3.14 \(\int \frac{a+b \text{csch}^{-1}(c x)}{x^7} \, dx\)

Optimal. Leaf size=98 \[ -\frac{a+b \text{csch}^{-1}(c x)}{6 x^6}+\frac{5 b c^5 \sqrt{\frac{1}{c^2 x^2}+1}}{96 x}-\frac{5 b c^3 \sqrt{\frac{1}{c^2 x^2}+1}}{144 x^3}+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1}}{36 x^5}-\frac{5}{96} b c^6 \text{csch}^{-1}(c x) \]

[Out]

(b*c*Sqrt[1 + 1/(c^2*x^2)])/(36*x^5) - (5*b*c^3*Sqrt[1 + 1/(c^2*x^2)])/(144*x^3) + (5*b*c^5*Sqrt[1 + 1/(c^2*x^
2)])/(96*x) - (5*b*c^6*ArcCsch[c*x])/96 - (a + b*ArcCsch[c*x])/(6*x^6)

________________________________________________________________________________________

Rubi [A]  time = 0.0636193, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6284, 335, 321, 215} \[ -\frac{a+b \text{csch}^{-1}(c x)}{6 x^6}+\frac{5 b c^5 \sqrt{\frac{1}{c^2 x^2}+1}}{96 x}-\frac{5 b c^3 \sqrt{\frac{1}{c^2 x^2}+1}}{144 x^3}+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1}}{36 x^5}-\frac{5}{96} b c^6 \text{csch}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsch[c*x])/x^7,x]

[Out]

(b*c*Sqrt[1 + 1/(c^2*x^2)])/(36*x^5) - (5*b*c^3*Sqrt[1 + 1/(c^2*x^2)])/(144*x^3) + (5*b*c^5*Sqrt[1 + 1/(c^2*x^
2)])/(96*x) - (5*b*c^6*ArcCsch[c*x])/96 - (a + b*ArcCsch[c*x])/(6*x^6)

Rule 6284

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

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 321

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{a+b \text{csch}^{-1}(c x)}{x^7} \, dx &=-\frac{a+b \text{csch}^{-1}(c x)}{6 x^6}-\frac{b \int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}} x^8} \, dx}{6 c}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{6 x^6}+\frac{b \operatorname{Subst}\left (\int \frac{x^6}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{6 c}\\ &=\frac{b c \sqrt{1+\frac{1}{c^2 x^2}}}{36 x^5}-\frac{a+b \text{csch}^{-1}(c x)}{6 x^6}-\frac{1}{36} (5 b c) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{b c \sqrt{1+\frac{1}{c^2 x^2}}}{36 x^5}-\frac{5 b c^3 \sqrt{1+\frac{1}{c^2 x^2}}}{144 x^3}-\frac{a+b \text{csch}^{-1}(c x)}{6 x^6}+\frac{1}{48} \left (5 b c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{b c \sqrt{1+\frac{1}{c^2 x^2}}}{36 x^5}-\frac{5 b c^3 \sqrt{1+\frac{1}{c^2 x^2}}}{144 x^3}+\frac{5 b c^5 \sqrt{1+\frac{1}{c^2 x^2}}}{96 x}-\frac{a+b \text{csch}^{-1}(c x)}{6 x^6}-\frac{1}{96} \left (5 b c^5\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{b c \sqrt{1+\frac{1}{c^2 x^2}}}{36 x^5}-\frac{5 b c^3 \sqrt{1+\frac{1}{c^2 x^2}}}{144 x^3}+\frac{5 b c^5 \sqrt{1+\frac{1}{c^2 x^2}}}{96 x}-\frac{5}{96} b c^6 \text{csch}^{-1}(c x)-\frac{a+b \text{csch}^{-1}(c x)}{6 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0718543, size = 88, normalized size = 0.9 \[ -\frac{a}{6 x^6}+b \left (-\frac{5 c^3}{144 x^3}+\frac{5 c^5}{96 x}+\frac{c}{36 x^5}\right ) \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}-\frac{5}{96} b c^6 \sinh ^{-1}\left (\frac{1}{c x}\right )-\frac{b \text{csch}^{-1}(c x)}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCsch[c*x])/x^7,x]

