3.76 \(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x))}{x^6} \, dx\)

Optimal. Leaf size=166 \[ -\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{5 d x^5}-\frac {b c d \sqrt {d-c^2 d x^2}}{20 x^4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^5 d \log (x) \sqrt {d-c^2 d x^2}}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d \sqrt {d-c^2 d x^2}}{5 x^2 \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

-1/5*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^5-1/20*b*c*d*(-c^2*d*x^2+d)^(1/2)/x^4/(c*x-1)^(1/2)/(c*x+1)^(
1/2)+1/5*b*c^3*d*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/5*b*c^5*d*ln(x)*(-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 = 0.35, antiderivative size = 179, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5798, 5724, 266, 43} \[ -\frac {d (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac {b c^3 d \sqrt {d-c^2 d x^2}}{5 x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d \sqrt {d-c^2 d x^2}}{20 x^4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^5 d \log (x) \sqrt {d-c^2 d x^2}}{5 \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^6,x]

[Out]

-(b*c*d*Sqrt[d - c^2*d*x^2])/(20*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*Sqrt[d - c^2*d*x^2])/(5*x^2*Sqrt
[-1 + c*x]*Sqrt[1 + c*x]) - (d*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(5*x^5) + (b*
c^5*d*Sqrt[d - c^2*d*x^2]*Log[x])/(5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5724

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)/(d
1*d2*f*(m + 1)), 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, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2,
0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1] && IntegerQ[p + 1/2]

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 {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^6} \, 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^6} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-1+c^2 x^2\right )^2}{x^5} \, dx}{5 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+c^2 x\right )^2}{x^3} \, dx,x,x^2\right )}{10 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^3}-\frac {2 c^2}{x^2}+\frac {c^4}{x}\right ) \, dx,x,x^2\right )}{10 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d \sqrt {d-c^2 d x^2}}{20 x^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d \sqrt {d-c^2 d x^2}}{5 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac {b c^5 d \sqrt {d-c^2 d x^2} \log (x)}{5 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 94, normalized size = 0.57 \[ -\frac {d \sqrt {d-c^2 d x^2} \left (4 (c x-1)^{5/2} (c x+1)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+b c x \left (-4 c^4 x^4 \log (x)-4 c^2 x^2+1\right )\right )}{20 x^5 \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^6,x]

[Out]

-1/20*(d*Sqrt[d - c^2*d*x^2]*(4*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + b*c*x*(1 - 4*c^2*x^2 -
4*c^4*x^4*Log[x])))/(x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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fricas [A]  time = 0.62, size = 572, normalized size = 3.45 \[ \left [-\frac {4 \, {\left (b c^{6} d x^{6} - 3 \, b c^{4} d x^{4} + 3 \, b c^{2} d x^{2} - b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, {\left (b c^{7} d x^{7} - b c^{5} d x^{5}\right )} \sqrt {-d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} - \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) - {\left (4 \, b c^{3} d x^{3} - {\left (4 \, b c^{3} - b c\right )} d x^{5} - b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 4 \, {\left (a c^{6} d x^{6} - 3 \, a c^{4} d x^{4} + 3 \, a c^{2} d x^{2} - a d\right )} \sqrt {-c^{2} d x^{2} + d}}{20 \, {\left (c^{2} x^{7} - x^{5}\right )}}, \frac {4 \, {\left (b c^{7} d x^{7} - b c^{5} d x^{5}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} + 1\right )} \sqrt {d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) - 4 \, {\left (b c^{6} d x^{6} - 3 \, b c^{4} d x^{4} + 3 \, b c^{2} d x^{2} - b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (4 \, b c^{3} d x^{3} - {\left (4 \, b c^{3} - b c\right )} d x^{5} - b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 4 \, {\left (a c^{6} d x^{6} - 3 \, a c^{4} d x^{4} + 3 \, a c^{2} d x^{2} - a d\right )} \sqrt {-c^{2} d x^{2} + d}}{20 \, {\left (c^{2} x^{7} - x^{5}\right )}}\right ] \]

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^6,x, algorithm="fricas")

[Out]

[-1/20*(4*(b*c^6*d*x^6 - 3*b*c^4*d*x^4 + 3*b*c^2*d*x^2 - b*d)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)
) - 2*(b*c^7*d*x^7 - b*c^5*d*x^5)*sqrt(-d)*log((c^2*d*x^6 + c^2*d*x^2 - d*x^4 - sqrt(-c^2*d*x^2 + d)*sqrt(c^2*
x^2 - 1)*(x^4 - 1)*sqrt(-d) - d)/(c^2*x^4 - x^2)) - (4*b*c^3*d*x^3 - (4*b*c^3 - b*c)*d*x^5 - b*c*d*x)*sqrt(-c^
2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 4*(a*c^6*d*x^6 - 3*a*c^4*d*x^4 + 3*a*c^2*d*x^2 - a*d)*sqrt(-c^2*d*x^2 + d))/(
c^2*x^7 - x^5), 1/20*(4*(b*c^7*d*x^7 - b*c^5*d*x^5)*sqrt(d)*arctan(sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x^2
+ 1)*sqrt(d)/(c^2*d*x^4 - (c^2 + 1)*d*x^2 + d)) - 4*(b*c^6*d*x^6 - 3*b*c^4*d*x^4 + 3*b*c^2*d*x^2 - b*d)*sqrt(
-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) + (4*b*c^3*d*x^3 - (4*b*c^3 - b*c)*d*x^5 - b*c*d*x)*sqrt(-c^2*d*x
^2 + d)*sqrt(c^2*x^2 - 1) - 4*(a*c^6*d*x^6 - 3*a*c^4*d*x^4 + 3*a*c^2*d*x^2 - a*d)*sqrt(-c^2*d*x^2 + d))/(c^2*x
^7 - x^5)]

