3.70 \(\int \frac {\sqrt {d-c^2 d x^2} (a+b \cosh ^{-1}(c x))}{x^5} \, dx\)

Optimal. Leaf size=315 \[ \frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac {c^4 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {i b c^4 \sqrt {d-c^2 d x^2} \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {i b c^4 \sqrt {d-c^2 d x^2} \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{8 x \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

-1/4*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^4+1/8*c^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^2-1/12*b*c*
(-c^2*d*x^2+d)^(1/2)/x^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/8*b*c^3*(-c^2*d*x^2+d)^(1/2)/x/(c*x-1)^(1/2)/(c*x+1)^(1
/2)+1/4*c^4*(a+b*arccosh(c*x))*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x
+1)^(1/2)-1/8*I*b*c^4*polylog(2,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+
1)^(1/2)+1/8*I*b*c^4*polylog(2,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)
^(1/2)

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Rubi [A]  time = 0.74, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {5798, 5738, 30, 5748, 5761, 4180, 2279, 2391} \[ -\frac {i b c^4 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {i b c^4 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac {c^4 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{8 x \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^5,x]

[Out]

-(b*c*Sqrt[d - c^2*d*x^2])/(12*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*Sqrt[d - c^2*d*x^2])/(8*x*Sqrt[-1 +
c*x]*Sqrt[1 + c*x]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(4*x^4) + (c^2*Sqrt[d - c^2*d*x^2]*(a + b*Arc
Cosh[c*x]))/(8*x^2) + (c^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(4*Sqrt[-1 + c*x]*
Sqrt[1 + c*x]) - ((I/8)*b*c^4*Sqrt[d - c^2*d*x^2]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*
x]) + ((I/8)*b*c^4*Sqrt[d - c^2*d*x^2]*PolyLog[2, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 4180

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

Rule 5738

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*
(x_)], x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(f*(m + 1)), x
] + (-Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)
*(a + b*ArcCosh[c*x])^(n - 1), x], x] - Dist[(c^2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f^2*(m + 1)*Sqrt[1 + c*x]*
Sqrt[-1 + c*x]), Int[((f*x)^(m + 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x]) /; FreeQ[{
a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[m, -1]

Rule 5748

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d1*
d2*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a
+ b*ArcCosh[c*x])^n, x], x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p
])/(f*(m + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b
*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 +
c*d2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m] && IntegerQ[p + 1/2]

Rule 5761

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[1/(c^(m + 1)*Sqrt[-(d1*d2)]), Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /
; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[n, 0] && GtQ[d1, 0] &&
LtQ[d2, 0] && IntegerQ[m]

Rule 5798

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

Rubi steps

\begin {align*} \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x^5} \, dx &=\frac {\sqrt {d-c^2 d x^2} \int \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{x^5} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{x^4} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {\left (b c^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{x^2} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c^4 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{8 x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}+\frac {\left (c^4 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{8 x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}+\frac {c^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (i b c^4 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (i b c^4 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{8 x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}+\frac {c^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (i b c^4 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (i b c^4 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{8 x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}+\frac {c^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b c^4 \sqrt {d-c^2 d x^2} \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {i b c^4 \sqrt {d-c^2 d x^2} \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]  time = 1.05, size = 290, normalized size = 0.92 \[ \frac {1}{24} \left (-3 a c^4 \sqrt {d} \log (x)+\frac {3 a \left (c^2 x^2-2\right ) \sqrt {d-c^2 d x^2}}{x^4}+3 a c^4 \sqrt {d} \log \left (\sqrt {d} \sqrt {d-c^2 d x^2}+d\right )+\frac {b \sqrt {d-c^2 d x^2} \left (-3 i c^4 x^4 \left (\text {Li}_2\left (-i e^{-\cosh ^{-1}(c x)}\right )-\text {Li}_2\left (i e^{-\cosh ^{-1}(c x)}\right )\right )-3 i c^4 x^4 \cosh ^{-1}(c x) \left (\log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(c x)}\right )\right )+3 c^3 x^3+3 c^2 x^2 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-2 c x-6 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)\right )}{x^4 \sqrt {\frac {c x-1}{c x+1}} (c x+1)}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^5,x]