[Out]

-a/(6*x^6) + b*(c/(36*x^5) - (5*c^3)/(144*x^3) + (5*c^5)/(96*x))*Sqrt[(1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsch[c*
x])/(6*x^6) - (5*b*c^6*ArcSinh[1/(c*x)])/96

________________________________________________________________________________________

Maple [A]  time = 0.197, size = 139, normalized size = 1.4 \begin{align*}{c}^{6} \left ( -{\frac{a}{6\,{c}^{6}{x}^{6}}}+b \left ( -{\frac{{\rm arccsch} \left (cx\right )}{6\,{c}^{6}{x}^{6}}}-{\frac{1}{288\,{c}^{7}{x}^{7}}\sqrt{{c}^{2}{x}^{2}+1} \left ( 15\,{\it Artanh} \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}} \right ){c}^{6}{x}^{6}-15\,{c}^{4}{x}^{4}\sqrt{{c}^{2}{x}^{2}+1}+10\,{c}^{2}{x}^{2}\sqrt{{c}^{2}{x}^{2}+1}-8\,\sqrt{{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}}}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^6*(-1/6*a/c^6/x^6+b*(-1/6/c^6/x^6*arccsch(c*x)-1/288*(c^2*x^2+1)^(1/2)*(15*arctanh(1/(c^2*x^2+1)^(1/2))*c^6*
x^6-15*c^4*x^4*(c^2*x^2+1)^(1/2)+10*c^2*x^2*(c^2*x^2+1)^(1/2)-8*(c^2*x^2+1)^(1/2))/((c^2*x^2+1)/c^2/x^2)^(1/2)
/c^7/x^7))

________________________________________________________________________________________

Maxima [B]  time = 1.05192, size = 250, normalized size = 2.55 \begin{align*} -\frac{1}{576} \, b{\left (\frac{15 \, c^{7} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right ) - 15 \, c^{7} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right ) - \frac{2 \,{\left (15 \, c^{12} x^{5}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} - 40 \, c^{10} x^{3}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 33 \, c^{8} x \sqrt{\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{6} x^{6}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{3} - 3 \, c^{4} x^{4}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{2} + 3 \, c^{2} x^{2}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} - 1}}{c} + \frac{96 \, \operatorname{arcsch}\left (c x\right )}{x^{6}}\right )} - \frac{a}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/576*b*((15*c^7*log(c*x*sqrt(1/(c^2*x^2) + 1) + 1) - 15*c^7*log(c*x*sqrt(1/(c^2*x^2) + 1) - 1) - 2*(15*c^12*
x^5*(1/(c^2*x^2) + 1)^(5/2) - 40*c^10*x^3*(1/(c^2*x^2) + 1)^(3/2) + 33*c^8*x*sqrt(1/(c^2*x^2) + 1))/(c^6*x^6*(
1/(c^2*x^2) + 1)^3 - 3*c^4*x^4*(1/(c^2*x^2) + 1)^2 + 3*c^2*x^2*(1/(c^2*x^2) + 1) - 1))/c + 96*arccsch(c*x)/x^6
) - 1/6*a/x^6

________________________________________________________________________________________

Fricas [A]  time = 2.17684, size = 225, normalized size = 2.3 \begin{align*} -\frac{3 \,{\left (5 \, b c^{6} x^{6} + 16 \, b\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) -{\left (15 \, b c^{5} x^{5} - 10 \, b c^{3} x^{3} + 8 \, b c x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 48 \, a}{288 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/288*(3*(5*b*c^6*x^6 + 16*b)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) - (15*b*c^5*x^5 - 10*b*c^3*x
^3 + 8*b*c*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 48*a)/x^6

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{acsch}{\left (c x \right )}}{x^{7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acsch(c*x))/x**7, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcsch}\left (c x\right ) + a}{x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccsch(c*x) + a)/x^7, x)