3.51 \(\int \frac{a+b \cosh ^{-1}(c x)}{x (d-c^2 d x^2)^3} \, dx\)

Optimal. Leaf size=171 \[ \frac{b \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}-\frac{b \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}+\frac{a+b \cosh ^{-1}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac{a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{2 \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^3}-\frac{2 b c x}{3 d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c x}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}} \]

[Out]

(b*c*x)/(12*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (2*b*c*x)/(3*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + b*Ar
cCosh[c*x])/(4*d^3*(1 - c^2*x^2)^2) + (a + b*ArcCosh[c*x])/(2*d^3*(1 - c^2*x^2)) + (2*(a + b*ArcCosh[c*x])*Arc
Tanh[E^(2*ArcCosh[c*x])])/d^3 + (b*PolyLog[2, -E^(2*ArcCosh[c*x])])/(2*d^3) - (b*PolyLog[2, E^(2*ArcCosh[c*x])
])/(2*d^3)

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Rubi [A]  time = 0.260796, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {5754, 5721, 5461, 4182, 2279, 2391, 39, 40} \[ \frac{b \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}-\frac{b \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}+\frac{a+b \cosh ^{-1}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac{a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{2 \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^3}-\frac{2 b c x}{3 d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c x}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])/(x*(d - c^2*d*x^2)^3),x]

[Out]

(b*c*x)/(12*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (2*b*c*x)/(3*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + b*Ar
cCosh[c*x])/(4*d^3*(1 - c^2*x^2)^2) + (a + b*ArcCosh[c*x])/(2*d^3*(1 - c^2*x^2)) + (2*(a + b*ArcCosh[c*x])*Arc
Tanh[E^(2*ArcCosh[c*x])])/d^3 + (b*PolyLog[2, -E^(2*ArcCosh[c*x])])/(2*d^3) - (b*PolyLog[2, E^(2*ArcCosh[c*x])
])/(2*d^3)

Rule 5754

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> -Simp
[((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*d*f*(p + 1)), x] + (Dist[(m + 2*p + 3)/(2*d*(p
+ 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-d)^p)/(2*f*(p + 1)), Int
[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b,
c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && IntegerQ[p]

Rule 5721

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

Rule 5461

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rule 40

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(x*(a + b*x)^(m + 1)*(c + d*x)^(m +
1))/(2*a*c*(m + 1)), x] + Dist[(2*m + 3)/(2*a*c*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /; F
reeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]

Rubi steps

\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx &=\frac{a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{(b c) \int \frac{1}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 d^3}+\frac{\int \frac{a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx}{d}\\ &=\frac{b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{a+b \cosh ^{-1}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac{(b c) \int \frac{1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{6 d^3}+\frac{(b c) \int \frac{1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^3}+\frac{\int \frac{a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )} \, dx}{d^2}\\ &=\frac{b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{2 b c x}{3 d^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{a+b \cosh ^{-1}(c x)}{2 d^3 \left (1-c^2 x^2\right )}-\frac{\operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}\\ &=\frac{b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{2 b c x}{3 d^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{a+b \cosh ^{-1}(c x)}{2 d^3 \left (1-c^2 x^2\right )}-\frac{2 \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}\\ &=\frac{b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{2 b c x}{3 d^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{a+b \cosh ^{-1}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac{2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^3}+\frac{b \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}-\frac{b \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}\\ &=\frac{b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{2 b c x}{3 d^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{a+b \cosh ^{-1}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac{2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^3}+\frac{b \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}-\frac{b \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}\\ &=\frac{b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{2 b c x}{3 d^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{a+b \cosh ^{-1}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac{2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^3}+\frac{b \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}-\frac{b \text{Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}\\ \end{align*}

Mathematica [A]  time = 1.42309, size = 210, normalized size = 1.23 \[ -\frac{b \left (6 \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )-6 \text{PolyLog}\left (2,e^{-2 \cosh ^{-1}(c x)}\right )+\frac{6 \cosh ^{-1}(c x)}{c^2 x^2-1}-\frac{3 \cosh ^{-1}(c x)}{\left (c^2 x^2-1\right )^2}-\frac{c x \left (\frac{c x-1}{c x+1}\right )^{3/2}}{(c x-1)^3}+\frac{8 c x \sqrt{\frac{c x-1}{c x+1}}}{c x-1}+12 \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )-12 \cosh ^{-1}(c x) \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )+\frac{6 a}{c^2 x^2-1}-\frac{3 a}{\left (c^2 x^2-1\right )^2}+6 a \log \left (1-c^2 x^2\right )-12 a \log (x)}{12 d^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[c*x])/(x*(d - c^2*d*x^2)^3),x]

[Out]

-((-3*a)/(-1 + c^2*x^2)^2 + (6*a)/(-1 + c^2*x^2) - 12*a*Log[x] + 6*a*Log[1 - c^2*x^2] + b*((8*c*x*Sqrt[(-1 + c
*x)/(1 + c*x)])/(-1 + c*x) - (c*x*((-1 + c*x)/(1 + c*x))^(3/2))/(-1 + c*x)^3 - (3*ArcCosh[c*x])/(-1 + c^2*x^2)
^2 + (6*ArcCosh[c*x])/(-1 + c^2*x^2) + 12*ArcCosh[c*x]*Log[1 - E^(-2*ArcCosh[c*x])] - 12*ArcCosh[c*x]*Log[1 +
E^(-2*ArcCosh[c*x])] + 6*PolyLog[2, -E^(-2*ArcCosh[c*x])] - 6*PolyLog[2, E^(-2*ArcCosh[c*x])]))/(12*d^3)

