3.105 \(\int \frac{a+b \tanh ^{-1}(c x^3)}{x^7} \, dx\)

Optimal. Leaf size=41 \[ -\frac{a+b \tanh ^{-1}\left (c x^3\right )}{6 x^6}+\frac{1}{6} b c^2 \tanh ^{-1}\left (c x^3\right )-\frac{b c}{6 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0270646, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6097, 275, 325, 206} \[ -\frac{a+b \tanh ^{-1}\left (c x^3\right )}{6 x^6}+\frac{1}{6} b c^2 \tanh ^{-1}\left (c x^3\right )-\frac{b c}{6 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^3])/x^7,x]

[Out]

-(b*c)/(6*x^3) + (b*c^2*ArcTanh[c*x^3])/6 - (a + b*ArcTanh[c*x^3])/(6*x^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 275

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 325

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

Rule 206

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

Rubi steps

\begin{align*} \int \frac{a+b \tanh ^{-1}\left (c x^3\right )}{x^7} \, dx &=-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{6 x^6}+\frac{1}{2} (b c) \int \frac{1}{x^4 \left (1-c^2 x^6\right )} \, dx\\ &=-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{6 x^6}+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx,x,x^3\right )\\ &=-\frac{b c}{6 x^3}-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{6 x^6}+\frac{1}{6} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,x^3\right )\\ &=-\frac{b c}{6 x^3}+\frac{1}{6} b c^2 \tanh ^{-1}\left (c x^3\right )-\frac{a+b \tanh ^{-1}\left (c x^3\right )}{6 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0182664, size = 65, normalized size = 1.59 \[ -\frac{a}{6 x^6}-\frac{1}{12} b c^2 \log \left (1-c x^3\right )+\frac{1}{12} b c^2 \log \left (c x^3+1\right )-\frac{b c}{6 x^3}-\frac{b \tanh ^{-1}\left (c x^3\right )}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x^3])/x^7,x]

[Out]

-a/(6*x^6) - (b*c)/(6*x^3) - (b*ArcTanh[c*x^3])/(6*x^6) - (b*c^2*Log[1 - c*x^3])/12 + (b*c^2*Log[1 + c*x^3])/1
2

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Maple [A]  time = 0.013, size = 55, normalized size = 1.3 \begin{align*} -{\frac{a}{6\,{x}^{6}}}-{\frac{b{\it Artanh} \left ( c{x}^{3} \right ) }{6\,{x}^{6}}}-{\frac{b{c}^{2}\ln \left ( c{x}^{3}-1 \right ) }{12}}+{\frac{b{c}^{2}\ln \left ( c{x}^{3}+1 \right ) }{12}}-{\frac{bc}{6\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^3))/x^7,x)

[Out]

-1/6*a/x^6-1/6*b/x^6*arctanh(c*x^3)-1/12*b*c^2*ln(c*x^3-1)+1/12*b*c^2*ln(c*x^3+1)-1/6*b*c/x^3

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Maxima [A]  time = 1.01283, size = 69, normalized size = 1.68 \begin{align*} \frac{1}{12} \,{\left ({\left (c \log \left (c x^{3} + 1\right ) - c \log \left (c x^{3} - 1\right ) - \frac{2}{x^{3}}\right )} c - \frac{2 \, \operatorname{artanh}\left (c x^{3}\right )}{x^{6}}\right )} b - \frac{a}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))/x^7,x, algorithm="maxima")

[Out]

1/12*((c*log(c*x^3 + 1) - c*log(c*x^3 - 1) - 2/x^3)*c - 2*arctanh(c*x^3)/x^6)*b - 1/6*a/x^6

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Fricas [A]  time = 1.93639, size = 104, normalized size = 2.54 \begin{align*} -\frac{2 \, b c x^{3} -{\left (b c^{2} x^{6} - b\right )} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + 2 \, a}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))/x^7,x, algorithm="fricas")

[Out]

-1/12*(2*b*c*x^3 - (b*c^2*x^6 - b)*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 2*a)/x^6

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x**3))/x**7,x)

[Out]

Timed out

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Giac [A]  time = 1.16228, size = 90, normalized size = 2.2 \begin{align*} \frac{1}{12} \, b c^{2} \log \left (c x^{3} + 1\right ) - \frac{1}{12} \, b c^{2} \log \left (c x^{3} - 1\right ) - \frac{b \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right )}{12 \, x^{6}} - \frac{b c x^{3} + a}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))/x^7,x, algorithm="giac")

[Out]

1/12*b*c^2*log(c*x^3 + 1) - 1/12*b*c^2*log(c*x^3 - 1) - 1/12*b*log(-(c*x^3 + 1)/(c*x^3 - 1))/x^6 - 1/6*(b*c*x^
3 + a)/x^6