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

Optimal. Leaf size=250 \[ -\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {d-c^2 d x^2}}{6 d^2 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c^3 \log (x) \sqrt {d-c^2 d x^2}}{3 d^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{2 d^2 \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

1/3*(-a-b*arccosh(c*x))/d/x^3/(-c^2*d*x^2+d)^(1/2)-4/3*c^2*(a+b*arccosh(c*x))/d/x/(-c^2*d*x^2+d)^(1/2)+8/3*c^4
*x*(a+b*arccosh(c*x))/d/(-c^2*d*x^2+d)^(1/2)-1/6*b*c*(-c^2*d*x^2+d)^(1/2)/d^2/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+
5/3*b*c^3*ln(x)*(-c^2*d*x^2+d)^(1/2)/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/2*b*c^3*ln(-c^2*x^2+1)*(-c^2*d*x^2+d)^(
1/2)/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)

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Rubi [A]  time = 0.46, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {5798, 103, 12, 39, 5733, 1251, 893} \[ \frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 d x^2 \sqrt {d-c^2 d x^2}}-\frac {5 b c^3 \sqrt {c x-1} \sqrt {c x+1} \log (x)}{3 d \sqrt {d-c^2 d x^2}}-\frac {b c^3 \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{2 d \sqrt {d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^(3/2)),x]

[Out]

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*d*x^2*Sqrt[d - c^2*d*x^2]) - (a + b*ArcCosh[c*x])/(3*d*x^3*Sqrt[d - c^2*
d*x^2]) - (4*c^2*(a + b*ArcCosh[c*x]))/(3*d*x*Sqrt[d - c^2*d*x^2]) + (8*c^4*x*(a + b*ArcCosh[c*x]))/(3*d*Sqrt[
d - c^2*d*x^2]) - (5*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[x])/(3*d*Sqrt[d - c^2*d*x^2]) - (b*c^3*Sqrt[-1 + c
*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(2*d*Sqrt[d - c^2*d*x^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 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rule 103

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] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 5733

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sym
bol] :> With[{u = IntHide[x^m*(1 + c*x)^p*(-1 + c*x)^p, x]}, Dist[(-(d1*d2))^p*(a + b*ArcCosh[c*x]), u, x] - D
ist[b*c*(-(d1*d2))^p, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d
1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2, 0] || IL
tQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d1, 0] && LtQ[d2, 0]

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 {a+b \cosh ^{-1}(c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x^4 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {-1-4 c^2 x^2+8 c^4 x^4}{3 x^3 \left (1-c^2 x^2\right )} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {-1-4 c^2 x^2+8 c^4 x^4}{x^3 \left (1-c^2 x^2\right )} \, dx}{3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {-1-4 c^2 x+8 c^4 x^2}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )}{6 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{x^2}-\frac {5 c^2}{x}-\frac {3 c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )}{6 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2 \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {5 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \log (x)}{3 d \sqrt {d-c^2 d x^2}}-\frac {b c^3 \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{2 d \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 161, normalized size = 0.64 \[ \frac {16 a c^4 x^4-8 a c^2 x^2-2 a-10 b c^3 x^3 \sqrt {c x-1} \sqrt {c x+1} \log (x)+2 b \left (8 c^4 x^4-4 c^2 x^2-1\right ) \cosh ^{-1}(c x)-3 b c^3 x^3 \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )+b c x \sqrt {c x-1} \sqrt {c x+1}}{6 d x^3 \sqrt {d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^(3/2)),x]

[Out]

(-2*a - 8*a*c^2*x^2 + 16*a*c^4*x^4 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*b*(-1 - 4*c^2*x^2 + 8*c^4*x^4)*Arc
Cosh[c*x] - 10*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[x] - 3*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1
- c^2*x^2])/(6*d*x^3*Sqrt[d - c^2*d*x^2])

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fricas [F]  time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{c^{4} d^{2} x^{8} - 2 \, c^{2} d^{2} x^{6} + d^{2} x^{4}}, x\right ) \]

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)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(-c^2*d*x^2 + d)*(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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]

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)^(3/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.66, size = 1050, normalized size = 4.20 \[ -\frac {a}{3 d \,x^{3} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {4 a \,c^{2}}{3 d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {8 a \,c^{4} x}{3 d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {16 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) c^{3}}{3 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {32 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5} \left (c x +1\right ) \left (c x -1\right ) c^{8}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}-\frac {32 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{7} c^{10}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}-\frac {16 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \left (c x +1\right ) \left (c x -1\right ) c^{6}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}+\frac {16 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5} c^{8}}{d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}+\frac {64 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2} \mathrm {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}-\frac {64 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \mathrm {arccosh}\left (c x \right ) c^{6}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \left (c x +1\right ) \left (c x -1\right ) c^{4}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} c^{6}}{d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}+\frac {8 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}+\frac {8 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \,\mathrm {arccosh}\left (c x \right ) c^{4}}{d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \,c^{4}}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right )}+\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) c^{2}}{d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right ) x}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, c}{6 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right ) x^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{3 d^{2} \left (8 c^{4} x^{4}-7 c^{2} x^{2}-1\right ) x^{3}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) c^{3}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c^{3}}{3 d^{2} \left (c^{2} x^{2}-1\right )} \]

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)^(3/2),x)

[Out]

-1/3*a/d/x^3/(-c^2*d*x^2+d)^(1/2)-4/3*a*c^2/d/x/(-c^2*d*x^2+d)^(1/2)+8/3*a*c^4/d*x/(-c^2*d*x^2+d)^(1/2)-16/3*b
*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(c^2*x^2-1)*arccosh(c*x)*c^3+32/3*b*(-d*(c^2*x^2-1))^(
1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x^5*(c*x+1)*(c*x-1)*c^8-32/3*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^
2-1)*x^7*c^10-16/3*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x^3*(c*x+1)*(c*x-1)*c^6+16*b*(-d*(c^2*
x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x^5*c^8+64/3*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x^
2*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5-64/3*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x^3*a
rccosh(c*x)*c^6-4/3*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x*(c*x+1)*(c*x-1)*c^4-4*b*(-d*(c^2*x^
2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x^3*c^6+8/3*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*arcco
sh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3+8*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x*arccosh(c*x)*
c^4-4/3*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3-4/3*b*(-d*(c^2*x^
2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x*c^4+4*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)/x*arccosh
(c*x)*c^2-1/6*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*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^2/(8*c^4*x^4-7*c^2*x^2-1)/x^3*arccosh(c*x)+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1
)^(1/2)/d^2/(c^2*x^2-1)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*c^3+5/3*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/
2)*(c*x+1)^(1/2)/d^2/(c^2*x^2-1)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*c^3

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

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)^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]

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)^(3/2)),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{4} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

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

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