3.1.62 \(\int \frac {1}{x^4 (a+b \text {sech}^{-1}(c x))^2} \, dx\) [62]

Optimal. Leaf size=190 \[ \frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{4 b x \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}-\frac {3 c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2} \]

[Out]

-1/4*c^3*Chi(a/b+arcsech(c*x))*cosh(a/b)/b^2-3/4*c^3*Chi(3*a/b+3*arcsech(c*x))*cosh(3*a/b)/b^2+1/4*c^3*Shi(a/b
+arcsech(c*x))*sinh(a/b)/b^2+3/4*c^3*Shi(3*a/b+3*arcsech(c*x))*sinh(3*a/b)/b^2+1/4*c^3*sinh(3*arcsech(c*x))/b/
(a+b*arcsech(c*x))+1/4*c^2*(c*x+1)*((-c*x+1)/(c*x+1))^(1/2)/b/x/(a+b*arcsech(c*x))

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Rubi [A]
time = 0.23, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6420, 5556, 3378, 3384, 3379, 3382} \begin {gather*} -\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}-\frac {3 c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {c^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{4 b x \left (a+b \text {sech}^{-1}(c x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*ArcSech[c*x])^2),x]

[Out]

(c^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(4*b*x*(a + b*ArcSech[c*x])) - (c^3*Cosh[a/b]*CoshIntegral[a/b + Arc
Sech[c*x]])/(4*b^2) - (3*c^3*Cosh[(3*a)/b]*CoshIntegral[(3*a)/b + 3*ArcSech[c*x]])/(4*b^2) + (c^3*Sinh[3*ArcSe
ch[c*x]])/(4*b*(a + b*ArcSech[c*x])) + (c^3*Sinh[a/b]*SinhIntegral[a/b + ArcSech[c*x]])/(4*b^2) + (3*c^3*Sinh[
(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcSech[c*x]])/(4*b^2)

Rule 3378

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m
+ 1))), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 6420

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[-(c^(m + 1))^(-1), Subst[Int[(a + b
*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, ArcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] &
& (GtQ[n, 0] || LtQ[m, -1])

Rubi steps

\begin {align*} \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx &=-\left (c^3 \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{(a+b x)^2} \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\left (c^3 \text {Subst}\left (\int \left (\frac {\sinh (x)}{4 (a+b x)^2}+\frac {\sinh (3 x)}{4 (a+b x)^2}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\left (\frac {1}{4} c^3 \text {Subst}\left (\int \frac {\sinh (x)}{(a+b x)^2} \, dx,x,\text {sech}^{-1}(c x)\right )\right )-\frac {1}{4} c^3 \text {Subst}\left (\int \frac {\sinh (3 x)}{(a+b x)^2} \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{4 b x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {c^3 \text {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 b}-\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 b}\\ &=\frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{4 b x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {\left (c^3 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 b}-\frac {\left (3 c^3 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 b}+\frac {\left (c^3 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 b}+\frac {\left (3 c^3 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 b}\\ &=\frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{4 b x \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}-\frac {3 c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2}\\ \end {align*}

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Mathematica [A]
time = 0.38, size = 250, normalized size = 1.32 \begin {gather*} \frac {4 b \sqrt {\frac {1-c x}{1+c x}}+4 b c x \sqrt {\frac {1-c x}{1+c x}}-c^3 x^3 \left (a+b \text {sech}^{-1}(c x)\right ) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )-3 c^3 x^3 \left (a+b \text {sech}^{-1}(c x)\right ) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )+a c^3 x^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )+b c^3 x^3 \text {sech}^{-1}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )+3 a c^3 x^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )+3 b c^3 x^3 \text {sech}^{-1}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )}{4 b^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*ArcSech[c*x])^2),x]

[Out]

(4*b*Sqrt[(1 - c*x)/(1 + c*x)] + 4*b*c*x*Sqrt[(1 - c*x)/(1 + c*x)] - c^3*x^3*(a + b*ArcSech[c*x])*Cosh[a/b]*Co
shIntegral[a/b + ArcSech[c*x]] - 3*c^3*x^3*(a + b*ArcSech[c*x])*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSech[c*
x])] + a*c^3*x^3*Sinh[a/b]*SinhIntegral[a/b + ArcSech[c*x]] + b*c^3*x^3*ArcSech[c*x]*Sinh[a/b]*SinhIntegral[a/
b + ArcSech[c*x]] + 3*a*c^3*x^3*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSech[c*x])] + 3*b*c^3*x^3*ArcSech[c*x]*
Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSech[c*x])])/(4*b^2*x^3*(a + b*ArcSech[c*x]))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(419\) vs. \(2(176)=352\).
time = 0.55, size = 420, normalized size = 2.21