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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((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^6,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 [B]  time = 0.88, size = 2171, normalized size = 13.08 \[ \text {result too large to display} \]

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^6,x)

[Out]

3/10*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^3*c^8-1/20*b*(-d*(c^2*x^2-1))^
(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x*c^6+1/5*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x
^6+10*c^4*x^4-5*c^2*x^2+1)*x^7*c^12-9/20*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2
+1)*x^5*c^10+b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^8/(c*x+1)^(1/2)/(c*x-1
)^(1/2)*arccosh(c*x)*c^13-2*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^6/(c*x+
1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^11+2*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x
^2+1)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^9-b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4
*x^4-5*c^2*x^2+1)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^7-56/5*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-
10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^6+28/5*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x
^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^4-8/5*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8
*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^2-b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8
*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^9/(c*x+1)/(c*x-1)*arccosh(c*x)*c^14+1/5*b*(-d*(c^2*x^2-1))^(1/2)/(c*
x-1)^(1/2)/(c*x+1)^(1/2)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*d*c^5-2/5*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^
(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*d*c^5-3/2*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x
^2+1)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^5-1/5*a/d/x^5*(-c^2*d*x^2+d)^(5/2)+b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-1
0*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^11-9/4*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x
^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^9+5/2*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^
8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^7+9/20*b*(-d*(c^2*x^2-1))^(1/2)*d/(
5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3-1/20*b*(-d*(c^2*x^2-1))^(1/2)
*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c+1/5*b*(-d*(c^2*x^2-1))^(1/2
)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^5+7/20*b*(-d*(c^2
*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^3/(c*x+1)/(c*x-1)*c^8-1/20*b*(-d*(c^2*x^2-1))
^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x/(c*x+1)/(c*x-1)*c^6+1/5*b*(-d*(c^2*x^2-1))^(1/2)*d/(5
*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/x^5/(c*x+1)/(c*x-1)*arccosh(c*x)-1/5*b*(-d*(c^2*x^2-1))^(1/2)*d/(5
*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^9/(c*x+1)/(c*x-1)*c^14+13/20*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x
^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^7/(c*x+1)/(c*x-1)*c^12-3/4*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^
6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^5/(c*x+1)/(c*x-1)*c^10+5*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c
^4*x^4-5*c^2*x^2+1)*x^7/(c*x+1)/(c*x-1)*arccosh(c*x)*c^12-11*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+
10*c^4*x^4-5*c^2*x^2+1)*x^5/(c*x+1)/(c*x-1)*arccosh(c*x)*c^10+14*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*
x^6+10*c^4*x^4-5*c^2*x^2+1)*x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^8

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maxima [C]  time = 0.46, size = 189, normalized size = 1.14 \[ -\frac {{\left (2 \, c^{6} d^{3} \sqrt {-\frac {1}{c^{4} d}} \log \left (x^{2} - \frac {1}{c^{2}}\right ) + 2 i \, \left (-1\right )^{-2 \, c^{2} d x^{2} + 2 \, d} c^{4} d^{\frac {5}{2}} \log \left (-2 \, c^{2} d + \frac {2 \, d}{x^{2}}\right ) + \frac {3 \, \sqrt {-c^{4} d x^{4} + 2 \, c^{2} d x^{2} - d} c^{2} d^{2}}{x^{2}} - \frac {\sqrt {-c^{4} d x^{4} + 2 \, c^{2} d x^{2} - d} d^{2}}{x^{4}}\right )} b c}{20 \, d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b \operatorname {arcosh}\left (c x\right )}{5 \, d x^{5}} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a}{5 \, d x^{5}} \]

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^6,x, algorithm="maxima")

[Out]

-1/20*(2*c^6*d^3*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2) + 2*I*(-1)^(-2*c^2*d*x^2 + 2*d)*c^4*d^(5/2)*log(-2*c^2*d +
2*d/x^2) + 3*sqrt(-c^4*d*x^4 + 2*c^2*d*x^2 - d)*c^2*d^2/x^2 - sqrt(-c^4*d*x^4 + 2*c^2*d*x^2 - d)*d^2/x^4)*b*c/
d - 1/5*(-c^2*d*x^2 + d)^(5/2)*b*arccosh(c*x)/(d*x^5) - 1/5*(-c^2*d*x^2 + d)^(5/2)*a/(d*x^5)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^6} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^6,x)

[Out]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^6, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{x^{6}}\, dx \]

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**6,x)

[Out]

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acosh(c*x))/x**6, x)

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