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

Optimal. Leaf size=158 \[ -\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}+\frac {5}{96} b c^6 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c x} \sqrt {c x+1}\right )+\frac {5 b c^4 \sqrt {1-c x}}{96 x^2 \sqrt {\frac {1}{c x+1}}}+\frac {5 b c^2 \sqrt {1-c x}}{144 x^4 \sqrt {\frac {1}{c x+1}}}+\frac {b \sqrt {1-c x}}{36 x^6 \sqrt {\frac {1}{c x+1}}} \]

[Out]

1/6*(-a-b*arcsech(c*x))/x^6+1/36*b*(-c*x+1)^(1/2)/x^6/(1/(c*x+1))^(1/2)+5/144*b*c^2*(-c*x+1)^(1/2)/x^4/(1/(c*x
+1))^(1/2)+5/96*b*c^4*(-c*x+1)^(1/2)/x^2/(1/(c*x+1))^(1/2)+5/96*b*c^6*arctanh((-c*x+1)^(1/2)*(c*x+1)^(1/2))*(1
/(c*x+1))^(1/2)*(c*x+1)^(1/2)

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Rubi [A]  time = 0.07, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6283, 103, 12, 92, 208} \[ -\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}+\frac {5 b c^4 \sqrt {1-c x}}{96 x^2 \sqrt {\frac {1}{c x+1}}}+\frac {5 b c^2 \sqrt {1-c x}}{144 x^4 \sqrt {\frac {1}{c x+1}}}+\frac {5}{96} b c^6 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c x} \sqrt {c x+1}\right )+\frac {b \sqrt {1-c x}}{36 x^6 \sqrt {\frac {1}{c x+1}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSech[c*x])/x^7,x]

[Out]

(b*Sqrt[1 - c*x])/(36*x^6*Sqrt[(1 + c*x)^(-1)]) + (5*b*c^2*Sqrt[1 - c*x])/(144*x^4*Sqrt[(1 + c*x)^(-1)]) + (5*
b*c^4*Sqrt[1 - c*x])/(96*x^2*Sqrt[(1 + c*x)^(-1)]) - (a + b*ArcSech[c*x])/(6*x^6) + (5*b*c^6*Sqrt[(1 + c*x)^(-
1)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c*x]*Sqrt[1 + c*x]])/96

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 208

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

Rule 6283

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

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 157, normalized size = 0.99 \[ -\frac {a}{6 x^6}-\frac {5}{96} b c^6 \log (x)+\frac {5}{96} b c^6 \log \left (c x \sqrt {\frac {1-c x}{c x+1}}+\sqrt {\frac {1-c x}{c x+1}}+1\right )+b \left (\frac {5 c^5}{96 x}+\frac {5 c^4}{96 x^2}+\frac {5 c^3}{144 x^3}+\frac {5 c^2}{144 x^4}+\frac {c}{36 x^5}+\frac {1}{36 x^6}\right ) \sqrt {\frac {1-c x}{c x+1}}-\frac {b \text {sech}^{-1}(c x)}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSech[c*x])/x^7,x]

[Out]

-1/6*a/x^6 + b*(1/(36*x^6) + c/(36*x^5) + (5*c^2)/(144*x^4) + (5*c^3)/(144*x^3) + (5*c^4)/(96*x^2) + (5*c^5)/(
96*x))*Sqrt[(1 - c*x)/(1 + c*x)] - (b*ArcSech[c*x])/(6*x^6) - (5*b*c^6*Log[x])/96 + (5*b*c^6*Log[1 + Sqrt[(1 -
c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/96

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fricas [A]  time = 0.59, size = 100, normalized size = 0.63 \[ \frac {3 \, {\left (5 \, b c^{6} x^{6} - 16 \, b\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (15 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 8 \, b c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 48 \, a}{288 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arsech}\left (c x\right ) + a}{x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.07, size = 155, normalized size = 0.98 \[ c^{6} \left (-\frac {a}{6 c^{6} x^{6}}+b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{6 c^{6} x^{6}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (15 \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{6} x^{6}+15 \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}+10 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}+8 \sqrt {-c^{2} x^{2}+1}\right )}{288 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^6*(-1/6*a/c^6/x^6+b*(-1/6/c^6/x^6*arcsech(c*x)+1/288*(-(c*x-1)/c/x)^(1/2)/c^5/x^5*((c*x+1)/c/x)^(1/2)*(15*ar
ctanh(1/(-c^2*x^2+1)^(1/2))*c^6*x^6+15*(-c^2*x^2+1)^(1/2)*c^4*x^4+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)^(1/2)))

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maxima [A]  time = 0.32, size = 185, normalized size = 1.17 \[ \frac {1}{576} \, b {\left (\frac {15 \, c^{7} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} - 1} + 1\right ) - 15 \, c^{7} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} - 1} - 1\right ) - \frac {2 \, {\left (15 \, c^{12} x^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} - 40 \, c^{10} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 33 \, c^{8} x \sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}}{c^{6} x^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} - 3 \, c^{4} x^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} - 1}}{c} - \frac {96 \, \operatorname {arsech}\left (c x\right )}{x^{6}}\right )} - \frac {a}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(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*arcsech(c*x)/x^6)
- 1/6*a/x^6

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x^7} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(1/(c*x)))/x^7,x)

[Out]

int((a + b*acosh(1/(c*x)))/x^7, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*asech(c*x))/x**7, x)

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