3.1.86 \(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x))}{x^5} \, dx\) [86]

Optimal. Leaf size=321 \[ -\frac {b c d \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{8 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac {3 c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {ArcTan}\left (e^{\cosh ^{-1}(c x)}\right )}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 i b c^4 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 i b c^4 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

-1/4*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^4+3/8*c^2*d*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^2-1/12*b*
c*d*(-c^2*d*x^2+d)^(1/2)/x^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/8*b*c^3*d*(-c^2*d*x^2+d)^(1/2)/x/(c*x-1)^(1/2)/(c*x
+1)^(1/2)-3/4*c^4*d*(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)+3/8*I*b*c^4*d*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)-3/8*I*b*c^4*d*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 = 0.31, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5928, 5924, 30, 5947, 4265, 2317, 2438, 74, 14} \begin {gather*} -\frac {3 c^4 d \sqrt {d-c^2 d x^2} \text {ArcTan}\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 {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac {b c d \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 i b c^4 d \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 {3 i b c^4 d \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 {5 b c^3 d \sqrt {d-c^2 d x^2}}{8 x \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^5,x]

[Out]

-1/12*(b*c*d*Sqrt[d - c^2*d*x^2])/(x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c^3*d*Sqrt[d - c^2*d*x^2])/(8*x*Sq
rt[-1 + c*x]*Sqrt[1 + c*x]) + (3*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*x^2) - ((d - c^2*d*x^2)^(3
/2)*(a + b*ArcCosh[c*x]))/(4*x^4) - (3*c^4*d*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]) + (((3*I)/8)*b*c^4*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(Sq
rt[-1 + c*x]*Sqrt[1 + c*x]) - (((3*I)/8)*b*c^4*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, I*E^ArcCosh[c*x]])/(Sqrt[-1 +
c*x]*Sqrt[1 + c*x])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

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

Rule 74

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

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 4265

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 5924

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

Rule 5928

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

Rule 5947

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[(1/c^(m + 1))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], S
ubst[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*d
1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^5} \, dx &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^5} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {-1+c^2 x^2}{x^4} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{x^3} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {1}{x^4}+\frac {c^2}{x^2}\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b c^3 d \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 (3 c^4 d \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 d \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{8 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac {\left (3 c^4 d \sqrt {d-c^2 d x^2}\right ) \text {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 d \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{8 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac {3 c^4 d \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 (3 i b c^4 d \sqrt {d-c^2 d x^2}\right ) \text {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 (3 i b c^4 d \sqrt {d-c^2 d x^2}\right ) \text {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 d \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{8 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac {3 c^4 d \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 (3 i b c^4 d \sqrt {d-c^2 d x^2}\right ) \text {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 (3 i b c^4 d \sqrt {d-c^2 d x^2}\right ) \text {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 d \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{8 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac {3 c^4 d \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 {3 i b c^4 d \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 {3 i b c^4 d \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 = 0.82, size = 574, normalized size = 1.79 \begin {gather*} \frac {-2 b c d^2 x+2 b c^2 d^2 x^2+15 b c^3 d^2 x^3-15 b c^4 d^2 x^4-6 a d^2 \sqrt {\frac {-1+c x}{1+c x}}+21 a c^2 d^2 x^2 \sqrt {\frac {-1+c x}{1+c x}}-15 a c^4 d^2 x^4 \sqrt {\frac {-1+c x}{1+c x}}-6 b d^2 \sqrt {\frac {-1+c x}{1+c x}} \cosh ^{-1}(c x)+21 b c^2 d^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \cosh ^{-1}(c x)-15 b c^4 d^2 x^4 \sqrt {\frac {-1+c x}{1+c x}} \cosh ^{-1}(c x)+9 i b c^4 d^2 x^4 \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-9 i b c^5 d^2 x^5 \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-9 i b c^4 d^2 x^4 \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+9 i b c^5 d^2 x^5 \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+9 a c^4 d^{3/2} x^4 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \log (x)-9 a c^4 d^{3/2} x^4 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )-9 i b c^4 d^2 x^4 (-1+c x) \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )+9 i b c^4 d^2 x^4 (-1+c x) \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )}{24 x^4 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^5,x]

