3.57 \(\int \frac{a+b \tanh ^{-1}(c x^2)}{x^7} \, dx\)

Optimal. Leaf size=56 \[ -\frac{a+b \tanh ^{-1}\left (c x^2\right )}{6 x^6}-\frac{1}{12} b c^3 \log \left (1-c^2 x^4\right )+\frac{1}{3} b c^3 \log (x)-\frac{b c}{12 x^4} \]

[Out]

-(b*c)/(12*x^4) - (a + b*ArcTanh[c*x^2])/(6*x^6) + (b*c^3*Log[x])/3 - (b*c^3*Log[1 - c^2*x^4])/12

________________________________________________________________________________________

Rubi [A]  time = 0.0340155, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {6097, 266, 44} \[ -\frac{a+b \tanh ^{-1}\left (c x^2\right )}{6 x^6}-\frac{1}{12} b c^3 \log \left (1-c^2 x^4\right )+\frac{1}{3} b c^3 \log (x)-\frac{b c}{12 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^2])/x^7,x]

[Out]

-(b*c)/(12*x^4) - (a + b*ArcTanh[c*x^2])/(6*x^6) + (b*c^3*Log[x])/3 - (b*c^3*Log[1 - c^2*x^4])/12

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 44

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

Rubi steps

\begin{align*} \int \frac{a+b \tanh ^{-1}\left (c x^2\right )}{x^7} \, dx &=-\frac{a+b \tanh ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{3} (b c) \int \frac{1}{x^5 \left (1-c^2 x^4\right )} \, dx\\ &=-\frac{a+b \tanh ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{12} (b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^4\right )\\ &=-\frac{a+b \tanh ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{12} (b c) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}+\frac{c^2}{x}-\frac{c^4}{-1+c^2 x}\right ) \, dx,x,x^4\right )\\ &=-\frac{b c}{12 x^4}-\frac{a+b \tanh ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{3} b c^3 \log (x)-\frac{1}{12} b c^3 \log \left (1-c^2 x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.0112476, size = 61, normalized size = 1.09 \[ -\frac{a}{6 x^6}-\frac{1}{12} b c^3 \log \left (1-c^2 x^4\right )+\frac{1}{3} b c^3 \log (x)-\frac{b c}{12 x^4}-\frac{b \tanh ^{-1}\left (c x^2\right )}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x^2])/x^7,x]

[Out]

-a/(6*x^6) - (b*c)/(12*x^4) - (b*ArcTanh[c*x^2])/(6*x^6) + (b*c^3*Log[x])/3 - (b*c^3*Log[1 - c^2*x^4])/12

________________________________________________________________________________________

Maple [A]  time = 0.015, size = 63, normalized size = 1.1 \begin{align*} -{\frac{a}{6\,{x}^{6}}}-{\frac{b{\it Artanh} \left ( c{x}^{2} \right ) }{6\,{x}^{6}}}-{\frac{b{c}^{3}\ln \left ( c{x}^{2}+1 \right ) }{12}}-{\frac{bc}{12\,{x}^{4}}}+{\frac{b{c}^{3}\ln \left ( x \right ) }{3}}-{\frac{b{c}^{3}\ln \left ( c{x}^{2}-1 \right ) }{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^2))/x^7,x)

[Out]

-1/6*a/x^6-1/6*b/x^6*arctanh(c*x^2)-1/12*b*c^3*ln(c*x^2+1)-1/12*b*c/x^4+1/3*b*c^3*ln(x)-1/12*b*c^3*ln(c*x^2-1)

________________________________________________________________________________________

Maxima [A]  time = 0.975046, size = 69, normalized size = 1.23 \begin{align*} -\frac{1}{12} \,{\left ({\left (c^{2} \log \left (c^{2} x^{4} - 1\right ) - c^{2} \log \left (x^{4}\right ) + \frac{1}{x^{4}}\right )} c + \frac{2 \, \operatorname{artanh}\left (c x^{2}\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^2))/x^7,x, algorithm="maxima")

[Out]

-1/12*((c^2*log(c^2*x^4 - 1) - c^2*log(x^4) + 1/x^4)*c + 2*arctanh(c*x^2)/x^6)*b - 1/6*a/x^6

________________________________________________________________________________________

Fricas [A]  time = 2.11026, size = 150, normalized size = 2.68 \begin{align*} -\frac{b c^{3} x^{6} \log \left (c^{2} x^{4} - 1\right ) - 4 \, b c^{3} x^{6} \log \left (x\right ) + b c x^{2} + b \log \left (-\frac{c x^{2} + 1}{c x^{2} - 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^2))/x^7,x, algorithm="fricas")

[Out]

-1/12*(b*c^3*x^6*log(c^2*x^4 - 1) - 4*b*c^3*x^6*log(x) + b*c*x^2 + b*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 2*a)/x^6

________________________________________________________________________________________

Sympy [A]  time = 34.4314, size = 97, normalized size = 1.73 \begin{align*} \begin{cases} - \frac{a}{6 x^{6}} + \frac{b c^{3} \log{\left (x \right )}}{3} - \frac{b c^{3} \log{\left (x - i \sqrt{\frac{1}{c}} \right )}}{6} - \frac{b c^{3} \log{\left (x + i \sqrt{\frac{1}{c}} \right )}}{6} + \frac{b c^{3} \operatorname{atanh}{\left (c x^{2} \right )}}{6} - \frac{b c}{12 x^{4}} - \frac{b \operatorname{atanh}{\left (c x^{2} \right )}}{6 x^{6}} & \text{for}\: c \neq 0 \\- \frac{a}{6 x^{6}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x**2))/x**7,x)

[Out]

Piecewise((-a/(6*x**6) + b*c**3*log(x)/3 - b*c**3*log(x - I*sqrt(1/c))/6 - b*c**3*log(x + I*sqrt(1/c))/6 + b*c
**3*atanh(c*x**2)/6 - b*c/(12*x**4) - b*atanh(c*x**2)/(6*x**6), Ne(c, 0)), (-a/(6*x**6), True))

________________________________________________________________________________________

Giac [A]  time = 1.33286, size = 88, normalized size = 1.57 \begin{align*} -\frac{1}{12} \, b c^{3} \log \left (c^{2} x^{4} - 1\right ) + \frac{1}{3} \, b c^{3} \log \left (x\right ) - \frac{b \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right )}{12 \, x^{6}} - \frac{b c x^{2} + 2 \, a}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^2))/x^7,x, algorithm="giac")

[Out]

-1/12*b*c^3*log(c^2*x^4 - 1) + 1/3*b*c^3*log(x) - 1/12*b*log(-(c*x^2 + 1)/(c*x^2 - 1))/x^6 - 1/12*(b*c*x^2 + 2
*a)/x^6