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

Optimal. Leaf size=106 \[ \frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac{2 b \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/2}}-\frac{2 b \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/2}}+\frac{4 b \sqrt{d x}}{3 c} \]

[Out]

(4*b*Sqrt[d*x])/(3*c) - (2*b*Sqrt[d]*ArcTan[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(3*c^(3/2)) + (2*(d*x)^(3/2)*(a + b*
ArcTanh[c*x]))/(3*d) - (2*b*Sqrt[d]*ArcTanh[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(3*c^(3/2))

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Rubi [A]  time = 0.0612043, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5916, 321, 329, 212, 208, 205} \[ \frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac{2 b \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/2}}-\frac{2 b \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/2}}+\frac{4 b \sqrt{d x}}{3 c} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*x]*(a + b*ArcTanh[c*x]),x]

[Out]

(4*b*Sqrt[d*x])/(3*c) - (2*b*Sqrt[d]*ArcTan[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(3*c^(3/2)) + (2*(d*x)^(3/2)*(a + b*
ArcTanh[c*x]))/(3*d) - (2*b*Sqrt[d]*ArcTanh[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(3*c^(3/2))

Rule 5916

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcT
anh[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(1 -
c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 212

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

Rule 208

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

Rule 205

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

Rubi steps

\begin{align*} \int \sqrt{d x} \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac{(2 b c) \int \frac{(d x)^{3/2}}{1-c^2 x^2} \, dx}{3 d}\\ &=\frac{4 b \sqrt{d x}}{3 c}+\frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac{(2 b d) \int \frac{1}{\sqrt{d x} \left (1-c^2 x^2\right )} \, dx}{3 c}\\ &=\frac{4 b \sqrt{d x}}{3 c}+\frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{1}{1-\frac{c^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{3 c}\\ &=\frac{4 b \sqrt{d x}}{3 c}+\frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac{(2 b d) \operatorname{Subst}\left (\int \frac{1}{d-c x^2} \, dx,x,\sqrt{d x}\right )}{3 c}-\frac{(2 b d) \operatorname{Subst}\left (\int \frac{1}{d+c x^2} \, dx,x,\sqrt{d x}\right )}{3 c}\\ &=\frac{4 b \sqrt{d x}}{3 c}-\frac{2 b \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/2}}+\frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac{2 b \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0502362, size = 114, normalized size = 1.08 \[ \frac{\sqrt{d x} \left (2 a c^{3/2} x^{3/2}+2 b c^{3/2} x^{3/2} \tanh ^{-1}(c x)+4 b \sqrt{c} \sqrt{x}+b \log \left (1-\sqrt{c} \sqrt{x}\right )-b \log \left (\sqrt{c} \sqrt{x}+1\right )-2 b \tan ^{-1}\left (\sqrt{c} \sqrt{x}\right )\right )}{3 c^{3/2} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d*x]*(4*b*Sqrt[c]*Sqrt[x] + 2*a*c^(3/2)*x^(3/2) - 2*b*ArcTan[Sqrt[c]*Sqrt[x]] + 2*b*c^(3/2)*x^(3/2)*ArcT
anh[c*x] + b*Log[1 - Sqrt[c]*Sqrt[x]] - b*Log[1 + Sqrt[c]*Sqrt[x]]))/(3*c^(3/2)*Sqrt[x])