[Out]

(-2*b*c*d^2*x + 2*b*c^2*d^2*x^2 + 15*b*c^3*d^2*x^3 - 15*b*c^4*d^2*x^4 - 6*a*d^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 2
1*a*c^2*d^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] - 15*a*c^4*d^2*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - 6*b*d^2*Sqrt[(-1 +
c*x)/(1 + c*x)]*ArcCosh[c*x] + 21*b*c^2*d^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] - 15*b*c^4*d^2*x^4*Sqr
t[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + (9*I)*b*c^4*d^2*x^4*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - (9*I)*b*c^
5*d^2*x^5*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - (9*I)*b*c^4*d^2*x^4*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]]
+ (9*I)*b*c^5*d^2*x^5*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + 9*a*c^4*d^(3/2)*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*
Sqrt[d - c^2*d*x^2]*Log[x] - 9*a*c^4*d^(3/2)*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*Log[d + Sqrt[d
]*Sqrt[d - c^2*d*x^2]] - (9*I)*b*c^4*d^2*x^4*(-1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (9*I)*b*c^4*d^2*x^4*
(-1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]])/(24*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])

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Maple [A]
time = 5.77, size = 570, normalized size = 1.78

method	result	size
default	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{4 d \,x^{4}}+\frac {a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{8 d \,x^{2}}+\frac {a \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{8}-\frac {3 a \,c^{4} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right ) d^{\frac {3}{2}}}{8}+\frac {3 a \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}\, d}{8}+\frac {5 b d \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) c^{4}}{8 \left (c x +1\right ) \left (c x -1\right )}+\frac {5 b d \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3}}{8 \sqrt {c x +1}\, x \sqrt {c x -1}}-\frac {7 b d \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) c^{2}}{8 \left (c x +1\right ) x^{2} \left (c x -1\right )}-\frac {b d \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c}{12 \sqrt {c x +1}\, x^{3} \sqrt {c x -1}}+\frac {b d \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{4 \left (c x +1\right ) x^{4} \left (c x -1\right )}+\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4} d}{8 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4} d}{8 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4} d}{8 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4} d}{8 \sqrt {c x -1}\, \sqrt {c x +1}}\)	\(570\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*a/d/x^4*(-c^2*d*x^2+d)^(5/2)+1/8*a*c^2/d/x^2*(-c^2*d*x^2+d)^(5/2)+1/8*a*c^4*(-c^2*d*x^2+d)^(3/2)-3/8*a*c^
4*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)*d^(3/2)+3/8*a*c^4*(-c^2*d*x^2+d)^(1/2)*d+5/8*b*d*(-d*(c^2*x^2-1))
^(1/2)/(c*x+1)/(c*x-1)*arccosh(c*x)*c^4+5/8*b*d*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)^(1/2)/x/(c*x-1)^(1/2)*c^3-7/8*b
*d*(-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*(-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*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)/x^4/(c*x-1)*arccosh(c*x)+3/8*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*d-3/8*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*d+3/8*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*d-3/8*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*d

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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((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^5,x, algorithm="maxima")

[Out]

-1/8*(3*c^4*d^(3/2)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x)) - (-c^2*d*x^2 + d)^(3/2)*c^4 - 3*s
qrt(-c^2*d*x^2 + d)*c^4*d - (-c^2*d*x^2 + d)^(5/2)*c^2/(d*x^2) + 2*(-c^2*d*x^2 + d)^(5/2)/(d*x^4))*a + b*integ
rate((-c^2*d*x^2 + d)^(3/2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x^5, 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((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))/x**5,x)

[Out]

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acosh(c*x))/x**5, x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^5,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^5,x)

[Out]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^5, x)

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