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

Optimal. Leaf size=44 \[ a x+b x \tanh ^{-1}\left (c x^2\right )+\frac{b \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-\frac{b \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.0244469, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6091, 298, 203, 206} \[ a x+b x \tanh ^{-1}\left (c x^2\right )+\frac{b \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-\frac{b \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcTanh[c*x^2],x]

[Out]

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

Rule 6091

Int[ArcTanh[(c_.)*(x_)^(n_)], x_Symbol] :> Simp[x*ArcTanh[c*x^n], x] - Dist[c*n, Int[x^n/(1 - c^2*x^(2*n)), x]
, x] /; FreeQ[{c, n}, x]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 203

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

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 \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \, dx &=a x+b \int \tanh ^{-1}\left (c x^2\right ) \, dx\\ &=a x+b x \tanh ^{-1}\left (c x^2\right )-(2 b c) \int \frac{x^2}{1-c^2 x^4} \, dx\\ &=a x+b x \tanh ^{-1}\left (c x^2\right )-b \int \frac{1}{1-c x^2} \, dx+b \int \frac{1}{1+c x^2} \, dx\\ &=a x+\frac{b \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-\frac{b \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}+b x \tanh ^{-1}\left (c x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0196384, size = 57, normalized size = 1.3 \[ a x+b x \tanh ^{-1}\left (c x^2\right )+\frac{b \left (\log \left (1-\sqrt{c} x\right )-\log \left (\sqrt{c} x+1\right )+2 \tan ^{-1}\left (\sqrt{c} x\right )\right )}{2 \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcTanh[c*x^2],x]

[Out]

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

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Maple [A]  time = 0.007, size = 37, normalized size = 0.8 \begin{align*} ax+bx{\it Artanh} \left ( c{x}^{2} \right ) +{b\arctan \left ( x\sqrt{c} \right ){\frac{1}{\sqrt{c}}}}-{b{\it Artanh} \left ( x\sqrt{c} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.07371, size = 394, normalized size = 8.95 \begin{align*} \left [\frac{b c x \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a c x + 2 \, b \sqrt{c} \arctan \left (\sqrt{c} x\right ) + b \sqrt{c} \log \left (\frac{c x^{2} - 2 \, \sqrt{c} x + 1}{c x^{2} - 1}\right )}{2 \, c}, \frac{b c x \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a c x + 2 \, b \sqrt{-c} \arctan \left (\sqrt{-c} x\right ) - b \sqrt{-c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-c} x - 1}{c x^{2} + 1}\right )}{2 \, c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(b*c*x*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 2*a*c*x + 2*b*sqrt(c)*arctan(sqrt(c)*x) + b*sqrt(c)*log((c*x^2 - 2
*sqrt(c)*x + 1)/(c*x^2 - 1)))/c, 1/2*(b*c*x*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 2*a*c*x + 2*b*sqrt(-c)*arctan(sqrt
(-c)*x) - b*sqrt(-c)*log((c*x^2 - 2*sqrt(-c)*x - 1)/(c*x^2 + 1)))/c]

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Sympy [A]  time = 8.39401, size = 139, normalized size = 3.16 \begin{align*} a x + b \left (\begin{cases} x \operatorname{atanh}{\left (c x^{2} \right )} - \frac{\log{\left (x - i \sqrt{\frac{1}{c}} \right )}}{2 c \sqrt{\frac{1}{c}}} - \frac{i \log{\left (x - i \sqrt{\frac{1}{c}} \right )}}{2 c \sqrt{\frac{1}{c}}} - \frac{\log{\left (x + i \sqrt{\frac{1}{c}} \right )}}{2 c \sqrt{\frac{1}{c}}} + \frac{i \log{\left (x + i \sqrt{\frac{1}{c}} \right )}}{2 c \sqrt{\frac{1}{c}}} + \frac{\log{\left (x - \sqrt{\frac{1}{c}} \right )}}{c \sqrt{\frac{1}{c}}} + \frac{\operatorname{atanh}{\left (c x^{2} \right )}}{c \sqrt{\frac{1}{c}}} & \text{for}\: c \neq 0 \\0 & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*Piecewise((x*atanh(c*x**2) - log(x - I*sqrt(1/c))/(2*c*sqrt(1/c)) - I*log(x - I*sqrt(1/c))/(2*c*sqrt(1
/c)) - log(x + I*sqrt(1/c))/(2*c*sqrt(1/c)) + I*log(x + I*sqrt(1/c))/(2*c*sqrt(1/c)) + log(x - sqrt(1/c))/(c*s
qrt(1/c)) + atanh(c*x**2)/(c*sqrt(1/c)), Ne(c, 0)), (0, True))

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Giac [B]  time = 1.19516, size = 112, normalized size = 2.55 \begin{align*} \frac{1}{2} \,{\left (c{\left (\frac{2 \, \sqrt{{\left | c \right |}} \arctan \left (x \sqrt{{\left | c \right |}}\right )}{c^{2}} - \frac{\sqrt{{\left | c \right |}} \log \left ({\left | x + \frac{1}{\sqrt{{\left | c \right |}}} \right |}\right )}{c^{2}} + \frac{\sqrt{{\left | c \right |}} \log \left ({\left | x - \frac{1}{\sqrt{{\left | c \right |}}} \right |}\right )}{c^{2}}\right )} + x \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right )\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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