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Maple [A]  time = 0.012, size = 89, normalized size = 0.8 \begin{align*}{\frac{2\,a}{3\,d} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{2\,b{\it Artanh} \left ( cx \right ) }{3\,d} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{4\,b}{3\,c}\sqrt{dx}}-{\frac{2\,db}{3\,c}\arctan \left ({c\sqrt{dx}{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{2\,db}{3\,c}{\it Artanh} \left ({c\sqrt{dx}{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3/d*(d*x)^(3/2)*a+2/3/d*b*(d*x)^(3/2)*arctanh(c*x)+4/3*b*(d*x)^(1/2)/c-2/3*d*b/c/(c*d)^(1/2)*arctan(c*(d*x)^
(1/2)/(c*d)^(1/2))-2/3*d*b/c/(c*d)^(1/2)*arctanh(c*(d*x)^(1/2)/(c*d)^(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((d*x)^(1/2)*(a+b*arctanh(c*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.22862, size = 509, normalized size = 4.8 \begin{align*} \left [-\frac{2 \, b \sqrt{\frac{d}{c}} \arctan \left (\frac{\sqrt{d x} c \sqrt{\frac{d}{c}}}{d}\right ) - b \sqrt{\frac{d}{c}} \log \left (\frac{c d x - 2 \, \sqrt{d x} c \sqrt{\frac{d}{c}} + d}{c x - 1}\right ) -{\left (b c x \log \left (-\frac{c x + 1}{c x - 1}\right ) + 2 \, a c x + 4 \, b\right )} \sqrt{d x}}{3 \, c}, \frac{2 \, b \sqrt{-\frac{d}{c}} \arctan \left (\frac{\sqrt{d x} c \sqrt{-\frac{d}{c}}}{d}\right ) + b \sqrt{-\frac{d}{c}} \log \left (\frac{c d x - 2 \, \sqrt{d x} c \sqrt{-\frac{d}{c}} - d}{c x + 1}\right ) +{\left (b c x \log \left (-\frac{c x + 1}{c x - 1}\right ) + 2 \, a c x + 4 \, b\right )} \sqrt{d x}}{3 \, c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/3*(2*b*sqrt(d/c)*arctan(sqrt(d*x)*c*sqrt(d/c)/d) - b*sqrt(d/c)*log((c*d*x - 2*sqrt(d*x)*c*sqrt(d/c) + d)/(
c*x - 1)) - (b*c*x*log(-(c*x + 1)/(c*x - 1)) + 2*a*c*x + 4*b)*sqrt(d*x))/c, 1/3*(2*b*sqrt(-d/c)*arctan(sqrt(d*
x)*c*sqrt(-d/c)/d) + b*sqrt(-d/c)*log((c*d*x - 2*sqrt(d*x)*c*sqrt(-d/c) - d)/(c*x + 1)) + (b*c*x*log(-(c*x + 1
)/(c*x - 1)) + 2*a*c*x + 4*b)*sqrt(d*x))/c]

