3.3.18 \(\int \frac {a+b \tanh ^{-1}(c x^{3/2})}{x^2} \, dx\) [218]

Optimal. Leaf size=172 \[ -\frac {1}{2} \sqrt {3} b c^{2/3} \text {ArcTan}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )+\frac {1}{2} \sqrt {3} b c^{2/3} \text {ArcTan}\left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )+b c^{2/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )-\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{x}-\frac {1}{4} b c^{2/3} \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )+\frac {1}{4} b c^{2/3} \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right ) \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6037, 335, 216, 648, 632, 210, 642, 212} \begin {gather*} -\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{x}-\frac {1}{2} \sqrt {3} b c^{2/3} \text {ArcTan}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )+\frac {1}{2} \sqrt {3} b c^{2/3} \text {ArcTan}\left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )-\frac {1}{4} b c^{2/3} \log \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )+\frac {1}{4} b c^{2/3} \log \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )+b c^{2/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^(3/2)])/x^2,x]

[Out]

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

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 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 216

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a/b, n]], s = Denominator[Rt[-a/b, n
]], k, u}, Simp[u = Int[(r - s*Cos[(2*k*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*C
os[(2*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n))*Int[1/(r^2 - s^2*x^2), x] + Dis
t[2*(r/(a*n)), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]

Rule 335

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 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 6037

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

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 205, normalized size = 1.19 \begin {gather*} -\frac {a}{x}+\frac {1}{2} \sqrt {3} b c^{2/3} \text {ArcTan}\left (\frac {-1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )+\frac {1}{2} \sqrt {3} b c^{2/3} \text {ArcTan}\left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )-\frac {b \tanh ^{-1}\left (c x^{3/2}\right )}{x}-\frac {1}{2} b c^{2/3} \log \left (1-\sqrt [3]{c} \sqrt {x}\right )+\frac {1}{2} b c^{2/3} \log \left (1+\sqrt [3]{c} \sqrt {x}\right )-\frac {1}{4} b c^{2/3} \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )+\frac {1}{4} b c^{2/3} \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x^(3/2)])/x^2,x]

[Out]

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

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Maple [A]
time = 0.05, size = 167, normalized size = 0.97

method	result	size
derivativedivides	\(-\frac {a}{x}-\frac {b \arctanh \left (c \,x^{\frac {3}{2}}\right )}{x}+\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}\)	\(167\)
default	\(-\frac {a}{x}-\frac {b \arctanh \left (c \,x^{\frac {3}{2}}\right )}{x}+\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}\)	\(167\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^(3/2)))/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.48, size = 163, normalized size = 0.95 \begin {gather*} \frac {1}{4} \, {\left ({\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} + \frac {\log \left (c^{\frac {2}{3}} x + c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {1}{3}}} - \frac {\log \left (c^{\frac {2}{3}} x - c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {1}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} + 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} - 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}}\right )} c - \frac {4 \, \operatorname {artanh}\left (c x^{\frac {3}{2}}\right )}{x}\right )} b - \frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(3/2)))/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 234, normalized size = 1.36 \begin {gather*} -\frac {2 \, \sqrt {3} \left (-c^{2}\right )^{\frac {1}{3}} b x \arctan \left (\frac {2 \, \sqrt {3} \left (-c^{2}\right )^{\frac {2}{3}} \sqrt {x} + \sqrt {3} c}{3 \, c}\right ) - 2 \, \sqrt {3} b {\left (c^{2}\right )}^{\frac {1}{3}} x \arctan \left (\frac {2 \, \sqrt {3} {\left (c^{2}\right )}^{\frac {2}{3}} \sqrt {x} - \sqrt {3} c}{3 \, c}\right ) + \left (-c^{2}\right )^{\frac {1}{3}} b x \log \left (c^{2} x - \left (-c^{2}\right )^{\frac {1}{3}} c \sqrt {x} + \left (-c^{2}\right )^{\frac {2}{3}}\right ) + b {\left (c^{2}\right )}^{\frac {1}{3}} x \log \left (c^{2} x - {\left (c^{2}\right )}^{\frac {1}{3}} c \sqrt {x} + {\left (c^{2}\right )}^{\frac {2}{3}}\right ) - 2 \, \left (-c^{2}\right )^{\frac {1}{3}} b x \log \left (c \sqrt {x} + \left (-c^{2}\right )^{\frac {1}{3}}\right ) - 2 \, b {\left (c^{2}\right )}^{\frac {1}{3}} x \log \left (c \sqrt {x} + {\left (c^{2}\right )}^{\frac {1}{3}}\right ) + 2 \, b \log \left (-\frac {c^{2} x^{3} + 2 \, c x^{\frac {3}{2}} + 1}{c^{2} x^{3} - 1}\right ) + 4 \, a}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(3/2)))/x^2,x, algorithm="fricas")

