3.2.8 \(\int (a+b \tanh ^{-1}(c x^3)) \, dx\) [108]

Optimal. Leaf size=101 \[ a x+\frac {\sqrt {3} b \text {ArcTan}\left (\frac {1+2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}+b x \tanh ^{-1}\left (c x^3\right )+\frac {b \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {b \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}} \]

[Out]

a*x+b*x*arctanh(c*x^3)+1/2*b*ln(1-c^(2/3)*x^2)/c^(1/3)-1/4*b*ln(1+c^(2/3)*x^2+c^(4/3)*x^4)/c^(1/3)+1/2*b*arcta
n(1/3*(1+2*c^(2/3)*x^2)*3^(1/2))*3^(1/2)/c^(1/3)

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Rubi [A]
time = 0.07, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6021, 281, 298, 31, 648, 631, 210, 642} \begin {gather*} a x+\frac {\sqrt {3} b \text {ArcTan}\left (\frac {2 c^{2/3} x^2+1}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}+\frac {b \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {b \log \left (c^{4/3} x^4+c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}}+b x \tanh ^{-1}\left (c x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a + b*ArcTanh[c*x^3],x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

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

Rule 281

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

Rule 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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 x^3\right )\right ) \, dx &=a x+b \int \tanh ^{-1}\left (c x^3\right ) \, dx\\ &=a x+b x \tanh ^{-1}\left (c x^3\right )-(3 b c) \int \frac {x^3}{1-c^2 x^6} \, dx\\ &=a x+b x \tanh ^{-1}\left (c x^3\right )-\frac {1}{2} (3 b c) \text {Subst}\left (\int \frac {x}{1-c^2 x^3} \, dx,x,x^2\right )\\ &=a x+b x \tanh ^{-1}\left (c x^3\right )-\frac {1}{2} \left (b \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {1}{1-c^{2/3} x} \, dx,x,x^2\right )+\frac {1}{2} \left (b \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {1-c^{2/3} x}{1+c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )\\ &=a x+b x \tanh ^{-1}\left (c x^3\right )+\frac {b \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {b \text {Subst}\left (\int \frac {c^{2/3}+2 c^{4/3} x}{1+c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )}{4 \sqrt [3]{c}}+\frac {1}{4} \left (3 b \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {1}{1+c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )\\ &=a x+b x \tanh ^{-1}\left (c x^3\right )+\frac {b \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {b \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 c^{2/3} x^2\right )}{2 \sqrt [3]{c}}\\ &=a x+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1+2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}+b x \tanh ^{-1}\left (c x^3\right )+\frac {b \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {b \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 136, normalized size = 1.35 \begin {gather*} a x+b x \tanh ^{-1}\left (c x^3\right )-\frac {b \left (-2 \sqrt {3} \text {ArcTan}\left (\frac {-1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )+2 \sqrt {3} \text {ArcTan}\left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )-2 \log \left (1-\sqrt [3]{c} x\right )-2 \log \left (1+\sqrt [3]{c} x\right )+\log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )+\log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )\right )}{4 \sqrt [3]{c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcTanh[c*x^3],x]

[Out]

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

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Maple [A]
time = 0.03, size = 99, normalized size = 0.98

method	result	size
default	\(a x +b x \arctanh \left (c \,x^{3}\right )+\frac {b \ln \left (x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x^{4}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}} x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}\)	\(99\)
risch	\(a x +\frac {b x \ln \left (c \,x^{3}+1\right )}{2}+\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {1}{c}\right )^{\frac {1}{3}}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b x \ln \left (-c \,x^{3}+1\right )}{2}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {1}{c}\right )^{\frac {1}{3}}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}\)	\(195\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*arctanh(c*x^3),x,method=_RETURNVERBOSE)

[Out]

a*x+b*x*arctanh(c*x^3)+1/2*b/c/(1/c^2)^(1/3)*ln(x^2-(1/c^2)^(1/3))-1/4*b/c/(1/c^2)^(1/3)*ln(x^4+(1/c^2)^(1/3)*
x^2+(1/c^2)^(2/3))+1/2*b*3^(1/2)/c/(1/c^2)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c^2)^(1/3)*x^2+1))

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Maxima [A]
time = 0.46, size = 90, normalized size = 0.89 \begin {gather*} \frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} + c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}} - \frac {\log \left (c^{\frac {4}{3}} x^{4} + c^{\frac {2}{3}} x^{2} + 1\right )}{c^{\frac {4}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {2}{3}} x^{2} - 1}{c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}}\right )} + 4 \, x \operatorname {artanh}\left (c x^{3}\right )\right )} b + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(4/3)*x^2 + c^(2/3))/c^(2/3))/c^(4/3) - log(c^(4/3)*x^4 + c^(2/3)*x^
2 + 1)/c^(4/3) + 2*log((c^(2/3)*x^2 - 1)/c^(2/3))/c^(4/3)) + 4*x*arctanh(c*x^3))*b + a*x

