3.1.91 \(\int \frac {(d+e x^2)^2 (a+b \text {csch}^{-1}(c x))}{x^4} \, dx\) [91]

Optimal. Leaf size=164 \[ -\frac {2 b c d \left (c^2 d-9 e\right ) \sqrt {-1-c^2 x^2}}{9 \sqrt {-c^2 x^2}}+\frac {b c d^2 \sqrt {-1-c^2 x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b e^2 x \text {ArcTan}\left (\frac {c x}{\sqrt {-1-c^2 x^2}}\right )}{\sqrt {-c^2 x^2}} \]

[Out]

-1/3*d^2*(a+b*arccsch(c*x))/x^3-2*d*e*(a+b*arccsch(c*x))/x+e^2*x*(a+b*arccsch(c*x))-b*e^2*x*arctan(c*x/(-c^2*x
^2-1)^(1/2))/(-c^2*x^2)^(1/2)-2/9*b*c*d*(c^2*d-9*e)*(-c^2*x^2-1)^(1/2)/(-c^2*x^2)^(1/2)+1/9*b*c*d^2*(-c^2*x^2-
1)^(1/2)/x^2/(-c^2*x^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.10, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {276, 6437, 12, 1279, 462, 223, 209} \begin {gather*} -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b e^2 x \text {ArcTan}\left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right )}{\sqrt {-c^2 x^2}}+\frac {b c d^2 \sqrt {-c^2 x^2-1}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {2 b c d \sqrt {-c^2 x^2-1} \left (c^2 d-9 e\right )}{9 \sqrt {-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^4,x]

[Out]

(-2*b*c*d*(c^2*d - 9*e)*Sqrt[-1 - c^2*x^2])/(9*Sqrt[-(c^2*x^2)]) + (b*c*d^2*Sqrt[-1 - c^2*x^2])/(9*x^2*Sqrt[-(
c^2*x^2)]) - (d^2*(a + b*ArcCsch[c*x]))/(3*x^3) - (2*d*e*(a + b*ArcCsch[c*x]))/x + e^2*x*(a + b*ArcCsch[c*x])
- (b*e^2*x*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/Sqrt[-(c^2*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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 462

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

Rule 1279

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Wit
h[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x,
x]}, Simp[R*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(d*f*(m + 1))), x] + Dist[1/(d*f^2*(m + 1)), Int[(f*x)^(m + 2)
*(d + e*x^2)^q*ExpandToSum[d*f*(m + 1)*(Qx/x) - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q},
x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]

Rule 6437

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u
= IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCsch[c*x], u, x] - Dist[b*c*(x/Sqrt[(-c^2)*x^2]), Int[Simp
lifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&
!(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0]))
|| (ILtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {-d^2-6 d e x^2+3 e^2 x^4}{3 x^4 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {-d^2-6 d e x^2+3 e^2 x^4}{x^4 \sqrt {-1-c^2 x^2}} \, dx}{3 \sqrt {-c^2 x^2}}\\ &=\frac {b c d^2 \sqrt {-1-c^2 x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {2 d \left (c^2 d-9 e\right )+9 e^2 x^2}{x^2 \sqrt {-1-c^2 x^2}} \, dx}{9 \sqrt {-c^2 x^2}}\\ &=-\frac {2 b c d \left (c^2 d-9 e\right ) \sqrt {-1-c^2 x^2}}{9 \sqrt {-c^2 x^2}}+\frac {b c d^2 \sqrt {-1-c^2 x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (b c e^2 x\right ) \int \frac {1}{\sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {2 b c d \left (c^2 d-9 e\right ) \sqrt {-1-c^2 x^2}}{9 \sqrt {-c^2 x^2}}+\frac {b c d^2 \sqrt {-1-c^2 x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (b c e^2 x\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1-c^2 x^2}}\right )}{\sqrt {-c^2 x^2}}\\ &=-\frac {2 b c d \left (c^2 d-9 e\right ) \sqrt {-1-c^2 x^2}}{9 \sqrt {-c^2 x^2}}+\frac {b c d^2 \sqrt {-1-c^2 x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b e^2 x \tan ^{-1}\left (\frac {c x}{\sqrt {-1-c^2 x^2}}\right )}{\sqrt {-c^2 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.16, size = 123, normalized size = 0.75 \begin {gather*} \frac {b c d \sqrt {1+\frac {1}{c^2 x^2}} x \left (d-2 c^2 d x^2+18 e x^2\right )-3 a \left (d^2+6 d e x^2-3 e^2 x^4\right )}{9 x^3}-\frac {b \left (d^2+6 d e x^2-3 e^2 x^4\right ) \text {csch}^{-1}(c x)}{3 x^3}+\frac {b e^2 \log \left (\left (1+\sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^4,x]

[Out]

(b*c*d*Sqrt[1 + 1/(c^2*x^2)]*x*(d - 2*c^2*d*x^2 + 18*e*x^2) - 3*a*(d^2 + 6*d*e*x^2 - 3*e^2*x^4))/(9*x^3) - (b*
(d^2 + 6*d*e*x^2 - 3*e^2*x^4)*ArcCsch[c*x])/(3*x^3) + (b*e^2*Log[(1 + Sqrt[1 + 1/(c^2*x^2)])*x])/c

