3.104 \(\int \frac{a+b \tanh ^{-1}(c x^3)}{x^4} \, dx\)

Optimal. Leaf size=40 \[ -\frac{a+b \tanh ^{-1}\left (c x^3\right )}{3 x^3}-\frac{1}{6} b c \log \left (1-c^2 x^6\right )+b c \log (x) \]

[Out]

-(a + b*ArcTanh[c*x^3])/(3*x^3) + b*c*Log[x] - (b*c*Log[1 - c^2*x^6])/6

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Rubi [A]  time = 0.0270264, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {6097, 266, 36, 29, 31} \[ -\frac{a+b \tanh ^{-1}\left (c x^3\right )}{3 x^3}-\frac{1}{6} b c \log \left (1-c^2 x^6\right )+b c \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^3])/x^4,x]

[Out]

-(a + b*ArcTanh[c*x^3])/(3*x^3) + b*c*Log[x] - (b*c*Log[1 - c^2*x^6])/6

Rule 6097

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
nh[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 - c^2*x^(2*n)), x], x
] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

Rule 266

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

Rule 36

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rubi steps

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

Mathematica [A]  time = 0.0122482, size = 45, normalized size = 1.12 \[ -\frac{a}{3 x^3}-\frac{1}{6} b c \log \left (1-c^2 x^6\right )-\frac{b \tanh ^{-1}\left (c x^3\right )}{3 x^3}+b c \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x^3])/x^4,x]

[Out]

-a/(3*x^3) - (b*ArcTanh[c*x^3])/(3*x^3) + b*c*Log[x] - (b*c*Log[1 - c^2*x^6])/6

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Maple [A]  time = 0.013, size = 49, normalized size = 1.2 \begin{align*} -{\frac{a}{3\,{x}^{3}}}-{\frac{b{\it Artanh} \left ( c{x}^{3} \right ) }{3\,{x}^{3}}}+bc\ln \left ( x \right ) -{\frac{bc\ln \left ( c{x}^{3}-1 \right ) }{6}}-{\frac{bc\ln \left ( c{x}^{3}+1 \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^3))/x^4,x)

[Out]

-1/3*a/x^3-1/3*b/x^3*arctanh(c*x^3)+b*c*ln(x)-1/6*b*c*ln(c*x^3-1)-1/6*b*c*ln(c*x^3+1)

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Maxima [A]  time = 1.02135, size = 55, normalized size = 1.38 \begin{align*} -\frac{1}{6} \,{\left (c{\left (\log \left (c^{2} x^{6} - 1\right ) - \log \left (x^{6}\right )\right )} + \frac{2 \, \operatorname{artanh}\left (c x^{3}\right )}{x^{3}}\right )} b - \frac{a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))/x^4,x, algorithm="maxima")

[Out]

-1/6*(c*(log(c^2*x^6 - 1) - log(x^6)) + 2*arctanh(c*x^3)/x^3)*b - 1/3*a/x^3

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Fricas [A]  time = 1.98002, size = 130, normalized size = 3.25 \begin{align*} -\frac{b c x^{3} \log \left (c^{2} x^{6} - 1\right ) - 6 \, b c x^{3} \log \left (x\right ) + b \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + 2 \, a}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))/x^4,x, algorithm="fricas")

[Out]

-1/6*(b*c*x^3*log(c^2*x^6 - 1) - 6*b*c*x^3*log(x) + b*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 2*a)/x^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x**3))/x**4,x)

[Out]

Exception raised: KeyError

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Giac [A]  time = 1.13407, size = 69, normalized size = 1.72 \begin{align*} -\frac{1}{6} \, b c \log \left (c^{2} x^{6} - 1\right ) + b c \log \left (x\right ) - \frac{b \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right )}{6 \, x^{3}} - \frac{a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))/x^4,x, algorithm="giac")

[Out]

-1/6*b*c*log(c^2*x^6 - 1) + b*c*log(x) - 1/6*b*log(-(c*x^3 + 1)/(c*x^3 - 1))/x^3 - 1/3*a/x^3