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Sympy [C]  time = 20.0057, size = 586, normalized size = 5.53 \begin{align*} \frac{2 a \left (d x\right )^{\frac{3}{2}}}{3 d} + \frac{2 b \left (\begin{cases} - \frac{i c d^{\frac{7}{2}} \left (\frac{1}{c}\right )^{\frac{3}{2}} \log{\left (- \sqrt{d} \sqrt{\frac{1}{c}} + \sqrt{d x} \right )}}{- \frac{3 i c^{3} d^{2}}{c^{2}} + \frac{15 i c^{2} d^{2}}{c}} - \frac{c d^{\frac{7}{2}} \left (\frac{1}{c}\right )^{\frac{3}{2}} \log{\left (i \sqrt{d} \sqrt{\frac{1}{c}} + \sqrt{d x} \right )}}{- \frac{3 i c^{3} d^{2}}{c^{2}} + \frac{15 i c^{2} d^{2}}{c}} + \frac{i c d^{\frac{7}{2}} \left (\frac{1}{c}\right )^{\frac{3}{2}} \log{\left (i \sqrt{d} \sqrt{\frac{1}{c}} + \sqrt{d x} \right )}}{- \frac{3 i c^{3} d^{2}}{c^{2}} + \frac{15 i c^{2} d^{2}}{c}} + \frac{5 i d^{\frac{7}{2}} \sqrt{\frac{1}{c}} \log{\left (- \sqrt{d} \sqrt{\frac{1}{c}} + \sqrt{d x} \right )}}{- \frac{3 i c^{3} d^{2}}{c^{2}} + \frac{15 i c^{2} d^{2}}{c}} - \frac{2 d^{\frac{7}{2}} \sqrt{\frac{1}{c}} \log{\left (- i \sqrt{d} \sqrt{\frac{1}{c}} + \sqrt{d x} \right )}}{- \frac{3 i c^{3} d^{2}}{c^{2}} + \frac{15 i c^{2} d^{2}}{c}} - \frac{2 i d^{\frac{7}{2}} \sqrt{\frac{1}{c}} \log{\left (- i \sqrt{d} \sqrt{\frac{1}{c}} + \sqrt{d x} \right )}}{- \frac{3 i c^{3} d^{2}}{c^{2}} + \frac{15 i c^{2} d^{2}}{c}} + \frac{3 d^{\frac{7}{2}} \sqrt{\frac{1}{c}} \log{\left (i \sqrt{d} \sqrt{\frac{1}{c}} + \sqrt{d x} \right )}}{- \frac{3 i c^{3} d^{2}}{c^{2}} + \frac{15 i c^{2} d^{2}}{c}} - \frac{3 i d^{\frac{7}{2}} \sqrt{\frac{1}{c}} \log{\left (i \sqrt{d} \sqrt{\frac{1}{c}} + \sqrt{d x} \right )}}{- \frac{3 i c^{3} d^{2}}{c^{2}} + \frac{15 i c^{2} d^{2}}{c}} + \frac{4 i d^{\frac{7}{2}} \sqrt{\frac{1}{c}} \operatorname{atanh}{\left (c x \right )}}{- \frac{3 i c^{3} d^{2}}{c^{2}} + \frac{15 i c^{2} d^{2}}{c}} + \frac{\left (d x\right )^{\frac{3}{2}} \operatorname{atanh}{\left (c x \right )}}{3} + \frac{2 d \sqrt{d x}}{3 c} & \text{for}\: c \neq 0 \\0 & \text{otherwise} \end{cases}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*(d*x)**(3/2)/(3*d) + 2*b*Piecewise((-I*c*d**(7/2)*(1/c)**(3/2)*log(-sqrt(d)*sqrt(1/c) + sqrt(d*x))/(-3*I*c
**3*d**2/c**2 + 15*I*c**2*d**2/c) - c*d**(7/2)*(1/c)**(3/2)*log(I*sqrt(d)*sqrt(1/c) + sqrt(d*x))/(-3*I*c**3*d*
*2/c**2 + 15*I*c**2*d**2/c) + I*c*d**(7/2)*(1/c)**(3/2)*log(I*sqrt(d)*sqrt(1/c) + sqrt(d*x))/(-3*I*c**3*d**2/c
**2 + 15*I*c**2*d**2/c) + 5*I*d**(7/2)*sqrt(1/c)*log(-sqrt(d)*sqrt(1/c) + sqrt(d*x))/(-3*I*c**3*d**2/c**2 + 15
*I*c**2*d**2/c) - 2*d**(7/2)*sqrt(1/c)*log(-I*sqrt(d)*sqrt(1/c) + sqrt(d*x))/(-3*I*c**3*d**2/c**2 + 15*I*c**2*
d**2/c) - 2*I*d**(7/2)*sqrt(1/c)*log(-I*sqrt(d)*sqrt(1/c) + sqrt(d*x))/(-3*I*c**3*d**2/c**2 + 15*I*c**2*d**2/c
) + 3*d**(7/2)*sqrt(1/c)*log(I*sqrt(d)*sqrt(1/c) + sqrt(d*x))/(-3*I*c**3*d**2/c**2 + 15*I*c**2*d**2/c) - 3*I*d
**(7/2)*sqrt(1/c)*log(I*sqrt(d)*sqrt(1/c) + sqrt(d*x))/(-3*I*c**3*d**2/c**2 + 15*I*c**2*d**2/c) + 4*I*d**(7/2)
*sqrt(1/c)*atanh(c*x)/(-3*I*c**3*d**2/c**2 + 15*I*c**2*d**2/c) + (d*x)**(3/2)*atanh(c*x)/3 + 2*d*sqrt(d*x)/(3*
c), Ne(c, 0)), (0, True))/d

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d x}{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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