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Maple [B]  time = 0.182, size = 508, normalized size = 3. \begin{align*}{\frac{a}{16\,{d}^{3} \left ( cx-1 \right ) ^{2}}}-{\frac{5\,a}{16\,{d}^{3} \left ( cx-1 \right ) }}-{\frac{a\ln \left ( cx-1 \right ) }{2\,{d}^{3}}}+{\frac{a\ln \left ( cx \right ) }{{d}^{3}}}+{\frac{a}{16\,{d}^{3} \left ( cx+1 \right ) ^{2}}}+{\frac{5\,a}{16\,{d}^{3} \left ( cx+1 \right ) }}-{\frac{a\ln \left ( cx+1 \right ) }{2\,{d}^{3}}}-{\frac{2\,{x}^{3}b{c}^{3}}{3\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}+{\frac{2\,{x}^{4}b{c}^{4}}{3\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }}-{\frac{b{\rm arccosh} \left (cx\right ){c}^{2}{x}^{2}}{2\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }}+{\frac{3\,xbc}{4\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}-{\frac{4\,{x}^{2}b{c}^{2}}{3\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }}+{\frac{3\,b{\rm arccosh} \left (cx\right )}{4\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }}+{\frac{2\,b}{3\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }}-{\frac{b{\rm arccosh} \left (cx\right )}{{d}^{3}}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{b}{{d}^{3}}{\it polylog} \left ( 2,-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{b{\rm arccosh} \left (cx\right )}{{d}^{3}}\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) }+{\frac{b}{2\,{d}^{3}}{\it polylog} \left ( 2,- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2} \right ) }-{\frac{b{\rm arccosh} \left (cx\right )}{{d}^{3}}\ln \left ( 1-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{b}{{d}^{3}}{\it polylog} \left ( 2,cx+\sqrt{cx-1}\sqrt{cx+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))/x/(-c^2*d*x^2+d)^3,x)

[Out]

1/16*a/d^3/(c*x-1)^2-5/16*a/d^3/(c*x-1)-1/2*a/d^3*ln(c*x-1)+a/d^3*ln(c*x)+1/16*a/d^3/(c*x+1)^2+5/16*a/d^3/(c*x
+1)-1/2*a/d^3*ln(c*x+1)-2/3*b/d^3/(c^4*x^4-2*c^2*x^2+1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3*x^3+2/3*b/d^3/(c^4*x^4
-2*c^2*x^2+1)*c^4*x^4-1/2*b/d^3/(c^4*x^4-2*c^2*x^2+1)*arccosh(c*x)*c^2*x^2+3/4*b/d^3/(c^4*x^4-2*c^2*x^2+1)*(c*
x+1)^(1/2)*(c*x-1)^(1/2)*c*x-4/3*b/d^3/(c^4*x^4-2*c^2*x^2+1)*c^2*x^2+3/4*b/d^3/(c^4*x^4-2*c^2*x^2+1)*arccosh(c
*x)+2/3*b/d^3/(c^4*x^4-2*c^2*x^2+1)-b/d^3*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-b/d^3*polylog(2,-
c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+b/d^3*arccosh(c*x)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+1)+1/2*b*polylog(2,
-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d^3-b/d^3*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-b/d^3*polyl
og(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{4} \, a{\left (\frac{2 \, c^{2} x^{2} - 3}{c^{4} d^{3} x^{4} - 2 \, c^{2} d^{3} x^{2} + d^{3}} + \frac{2 \, \log \left (c x + 1\right )}{d^{3}} + \frac{2 \, \log \left (c x - 1\right )}{d^{3}} - \frac{4 \, \log \left (x\right )}{d^{3}}\right )} - b \int \frac{\log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{c^{6} d^{3} x^{7} - 3 \, c^{4} d^{3} x^{5} + 3 \, c^{2} d^{3} x^{3} - d^{3} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x/(-c^2*d*x^2+d)^3,x, algorithm="maxima")

[Out]

-1/4*a*((2*c^2*x^2 - 3)/(c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^3) + 2*log(c*x + 1)/d^3 + 2*log(c*x - 1)/d^3 - 4*log(
x)/d^3) - b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^6*d^3*x^7 - 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 - d^
3*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b \operatorname{arcosh}\left (c x\right ) + a}{c^{6} d^{3} x^{7} - 3 \, c^{4} d^{3} x^{5} + 3 \, c^{2} d^{3} x^{3} - d^{3} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x/(-c^2*d*x^2+d)^3,x, algorithm="fricas")

[Out]

integral(-(b*arccosh(c*x) + a)/(c^6*d^3*x^7 - 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 - d^3*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{a}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx + \int \frac{b \operatorname{acosh}{\left (c x \right )}}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))/x/(-c**2*d*x**2+d)**3,x)

[Out]

-(Integral(a/(c**6*x**7 - 3*c**4*x**5 + 3*c**2*x**3 - x), x) + Integral(b*acosh(c*x)/(c**6*x**7 - 3*c**4*x**5
+ 3*c**2*x**3 - x), x))/d**3

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{b \operatorname{arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x/(-c^2*d*x^2+d)^3,x, algorithm="giac")

[Out]

integrate(-(b*arccosh(c*x) + a)/((c^2*d*x^2 - d)^3*x), x)