[Out]

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

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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**(3/2)))/x**2,x)

[Out]

Timed out

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Giac [A]
time = 0.49, size = 172, normalized size = 1.00 \begin {gather*} \frac {1}{4} \, {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} - \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {\log \left (x + \frac {\sqrt {x}}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} - \frac {\log \left (x - \frac {\sqrt {x}}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \log \left (\sqrt {x} + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} - \frac {2 \, \log \left ({\left | \sqrt {x} - \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{{\left | c \right |}^{\frac {1}{3}}}\right )} b c - \frac {b \log \left (-\frac {c x^{\frac {3}{2}} + 1}{c x^{\frac {3}{2}} - 1}\right )}{2 \, x} - \frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(3/2)))/x^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 7.46, size = 220, normalized size = 1.28 \begin {gather*} \frac {b\,c^{2/3}\,\ln \left (\frac {c^{1/3}\,\sqrt {x}+1}{c^{1/3}\,\sqrt {x}-1}\right )}{2}-\frac {a}{x}+\frac {\ln \left (1-c\,x^{3/2}\right )\,\left (b\,x-b\,c^2\,x^4\right )}{2\,x^2-2\,c^2\,x^5}-\frac {b\,\ln \left (c\,x^{3/2}+1\right )}{2\,x}+\frac {b\,c^{2/3}\,\ln \left (\frac {\sqrt {3}+c^{2/3}\,x\,1{}\mathrm {i}-c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}\,c^{2/3}\,x+1{}\mathrm {i}}{2\,c^{2/3}\,x+1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\,1{}\mathrm {i}}{2}+\frac {b\,c^{2/3}\,\ln \left (\frac {\sqrt {3}\,c^{2/3}\,x+c^{2/3}\,x\,1{}\mathrm {i}+c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}+1{}\mathrm {i}}{2\,c^{2/3}\,x+1-\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^(3/2)))/x^2,x)

[Out]

(b*c^(2/3)*log((c^(1/3)*x^(1/2) + 1)/(c^(1/3)*x^(1/2) - 1)))/2 - a/x + (log(1 - c*x^(3/2))*(b*x - b*c^2*x^4))/
(2*x^2 - 2*c^2*x^5) - (b*log(c*x^(3/2) + 1))/(2*x) + (b*c^(2/3)*log((3^(1/2) + c^(2/3)*x*1i - c^(1/3)*x^(1/2)*
4i - 3^(1/2)*c^(2/3)*x + 1i)/(3^(1/2)*1i + 2*c^(2/3)*x + 1))*((3^(1/2)*1i)/2 + 1/2)^(1/2)*1i)/2 + (b*c^(2/3)*l
og((c^(2/3)*x*1i - 3^(1/2) + c^(1/3)*x^(1/2)*4i + 3^(1/2)*c^(2/3)*x + 1i)/(2*c^(2/3)*x - 3^(1/2)*1i + 1))*((3^
(1/2)*1i)/2 - 1/2)^(1/2))/2

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