3.1.45 \(\int \frac {a+b \cosh ^{-1}(c x)}{x^4 (d-c^2 d x^2)^2} \, dx\) [45]

Optimal. Leaf size=248 \[ -\frac {b c^3}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {13 b c^3 \text {ArcTan}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d^2}+\frac {5 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2}+\frac {5 b c^3 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {5 b c^3 \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{2 d^2} \]

[Out]

1/3*(-a-b*arccosh(c*x))/d^2/x^3/(-c^2*x^2+1)-5/3*c^2*(a+b*arccosh(c*x))/d^2/x/(-c^2*x^2+1)+5/2*c^4*x*(a+b*arcc
osh(c*x))/d^2/(-c^2*x^2+1)+13/6*b*c^3*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))/d^2+5*c^3*(a+b*arccosh(c*x))*arctanh
(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^2+5/2*b*c^3*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^2-5/2*b*c^3*poly
log(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^2-1/3*b*c^3/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/6*b*c/d^2/x^2/(c*x-1)^(
1/2)/(c*x+1)^(1/2)

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Rubi [A]
time = 0.21, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {5932, 5901, 5903, 4267, 2317, 2438, 75, 106, 21, 94, 211, 105, 12} \begin {gather*} \frac {5 c^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}-\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {5 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {13 b c^3 \text {ArcTan}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{6 d^2}+\frac {5 b c^3 \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {5 b c^3 \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {b c^3}{3 d^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c}{6 d^2 x^2 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^2),x]

[Out]

-1/3*(b*c^3)/(d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c)/(6*d^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (a + b*ArcC
osh[c*x])/(3*d^2*x^3*(1 - c^2*x^2)) - (5*c^2*(a + b*ArcCosh[c*x]))/(3*d^2*x*(1 - c^2*x^2)) + (5*c^4*x*(a + b*A
rcCosh[c*x]))/(2*d^2*(1 - c^2*x^2)) + (13*b*c^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(6*d^2) + (5*c^3*(a + b*
ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/d^2 + (5*b*c^3*PolyLog[2, -E^ArcCosh[c*x]])/(2*d^2) - (5*b*c^3*PolyLog[
2, E^ArcCosh[c*x]])/(2*d^2)

Rule 12

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 75

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

Rule 94

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 105

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] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 106

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] &&
IntegersQ[2*m, 2*n, 2*p]

Rule 211

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

Rule 2317

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 2438

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 4267

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 5901

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

Rule 5903

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

Rule 5932

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/(d*f*(m + 1))), x] + (Dist[c^2*((m + 2*p + 3)/(f^2*(
m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] + Dist[b*c*(n/(f*(m + 1)))*Simp[(d +
e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCos
h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx &=-\frac {a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {1}{3} \left (5 c^2\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx-\frac {(b c) \int \frac {1}{x^3 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2}\\ &=-\frac {b c}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\left (5 c^4\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx-\frac {(b c) \int \frac {3 c^2}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{6 d^2}-\frac {\left (5 b c^3\right ) \int \frac {1}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2}\\ &=\frac {5 b c^3}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {\left (5 b c^2\right ) \int \frac {c+c^2 x}{x \sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{3 d^2}-\frac {\left (b c^3\right ) \int \frac {1}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2}+\frac {\left (5 b c^5\right ) \int \frac {x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2}+\frac {\left (5 c^4\right ) \int \frac {a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 d}\\ &=-\frac {b c^3}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {\left (b c^2\right ) \int \frac {c+c^2 x}{x \sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{2 d^2}-\frac {\left (5 c^3\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}+\frac {\left (5 b c^3\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 d^2}\\ &=-\frac {b c^3}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {5 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2}+\frac {\left (b c^3\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 d^2}+\frac {\left (5 b c^3\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}-\frac {\left (5 b c^3\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}+\frac {\left (5 b c^4\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{3 d^2}\\ &=-\frac {b c^3}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {5 b c^3 \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{3 d^2}+\frac {5 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2}+\frac {\left (5 b c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {\left (5 b c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d^2}+\frac {\left (b c^4\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{2 d^2}\\ &=-\frac {b c^3}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {13 b c^3 \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d^2}+\frac {5 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2}+\frac {5 b c^3 \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {5 b c^3 \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{2 d^2}\\ \end {align*}

