\right ) - \frac{2}{\sqrt{x}}\right )} c - \frac{2 \, \operatorname{artanh}\left (c \sqrt{x}\right )}{x}\right )} b - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((c*log(c*sqrt(x) + 1) - c*log(c*sqrt(x) - 1) - 2/sqrt(x))*c - 2*arctanh(c*sqrt(x))/x)*b - a/x

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Fricas [A]  time = 1.74626, size = 122, normalized size = 3.05 \begin{align*} -\frac{2 \, b c \sqrt{x} -{\left (b c^{2} x - b\right )} \log \left (-\frac{c^{2} x + 2 \, c \sqrt{x} + 1}{c^{2} x - 1}\right ) + 2 \, a}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 40.8538, size = 231, normalized size = 5.78 \begin{align*} \begin{cases} - \frac{a}{x} + \frac{b \operatorname{atanh}{\left (\sqrt{x} \sqrt{\frac{1}{x}} \right )}}{x} & \text{for}\: c = - \sqrt{\frac{1}{x}} \\- \frac{a}{x} - \frac{b \operatorname{atanh}{\left (\sqrt{x} \sqrt{\frac{1}{x}} \right )}}{x} & \text{for}\: c = \sqrt{\frac{1}{x}} \\- \frac{a c^{4} x^{\frac{5}{2}}}{c^{2} x^{\frac{5}{2}} - x^{\frac{3}{2}}} + \frac{a \sqrt{x}}{c^{2} x^{\frac{5}{2}} - x^{\frac{3}{2}}} + \frac{b c^{4} x^{\frac{5}{2}} \operatorname{atanh}{\left (c \sqrt{x} \right )}}{c^{2} x^{\frac{5}{2}} - x^{\frac{3}{2}}} - \frac{b c^{3} x^{2}}{c^{2} x^{\frac{5}{2}} - x^{\frac{3}{2}}} - \frac{2 b c^{2} x^{\frac{3}{2}} \operatorname{atanh}{\left (c \sqrt{x} \right )}}{c^{2} x^{\frac{5}{2}} - x^{\frac{3}{2}}} + \frac{b c x}{c^{2} x^{\frac{5}{2}} - x^{\frac{3}{2}}} + \frac{b \sqrt{x} \operatorname{atanh}{\left (c \sqrt{x} \right )}}{c^{2} x^{\frac{5}{2}} - x^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a/x + b*atanh(sqrt(x)*sqrt(1/x))/x, Eq(c, -sqrt(1/x))), (-a/x - b*atanh(sqrt(x)*sqrt(1/x))/x, Eq(c
, sqrt(1/x))), (-a*c**4*x**(5/2)/(c**2*x**(5/2) - x**(3/2)) + a*sqrt(x)/(c**2*x**(5/2) - x**(3/2)) + b*c**4*x*
*(5/2)*atanh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) - b*c**3*x**2/(c**2*x**(5/2) - x**(3/2)) - 2*b*c**2*x**(3/2
)*atanh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) + b*c*x/(c**2*x**(5/2) - x**(3/2)) + b*sqrt(x)*atanh(c*sqrt(x))/
(c**2*x**(5/2) - x**(3/2)), True))

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Giac [A]  time = 1.33223, size = 90, normalized size = 2.25 \begin{align*} \frac{1}{2} \, b c^{2} \log \left (c \sqrt{x} + 1\right ) - \frac{1}{2} \, b c^{2} \log \left (c \sqrt{x} - 1\right ) - \frac{b \log \left (-\frac{c \sqrt{x} + 1}{c \sqrt{x} - 1}\right )}{2 \, x} - \frac{b c \sqrt{x} + a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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