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

Optimal. Leaf size=57 \[ -\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{6 x^6}-\frac{b \log \left (c^2-x^4\right )}{12 c^3}+\frac{b \log (x)}{3 c^3}-\frac{b}{12 c x^4} \]

[Out]

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

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Rubi [A]  time = 0.0400527, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6097, 263, 266, 44} \[ -\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{6 x^6}-\frac{b \log \left (c^2-x^4\right )}{12 c^3}+\frac{b \log (x)}{3 c^3}-\frac{b}{12 c x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x^2])/x^7,x]

[Out]

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

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 263

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

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 (\frac{c}{x^2}\right )}{x^7} \, dx &=-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{6 x^6}-\frac{1}{3} (b c) \int \frac{1}{\left (1-\frac{c^2}{x^4}\right ) x^9} \, dx\\ &=-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{6 x^6}-\frac{1}{3} (b c) \int \frac{1}{x^5 \left (-c^2+x^4\right )} \, dx\\ &=-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{6 x^6}-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (-c^2+x\right )} \, dx,x,x^4\right )\\ &=-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{6 x^6}-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4 \left (c^2-x\right )}-\frac{1}{c^2 x^2}-\frac{1}{c^4 x}\right ) \, dx,x,x^4\right )\\ &=-\frac{b}{12 c x^4}-\frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{6 x^6}+\frac{b \log (x)}{3 c^3}-\frac{b \log \left (c^2-x^4\right )}{12 c^3}\\ \end{align*}

Mathematica [A]  time = 0.012913, size = 62, normalized size = 1.09 \[ -\frac{a}{6 x^6}-\frac{b \log \left (x^4-c^2\right )}{12 c^3}+\frac{b \log (x)}{3 c^3}-\frac{b}{12 c x^4}-\frac{b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{6 x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 45, normalized size = 0.8 \begin{align*} -{\frac{a}{6\,{x}^{6}}}-{\frac{b}{6\,{x}^{6}}{\it Artanh} \left ({\frac{c}{{x}^{2}}} \right ) }-{\frac{b}{12\,c{x}^{4}}}-{\frac{b}{12\,{c}^{3}}\ln \left ({\frac{{c}^{2}}{{x}^{4}}}-1 \right ) } \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/x^4-1/12*b/c^3*ln(c^2/x^4-1)

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Maxima [A]  time = 0.979566, size = 74, normalized size = 1.3 \begin{align*} -\frac{1}{12} \,{\left (c{\left (\frac{\log \left (x^{4} - c^{2}\right )}{c^{4}} - \frac{\log \left (x^{4}\right )}{c^{4}} + \frac{1}{c^{2} x^{4}}\right )} + \frac{2 \, \operatorname{artanh}\left (\frac{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*(log(x^4 - c^2)/c^4 - log(x^4)/c^4 + 1/(c^2*x^4)) + 2*arctanh(c/x^2)/x^6)*b - 1/6*a/x^6

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Fricas [A]  time = 1.67821, size = 151, normalized size = 2.65 \begin{align*} -\frac{b x^{6} \log \left (x^{4} - c^{2}\right ) - 4 \, b x^{6} \log \left (x\right ) + b c^{2} x^{2} + b c^{3} \log \left (\frac{x^{2} + c}{x^{2} - c}\right ) + 2 \, a c^{3}}{12 \, c^{3} 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*x^6*log(x^4 - c^2) - 4*b*x^6*log(x) + b*c^2*x^2 + b*c^3*log((x^2 + c)/(x^2 - c)) + 2*a*c^3)/(c^3*x^6)

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Sympy [A]  time = 47.1008, size = 94, normalized size = 1.65 \begin{align*} \begin{cases} - \frac{a}{6 x^{6}} - \frac{b \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{6 x^{6}} - \frac{b}{12 c x^{4}} + \frac{b \log{\left (x \right )}}{3 c^{3}} - \frac{b \log{\left (- i \sqrt{c} + x \right )}}{6 c^{3}} - \frac{b \log{\left (i \sqrt{c} + x \right )}}{6 c^{3}} + \frac{b \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{6 c^{3}} & \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*atanh(c/x**2)/(6*x**6) - b/(12*c*x**4) + b*log(x)/(3*c**3) - b*log(-I*sqrt(c) + x)/
(6*c**3) - b*log(I*sqrt(c) + x)/(6*c**3) + b*atanh(c/x**2)/(6*c**3), Ne(c, 0)), (-a/(6*x**6), True))

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Giac [A]  time = 1.16046, size = 88, normalized size = 1.54 \begin{align*} -\frac{b \log \left (x^{4} - c^{2}\right )}{12 \, c^{3}} + \frac{b \log \left (x\right )}{3 \, c^{3}} - \frac{b \log \left (\frac{x^{2} + c}{x^{2} - c}\right )}{12 \, x^{6}} - \frac{b x^{2} + 2 \, a c}{12 \, c 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*log(x^4 - c^2)/c^3 + 1/3*b*log(x)/c^3 - 1/12*b*log((x^2 + c)/(x^2 - c))/x^6 - 1/12*(b*x^2 + 2*a*c)/(c*
x^6)