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Mathematica [A]
time = 1.03, size = 377, normalized size = 1.52 \begin {gather*} -\frac {\frac {4 a}{x^3}+\frac {24 a c^2}{x}-3 b c^3 \sqrt {\frac {-1+c x}{1+c x}}+\frac {3 b c^3 \sqrt {\frac {-1+c x}{1+c x}}}{-1+c x}+\frac {3 b c^4 x \sqrt {\frac {-1+c x}{1+c x}}}{-1+c x}-\frac {2 b c^3}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c}{x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {6 a c^4 x}{-1+c^2 x^2}+\frac {4 b \cosh ^{-1}(c x)}{x^3}+\frac {24 b c^2 \cosh ^{-1}(c x)}{x}+\frac {3 b c^3 \cosh ^{-1}(c x)}{-1+c x}+\frac {3 b c^3 \cosh ^{-1}(c x)}{1+c x}-\frac {26 b c^3 \sqrt {-1+c^2 x^2} \text {ArcTan}\left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+30 b c^3 \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )-30 b c^3 \cosh ^{-1}(c x) \log \left (1+e^{\cosh ^{-1}(c x)}\right )+15 a c^3 \log (1-c x)-15 a c^3 \log (1+c x)-30 b c^3 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )+30 b c^3 \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{12 d^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^2),x]

[Out]

-1/12*((4*a)/x^3 + (24*a*c^2)/x - 3*b*c^3*Sqrt[(-1 + c*x)/(1 + c*x)] + (3*b*c^3*Sqrt[(-1 + c*x)/(1 + c*x)])/(-
1 + c*x) + (3*b*c^4*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(-1 + c*x) - (2*b*c^3)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b
*c)/(x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (6*a*c^4*x)/(-1 + c^2*x^2) + (4*b*ArcCosh[c*x])/x^3 + (24*b*c^2*ArcCo
sh[c*x])/x + (3*b*c^3*ArcCosh[c*x])/(-1 + c*x) + (3*b*c^3*ArcCosh[c*x])/(1 + c*x) - (26*b*c^3*Sqrt[-1 + c^2*x^
2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + 30*b*c^3*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]]
- 30*b*c^3*ArcCosh[c*x]*Log[1 + E^ArcCosh[c*x]] + 15*a*c^3*Log[1 - c*x] - 15*a*c^3*Log[1 + c*x] - 30*b*c^3*Pol
yLog[2, -E^ArcCosh[c*x]] + 30*b*c^3*PolyLog[2, E^ArcCosh[c*x]])/d^2

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Maple [A]
time = 6.30, size = 335, normalized size = 1.35

method	result	size
derivativedivides	\(c^{3} \left (-\frac {a}{4 d^{2} \left (c x -1\right )}-\frac {5 a \ln \left (c x -1\right )}{4 d^{2}}-\frac {a}{4 d^{2} \left (c x +1\right )}+\frac {5 a \ln \left (c x +1\right )}{4 d^{2}}-\frac {a}{3 d^{2} c^{3} x^{3}}-\frac {2 a}{d^{2} c x}-\frac {5 b \,\mathrm {arccosh}\left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{3 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {5 b \,\mathrm {arccosh}\left (c x \right )}{3 d^{2} \left (c^{2} x^{2}-1\right ) c x}-\frac {b \sqrt {c x +1}\, \sqrt {c x -1}}{6 d^{2} \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}+\frac {b \,\mathrm {arccosh}\left (c x \right )}{3 d^{2} \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {13 b \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{2}}+\frac {5 b \dilog \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {5 b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {5 b \dilog \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}\right )\)	\(335\)
default	\(c^{3} \left (-\frac {a}{4 d^{2} \left (c x -1\right )}-\frac {5 a \ln \left (c x -1\right )}{4 d^{2}}-\frac {a}{4 d^{2} \left (c x +1\right )}+\frac {5 a \ln \left (c x +1\right )}{4 d^{2}}-\frac {a}{3 d^{2} c^{3} x^{3}}-\frac {2 a}{d^{2} c x}-\frac {5 b \,\mathrm {arccosh}\left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{3 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {5 b \,\mathrm {arccosh}\left (c x \right )}{3 d^{2} \left (c^{2} x^{2}-1\right ) c x}-\frac {b \sqrt {c x +1}\, \sqrt {c x -1}}{6 d^{2} \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}+\frac {b \,\mathrm {arccosh}\left (c x \right )}{3 d^{2} \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {13 b \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{2}}+\frac {5 b \dilog \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {5 b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {5 b \dilog \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}\right )\)	\(335\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^2,x,method=_RETURNVERBOSE)