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Fricas [A]
time = 0.39, size = 260, normalized size = 2.57 \begin {gather*} \left [\frac {\sqrt {3} b c \sqrt {-\frac {1}{c^{\frac {2}{3}}}} \log \left (\frac {2 \, c^{2} x^{6} - 3 \, c^{\frac {2}{3}} x^{2} + \sqrt {3} {\left (2 \, c^{\frac {5}{3}} x^{4} - c x^{2} - c^{\frac {1}{3}}\right )} \sqrt {-\frac {1}{c^{\frac {2}{3}}}} + 1}{c^{2} x^{6} - 1}\right ) + 2 \, b c x \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a c x - b c^{\frac {2}{3}} \log \left (c^{2} x^{4} + c^{\frac {4}{3}} x^{2} + c^{\frac {2}{3}}\right ) + 2 \, b c^{\frac {2}{3}} \log \left (c x^{2} - c^{\frac {1}{3}}\right )}{4 \, c}, \frac {2 \, b c x \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 2 \, \sqrt {3} b c^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, c x^{2} + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right ) + 4 \, a c x - b c^{\frac {2}{3}} \log \left (c^{2} x^{4} + c^{\frac {4}{3}} x^{2} + c^{\frac {2}{3}}\right ) + 2 \, b c^{\frac {2}{3}} \log \left (c x^{2} - c^{\frac {1}{3}}\right )}{4 \, c}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(sqrt(3)*b*c*sqrt(-1/c^(2/3))*log((2*c^2*x^6 - 3*c^(2/3)*x^2 + sqrt(3)*(2*c^(5/3)*x^4 - c*x^2 - c^(1/3))*
sqrt(-1/c^(2/3)) + 1)/(c^2*x^6 - 1)) + 2*b*c*x*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 4*a*c*x - b*c^(2/3)*log(c^2*x^4
+ c^(4/3)*x^2 + c^(2/3)) + 2*b*c^(2/3)*log(c*x^2 - c^(1/3)))/c, 1/4*(2*b*c*x*log(-(c*x^3 + 1)/(c*x^3 - 1)) +
2*sqrt(3)*b*c^(2/3)*arctan(1/3*sqrt(3)*(2*c*x^2 + c^(1/3))/c^(1/3)) + 4*a*c*x - b*c^(2/3)*log(c^2*x^4 + c^(4/3
)*x^2 + c^(2/3)) + 2*b*c^(2/3)*log(c*x^2 - c^(1/3)))/c]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*atanh(c*x**3),x)

[Out]

Timed out

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Giac [A]
time = 0.41, size = 109, normalized size = 1.08 \begin {gather*} \frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} {\left | c \right |}^{\frac {2}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )} {\left | c \right |}^{\frac {2}{3}}\right )}{c^{2}} - \frac {{\left | c \right |}^{\frac {2}{3}} \log \left (x^{4} + \frac {x^{2}}{{\left | c \right |}^{\frac {2}{3}}} + \frac {1}{{\left | c \right |}^{\frac {4}{3}}}\right )}{c^{2}} + \frac {2 \, \log \left ({\left | x^{2} - \frac {1}{{\left | c \right |}^{\frac {2}{3}}} \right |}\right )}{{\left | c \right |}^{\frac {4}{3}}}\right )} + 2 \, x \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )\right )} b + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c*(2*sqrt(3)*abs(c)^(2/3)*arctan(1/3*sqrt(3)*(2*x^2 + 1/abs(c)^(2/3))*abs(c)^(2/3))/c^2 - abs(c)^(2/3)*lo
g(x^4 + x^2/abs(c)^(2/3) + 1/abs(c)^(4/3))/c^2 + 2*log(abs(x^2 - 1/abs(c)^(2/3)))/abs(c)^(4/3)) + 2*x*log(-(c*
x^3 + 1)/(c*x^3 - 1)))*b + a*x

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Mupad [B]
time = 2.76, size = 107, normalized size = 1.06 \begin {gather*} a\,x+\frac {b\,\ln \left (c^{2/3}\,x^2-1\right )}{2\,c^{1/3}}-\frac {\ln \left (4\,c^{2/3}\,x^2+2-\sqrt {3}\,2{}\mathrm {i}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{4\,c^{1/3}}-\frac {\ln \left (4\,c^{2/3}\,x^2+2+\sqrt {3}\,2{}\mathrm {i}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{4\,c^{1/3}}+\frac {b\,x\,\ln \left (c\,x^3+1\right )}{2}-\frac {b\,x\,\ln \left (1-c\,x^3\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*atanh(c*x^3),x)

[Out]

a*x + (b*log(c^(2/3)*x^2 - 1))/(2*c^(1/3)) - (log(4*c^(2/3)*x^2 - 3^(1/2)*2i + 2)*(b + 3^(1/2)*b*1i))/(4*c^(1/
3)) - (log(3^(1/2)*2i + 4*c^(2/3)*x^2 + 2)*(b - 3^(1/2)*b*1i))/(4*c^(1/3)) + (b*x*log(c*x^3 + 1))/2 - (b*x*log
(1 - c*x^3))/2

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