[Out]

((3*a*(-2 + c^2*x^2)*Sqrt[d - c^2*d*x^2])/x^4 - 3*a*c^4*Sqrt[d]*Log[x] + 3*a*c^4*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[
d - c^2*d*x^2]] + (b*Sqrt[d - c^2*d*x^2]*(-2*c*x + 3*c^3*x^3 - 6*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[
c*x] + 3*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - (3*I)*c^4*x^4*ArcCosh[c*x]*(Log[1 - I/E^A
rcCosh[c*x]] - Log[1 + I/E^ArcCosh[c*x]]) - (3*I)*c^4*x^4*(PolyLog[2, (-I)/E^ArcCosh[c*x]] - PolyLog[2, I/E^Ar
cCosh[c*x]])))/(x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)))/24

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fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^5,x, algorithm="fricas")

[Out]

integral(sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/x^5, x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^5,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.75, size = 541, normalized size = 1.72 \[ -\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 d \,x^{4}}-\frac {a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{8 d \,x^{2}}+\frac {a \,c^{4} \sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{8}-\frac {a \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) c^{4}}{8 \left (c x +1\right ) \left (c x -1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3}}{8 \sqrt {c x +1}\, x \sqrt {c x -1}}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) c^{2}}{8 \left (c x +1\right ) x^{2} \left (c x -1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c}{12 \sqrt {c x +1}\, x^{3} \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{4 \left (c x +1\right ) x^{4} \left (c x -1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4}}{8 \sqrt {c x -1}\, \sqrt {c x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^5,x)

[Out]

-1/4*a/d/x^4*(-c^2*d*x^2+d)^(3/2)-1/8*a*c^2/d/x^2*(-c^2*d*x^2+d)^(3/2)+1/8*a*c^4*d^(1/2)*ln((2*d+2*d^(1/2)*(-c
^2*d*x^2+d)^(1/2))/x)-1/8*a*c^4*(-c^2*d*x^2+d)^(1/2)+1/8*b*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)/(c*x-1)*arccosh(c*x)
*c^4+1/8*b*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)^(1/2)/x/(c*x-1)^(1/2)*c^3-3/8*b*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)/x^2/(
c*x-1)*arccosh(c*x)*c^2-1/12*b*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)^(1/2)/x^3/(c*x-1)^(1/2)*c+1/4*b*(-d*(c^2*x^2-1))
^(1/2)/(c*x+1)/x^4/(c*x-1)*arccosh(c*x)-1/8*I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x
)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^4+1/8*I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arc
cosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^4-1/8*I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(
1/2)*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^4+1/8*I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/
2)*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^4

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, {\left (c^{4} \sqrt {d} \log \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {d}}{{\left | x \right |}} + \frac {2 \, d}{{\left | x \right |}}\right ) - \sqrt {-c^{2} d x^{2} + d} c^{4} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2}}{d x^{2}} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{d x^{4}}\right )} a + b \int \frac {\sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^5,x, algorithm="maxima")

[Out]

1/8*(c^4*sqrt(d)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x)) - sqrt(-c^2*d*x^2 + d)*c^4 - (-c^2*d*
x^2 + d)^(3/2)*c^2/(d*x^2) - 2*(-c^2*d*x^2 + d)^(3/2)/(d*x^4))*a + b*integrate(sqrt(-c^2*d*x^2 + d)*log(c*x +
sqrt(c*x + 1)*sqrt(c*x - 1))/x^5, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{x^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(1/2))/x^5,x)

[Out]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(1/2))/x^5, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))*(-c**2*d*x**2+d)**(1/2)/x**5,x)

[Out]

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*acosh(c*x))/x**5, x)

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