[Out]

c^3*(-1/4*a/d^2/(c*x-1)-5/4*a/d^2*ln(c*x-1)-1/4*a/d^2/(c*x+1)+5/4*a/d^2*ln(c*x+1)-1/3*a/d^2/c^3/x^3-2*a/d^2/c/
x-5/2*b/d^2/(c^2*x^2-1)*arccosh(c*x)*c*x-1/3*b/d^2/(c^2*x^2-1)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+5/3*b/d^2/(c^2*x^2-
1)/c/x*arccosh(c*x)-1/6*b/d^2/(c^2*x^2-1)/c^2/x^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)+1/3*b/d^2/(c^2*x^2-1)/c^3/x^3*ar
ccosh(c*x)+13/3*b/d^2*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+5/2*b/d^2*dilog(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2
))+5/2*b/d^2*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+5/2*b/d^2*dilog(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2
)))

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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((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^2,x, algorithm="maxima")

[Out]

1/12*(15*c^3*log(c*x + 1)/d^2 - 15*c^3*log(c*x - 1)/d^2 - 2*(15*c^4*x^4 - 10*c^2*x^2 - 2)/(c^2*d^2*x^5 - d^2*x
^3))*a + 1/192*(8640*c^7*integrate(1/24*x^5*log(c*x - 1)/(c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2), x) - 120*c^6
*(2*x/(c^4*d^2*x^2 - c^2*d^2) + log(c*x + 1)/(c^3*d^2) - log(c*x - 1)/(c^3*d^2)) - 2880*c^6*integrate(1/24*x^4
*log(c*x - 1)/(c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2), x) + 45*(c*(2/(c^4*d^2*x - c^3*d^2) - log(c*x + 1)/(c^3
*d^2) + log(c*x - 1)/(c^3*d^2)) + 4*log(c*x - 1)/(c^4*d^2*x^2 - c^2*d^2))*c^5 + 80*c^4*(2*x/(c^2*d^2*x^2 - d^2
) - log(c*x + 1)/(c*d^2) + log(c*x - 1)/(c*d^2)) + 2880*c^4*integrate(1/24*x^2*log(c*x - 1)/(c^4*d^2*x^6 - 2*c
^2*d^2*x^4 + d^2*x^2), x) + 16*c^2*(2*(3*c^2*x^2 - 2)/(c^2*d^2*x^3 - d^2*x) - 3*c*log(c*x + 1)/d^2 + 3*c*log(c
*x - 1)/d^2) - 4*(15*(c^5*x^5 - c^3*x^3)*log(c*x + 1)^2 + 30*(c^5*x^5 - c^3*x^3)*log(c*x + 1)*log(c*x - 1) + 4
*(30*c^4*x^4 - 20*c^2*x^2 - 15*(c^5*x^5 - c^3*x^3)*log(c*x + 1) + 15*(c^5*x^5 - c^3*x^3)*log(c*x - 1) - 4)*log
(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*d^2*x^5 - d^2*x^3) + 192*integrate(-1/12*(30*c^5*x^4 - 20*c^3*x^2 -
15*(c^6*x^5 - c^4*x^3)*log(c*x + 1) + 15*(c^6*x^5 - c^4*x^3)*log(c*x - 1) - 4*c)/(c^5*d^2*x^8 - 2*c^3*d^2*x^6
+ c*d^2*x^4 + (c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2*x^3)*sqrt(c*x + 1)*sqrt(c*x - 1)), x))*b

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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((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^2,x, algorithm="fricas")

[Out]

integral((b*arccosh(c*x) + a)/(c^4*d^2*x^8 - 2*c^2*d^2*x^6 + d^2*x^4), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx}{d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))/x**4/(-c**2*d*x**2+d)**2,x)

[Out]

(Integral(a/(c**4*x**8 - 2*c**2*x**6 + x**4), x) + Integral(b*acosh(c*x)/(c**4*x**8 - 2*c**2*x**6 + x**4), x))
/d**2

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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((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^2,x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)/((c^2*d*x^2 - d)^2*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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