3.2.90 \(\int (a+b \tanh ^{-1}(c \sqrt {x})) \, dx\) [190]

Optimal. Leaf size=39 \[ \frac {b \sqrt {x}}{c}+a x-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2}+b x \tanh ^{-1}\left (c \sqrt {x}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6021, 52, 65, 212} \begin {gather*} a x-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2}+\frac {b \sqrt {x}}{c}+b x \tanh ^{-1}\left (c \sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a + b*ArcTanh[c*Sqrt[x]],x]

[Out]

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

Rule 52

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

Rule 65

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 212

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

Rule 6021

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x^n])^p, x] - Dist[b
*c*n*p, Int[x^n*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p
, 0] && (EqQ[n, 1] || EqQ[p, 1])

Rubi steps

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

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Mathematica [A]
time = 0.11, size = 42, normalized size = 1.08 \begin {gather*} a x+b x \tanh ^{-1}\left (c \sqrt {x}\right )-b c \left (-\frac {\sqrt {x}}{c^2}+\frac {\tanh ^{-1}\left (c \sqrt {x}\right )}{c^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcTanh[c*Sqrt[x]],x]

[Out]

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

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Maple [A]
time = 0.07, size = 50, normalized size = 1.28

method	result	size
default	\(a x +b x \arctanh \left (c \sqrt {x}\right )+\frac {b \sqrt {x}}{c}+\frac {b \ln \left (c \sqrt {x}-1\right )}{2 c^{2}}-\frac {b \ln \left (1+c \sqrt {x}\right )}{2 c^{2}}\)	\(50\)
derivativedivides	\(\frac {a \,c^{2} x +b \,c^{2} x \arctanh \left (c \sqrt {x}\right )+b c \sqrt {x}+\frac {b \ln \left (c \sqrt {x}-1\right )}{2}-\frac {b \ln \left (1+c \sqrt {x}\right )}{2}}{c^{2}}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 53, normalized size = 1.36 \begin {gather*} \frac {1}{2} \, {\left (c {\left (\frac {2 \, \sqrt {x}}{c^{2}} - \frac {\log \left (c \sqrt {x} + 1\right )}{c^{3}} + \frac {\log \left (c \sqrt {x} - 1\right )}{c^{3}}\right )} + 2 \, x \operatorname {artanh}\left (c \sqrt {x}\right )\right )} b + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*(2*sqrt(x)/c^2 - log(c*sqrt(x) + 1)/c^3 + log(c*sqrt(x) - 1)/c^3) + 2*x*arctanh(c*sqrt(x)))*b + a*x

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Fricas [A]
time = 0.47, size = 56, normalized size = 1.44 \begin {gather*} \frac {2 \, a c^{2} x + 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 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(a + b*atanh(c*sqrt(x)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (33) = 66\).
time = 0.48, size = 174, normalized size = 4.46 \begin {gather*} 2 \, b c {\left (\frac {1}{c^{3} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}} + \frac {{\left (c \sqrt {x} + 1\right )} \log \left (-\frac {\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} + 1}{\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} - 1}\right )}{{\left (c \sqrt {x} - 1\right )} c^{3} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}^{2}}\right )} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*c*(1/(c^3*((c*sqrt(x) + 1)/(c*sqrt(x) - 1) - 1)) + (c*sqrt(x) + 1)*log(-(c*((c*sqrt(x) + 1)/(c*sqrt(x) - 1
) + 1)/((c*sqrt(x) + 1)*c/(c*sqrt(x) - 1) - c) + 1)/(c*((c*sqrt(x) + 1)/(c*sqrt(x) - 1) + 1)/((c*sqrt(x) + 1)*
c/(c*sqrt(x) - 1) - c) - 1))/((c*sqrt(x) - 1)*c^3*((c*sqrt(x) + 1)/(c*sqrt(x) - 1) - 1)^2)) + a*x

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Mupad [B]
time = 0.90, size = 32, normalized size = 0.82 \begin {gather*} a\,x+b\,x\,\mathrm {atanh}\left (c\,\sqrt {x}\right )-\frac {b\,\left (\mathrm {atanh}\left (c\,\sqrt {x}\right )-c\,\sqrt {x}\right )}{c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*atanh(c*x^(1/2)),x)

[Out]

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

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