method	result	size
derivativedivides	\(c^{3} \left (-\frac {\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x -3 c^{2} x^{2}+4}{8 c^{3} x^{3} b \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {3 \,{\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, \frac {3 a}{b}+3 \,\mathrm {arcsech}\left (c x \right )\right )}{8 b^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x -1}{8 c x b \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \frac {a}{b}+\mathrm {arcsech}\left (c x \right )\right )}{8 b^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +1}{8 b c x \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arcsech}\left (c x \right )-\frac {a}{b}\right )}{8 b^{2}}-\frac {\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +3 c^{2} x^{2}-4}{8 b \,c^{3} x^{3} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {3 \,{\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \,\mathrm {arcsech}\left (c x \right )-\frac {3 a}{b}\right )}{8 b^{2}}\right )\)	\(420\)
default	\(c^{3} \left (-\frac {\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x -3 c^{2} x^{2}+4}{8 c^{3} x^{3} b \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {3 \,{\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, \frac {3 a}{b}+3 \,\mathrm {arcsech}\left (c x \right )\right )}{8 b^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x -1}{8 c x b \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \frac {a}{b}+\mathrm {arcsech}\left (c x \right )\right )}{8 b^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +1}{8 b c x \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arcsech}\left (c x \right )-\frac {a}{b}\right )}{8 b^{2}}-\frac {\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +3 c^{2} x^{2}-4}{8 b \,c^{3} x^{3} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {3 \,{\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \,\mathrm {arcsech}\left (c x \right )-\frac {3 a}{b}\right )}{8 b^{2}}\right )\)	\(420\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(a+b*arcsech(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

c^3*(-1/8*(((c*x+1)/c/x)^(1/2)*(-(c*x-1)/c/x)^(1/2)*c^3*x^3-4*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)*c*x-3*c
^2*x^2+4)/c^3/x^3/b/(a+b*arcsech(c*x))+3/8/b^2*exp(3*a/b)*Ei(1,3*a/b+3*arcsech(c*x))+1/8*((-(c*x-1)/c/x)^(1/2)
*((c*x+1)/c/x)^(1/2)*c*x-1)/c/x/b/(a+b*arcsech(c*x))+1/8/b^2*exp(a/b)*Ei(1,a/b+arcsech(c*x))+1/8/b*((-(c*x-1)/
c/x)^(1/2)*((c*x+1)/c/x)^(1/2)*c*x+1)/c/x/(a+b*arcsech(c*x))+1/8/b^2*exp(-a/b)*Ei(1,-arcsech(c*x)-a/b)-1/8/b*(
((c*x+1)/c/x)^(1/2)*(-(c*x-1)/c/x)^(1/2)*c^3*x^3-4*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)*c*x+3*c^2*x^2-4)/c
^3/x^3/(a+b*arcsech(c*x))+3/8/b^2*exp(-3*a/b)*Ei(1,-3*arcsech(c*x)-3*a/b))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+b*arcsech(c*x))^2,x, algorithm="maxima")

[Out]

-(c^2*x^3 + (c^2*x^3 - x)*sqrt(c*x + 1)*sqrt(-c*x + 1) - x)/((b^2*c^2*x^2 - b^2)*x^4*log(x) + ((b^2*c^2*log(c)
- a*b*c^2)*x^2 - b^2*log(c) + a*b)*x^4 - (b^2*x^4*log(x) + (b^2*log(c) - a*b)*x^4)*sqrt(c*x + 1)*sqrt(-c*x +
1) + (sqrt(c*x + 1)*sqrt(-c*x + 1)*b^2*x^4 - (b^2*c^2*x^2 - b^2)*x^4)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)) -
integrate((3*c^4*x^4 - 6*c^2*x^2 + (c^2*x^2 - 3)*(c*x + 1)*(c*x - 1) + (2*c^4*x^4 - 7*c^2*x^2 + 6)*sqrt(c*x +
1)*sqrt(-c*x + 1) + 3)/((b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*x^4*log(x) + ((b^2*c^4*log(c) - a*b*c^4)*x^4 - 2*
(b^2*c^2*log(c) - a*b*c^2)*x^2 + b^2*log(c) - a*b)*x^4 - (b^2*x^4*log(x) + (b^2*log(c) - a*b)*x^4)*(c*x + 1)*(
c*x - 1) - 2*((b^2*c^2*x^2 - b^2)*x^4*log(x) + ((b^2*c^2*log(c) - a*b*c^2)*x^2 - b^2*log(c) + a*b)*x^4)*sqrt(c
*x + 1)*sqrt(-c*x + 1) + ((c*x + 1)*(c*x - 1)*b^2*x^4 + 2*(b^2*c^2*x^2 - b^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x^4
- (b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*x^4)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+b*arcsech(c*x))^2,x, algorithm="fricas")

[Out]

integral(1/(b^2*x^4*arcsech(c*x)^2 + 2*a*b*x^4*arcsech(c*x) + a^2*x^4), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(a+b*asech(c*x))**2,x)

[Out]

Integral(1/(x**4*(a + b*asech(c*x))**2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+b*arcsech(c*x))^2,x, algorithm="giac")

[Out]

integrate(1/((b*arcsech(c*x) + a)^2*x^4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^4\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*acosh(1/(c*x)))^2),x)

[Out]

int(1/(x^4*(a + b*acosh(1/(c*x)))^2), x)

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