3.1.75 \(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x))}{x^4} \, dx\) [75]

Optimal. Leaf size=203 \[ -\frac {b c d \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {c^3 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c^3 d \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

-1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^3+c^2*d*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x-1/6*b*c*d*(-c
^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*c^3*d*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/(c*x-1
)^(1/2)/(c*x+1)^(1/2)-4/3*b*c^3*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 = 0.21, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5928, 5924, 29, 5893, 74, 14} \begin {gather*} \frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {c^3 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {4 b c^3 d \log (x) \sqrt {d-c^2 d x^2}}{3 \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^4,x]

[Out]

-1/6*(b*c*d*Sqrt[d - c^2*d*x^2])/(x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos
h[c*x]))/x - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(3*x^3) - (c^3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[
c*x])^2)/(2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*b*c^3*d*Sqrt[d - c^2*d*x^2]*Log[x])/(3*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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

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]

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

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Mathematica [A]
time = 0.52, size = 259, normalized size = 1.28 \begin {gather*} \frac {-2 b d^2 \sqrt {\frac {-1+c x}{1+c x}} \left (1-5 c^2 x^2+4 c^4 x^4\right ) \cosh ^{-1}(c x)+3 b c^3 d^2 x^3 (-1+c x) \cosh ^{-1}(c x)^2-6 a c^3 d^{3/2} x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+d^2 \left (b c x (-1+c x)-2 a \sqrt {\frac {-1+c x}{1+c x}} \left (1-5 c^2 x^2+4 c^4 x^4\right )+8 b c^3 x^3 (-1+c x) \log (c x)\right )}{6 x^3 \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^4,x]

[Out]

(-2*b*d^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 - 5*c^2*x^2 + 4*c^4*x^4)*ArcCosh[c*x] + 3*b*c^3*d^2*x^3*(-1 + c*x)*Arc
Cosh[c*x]^2 - 6*a*c^3*d^(3/2)*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^
2])/(Sqrt[d]*(-1 + c^2*x^2))] + d^2*(b*c*x*(-1 + c*x) - 2*a*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 - 5*c^2*x^2 + 4*c^4*
x^4) + 8*b*c^3*x^3*(-1 + c*x)*Log[c*x]))/(6*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1180\) vs. \(2(173)=346\).
time = 5.97, size = 1181, normalized size = 5.82

method	result	size
default	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d \,x^{3}}+\frac {2 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d x}+\frac {2 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+a \,c^{4} d x \sqrt {-c^{2} d \,x^{2}+d}+\frac {a \,c^{4} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} c^{3} d}{2 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {8 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) c^{3} d}{3 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {32 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{4} \mathrm {arccosh}\left (c x \right ) c^{7}}{\left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {32 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{5} \mathrm {arccosh}\left (c x \right ) c^{8}}{\left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c x +1\right ) \left (c x -1\right )}-\frac {8 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{3} c^{6}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right )}+\frac {8 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{5} c^{8}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c x +1\right ) \left (c x -1\right )}+\frac {12 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{2} \mathrm {arccosh}\left (c x \right ) c^{5}}{\left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {52 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{3} \mathrm {arccosh}\left (c x \right ) c^{6}}{\left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c x +1\right ) \left (c x -1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d x \,c^{4}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right )}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{2} c^{5}}{\left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {10 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{3} c^{6}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c x +1\right ) \left (c x -1\right )}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\mathrm {arccosh}\left (c x \right ) c^{3}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {73 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d x \,\mathrm {arccosh}\left (c x \right ) c^{4}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c x +1\right ) \left (c x -1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{3}}{2 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d x \,c^{4}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c x +1\right ) \left (c x -1\right )}-\frac {14 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\mathrm {arccosh}\left (c x \right ) c^{2}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) x \left (c x +1\right ) \left (c x -1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c}{6 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) x^{2} \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\mathrm {arccosh}\left (c x \right )}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) x^{3} \left (c x +1\right ) \left (c x -1\right )}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c^{3} d}{3 \sqrt {c x -1}\, \sqrt {c x +1}}\)	\(1181\)

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^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*a/d/x^3*(-c^2*d*x^2+d)^(5/2)+2/3*a*c^2/d/x*(-c^2*d*x^2+d)^(5/2)+2/3*a*c^4*x*(-c^2*d*x^2+d)^(3/2)+a*c^4*d*
x*(-c^2*d*x^2+d)^(1/2)+a*c^4*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/2*b*(-d*(c^2*x^2
-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)^2*c^3*d+8/3*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)
^(1/2)*arccosh(c*x)*c^3*d-32*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/
2)*arccosh(c*x)*c^7+32*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^5/(c*x+1)/(c*x-1)*arccosh(c*x)*c^
8-8/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^3*c^6+8/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9
*c^2*x^2+1)*x^5/(c*x+1)/(c*x-1)*c^8+12*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^2/(c*x+1)^(1/2)/(
c*x-1)^(1/2)*arccosh(c*x)*c^5-52*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^3/(c*x+1)/(c*x-1)*arcco
sh(c*x)*c^6+2/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x*c^4-4*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4
*x^4-9*c^2*x^2+1)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^5-10/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)
*x^3/(c*x+1)/(c*x-1)*c^6-4/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*a
rccosh(c*x)*c^3+73/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^4+3/
2*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3+2/3*b*(-d*(c^2*x^2-1))^(
1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x/(c*x+1)/(c*x-1)*c^4-14/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)
/x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^2-1/6*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/x^2/(c*x+1)^(1/2)/
(c*x-1)^(1/2)*c+1/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/x^3/(c*x+1)/(c*x-1)*arccosh(c*x)-4/3*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)*c^3*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^4,x, algorithm="maxima")

[Out]

1/3*(3*sqrt(-c^2*d*x^2 + d)*c^4*d*x + 3*c^3*d^(3/2)*arcsin(c*x) + 2*(-c^2*d*x^2 + d)^(3/2)*c^2/x - (-c^2*d*x^2
+ d)^(5/2)/(d*x^3))*a + b*integrate((-c^2*d*x^2 + d)^(3/2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x^4, 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^4,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^4, 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^{4}}\, 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**4,x)

[Out]

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acosh(c*x))/x**4, 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^4,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^4} \,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^4,x)

[Out]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^4, x)

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