________________________________________________________________________________________

Maple [A]
time = 0.32, size = 190, normalized size = 1.16

method	result	size
derivativedivides	\(c^{3} \left (\frac {a \left (e^{2} c x -\frac {c \,d^{2}}{3 x^{3}}-\frac {2 c d e}{x}\right )}{c^{4}}+\frac {b \left (\mathrm {arccsch}\left (c x \right ) e^{2} c x -\frac {\mathrm {arccsch}\left (c x \right ) c \,d^{2}}{3 x^{3}}-\frac {2 \,\mathrm {arccsch}\left (c x \right ) c d e}{x}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (-2 \sqrt {c^{2} x^{2}+1}\, c^{6} d^{2} x^{2}+c^{4} d^{2} \sqrt {c^{2} x^{2}+1}+18 c^{4} d e \sqrt {c^{2} x^{2}+1}\, x^{2}+9 e^{2} \arcsinh \left (c x \right ) c^{3} x^{3}\right )}{9 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )}{c^{4}}\right )\)	\(190\)
default	\(c^{3} \left (\frac {a \left (e^{2} c x -\frac {c \,d^{2}}{3 x^{3}}-\frac {2 c d e}{x}\right )}{c^{4}}+\frac {b \left (\mathrm {arccsch}\left (c x \right ) e^{2} c x -\frac {\mathrm {arccsch}\left (c x \right ) c \,d^{2}}{3 x^{3}}-\frac {2 \,\mathrm {arccsch}\left (c x \right ) c d e}{x}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (-2 \sqrt {c^{2} x^{2}+1}\, c^{6} d^{2} x^{2}+c^{4} d^{2} \sqrt {c^{2} x^{2}+1}+18 c^{4} d e \sqrt {c^{2} x^{2}+1}\, x^{2}+9 e^{2} \arcsinh \left (c x \right ) c^{3} x^{3}\right )}{9 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )}{c^{4}}\right )\)	\(190\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^2*(a+b*arccsch(c*x))/x^4,x,method=_RETURNVERBOSE)

[Out]

c^3*(a/c^4*(e^2*c*x-1/3*c*d^2/x^3-2*c*d*e/x)+b/c^4*(arccsch(c*x)*e^2*c*x-1/3*arccsch(c*x)*c*d^2/x^3-2*arccsch(
c*x)*c*d*e/x+1/9*(c^2*x^2+1)^(1/2)*(-2*(c^2*x^2+1)^(1/2)*c^6*d^2*x^2+c^4*d^2*(c^2*x^2+1)^(1/2)+18*c^4*d*e*(c^2
*x^2+1)^(1/2)*x^2+9*e^2*arcsinh(c*x)*c^3*x^3)/((c^2*x^2+1)/c^2/x^2)^(1/2)/c^4/x^4))

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 152, normalized size = 0.93 \begin {gather*} \frac {1}{9} \, b d^{2} {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arcsch}\left (c x\right )}{x^{3}}\right )} + 2 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsch}\left (c x\right )}{x}\right )} b d e + a x e^{2} + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b e^{2}}{2 \, c} - \frac {2 \, a d e}{x} - \frac {a d^{2}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2*(a+b*arccsch(c*x))/x^4,x, algorithm="maxima")

[Out]

1/9*b*d^2*((c^4*(1/(c^2*x^2) + 1)^(3/2) - 3*c^4*sqrt(1/(c^2*x^2) + 1))/c - 3*arccsch(c*x)/x^3) + 2*(c*sqrt(1/(
c^2*x^2) + 1) - arccsch(c*x)/x)*b*d*e + a*x*e^2 + 1/2*(2*c*x*arccsch(c*x) + log(sqrt(1/(c^2*x^2) + 1) + 1) - l
og(sqrt(1/(c^2*x^2) + 1) - 1))*b*e^2/c - 2*a*d*e/x - 1/3*a*d^2/x^3

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (146) = 292\).
time = 0.49, size = 559, normalized size = 3.41 \begin {gather*} -\frac {2 \, b c^{4} d^{2} x^{3} - 9 \, a c x^{4} \cosh \left (1\right )^{2} - 9 \, a c x^{4} \sinh \left (1\right )^{2} + 3 \, a c d^{2} - 18 \, {\left (b c^{2} d x^{3} - a c d x^{2}\right )} \cosh \left (1\right ) + 3 \, {\left (b c d^{2} x^{3} + 6 \, b c d x^{3} \cosh \left (1\right ) - 3 \, b c x^{3} \cosh \left (1\right )^{2} - 3 \, b c x^{3} \sinh \left (1\right )^{2} + 6 \, {\left (b c d x^{3} - b c x^{3} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + 9 \, {\left (b x^{3} \cosh \left (1\right )^{2} + 2 \, b x^{3} \cosh \left (1\right ) \sinh \left (1\right ) + b x^{3} \sinh \left (1\right )^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 3 \, {\left (b c d^{2} x^{3} + 6 \, b c d x^{3} \cosh \left (1\right ) - 3 \, b c x^{3} \cosh \left (1\right )^{2} - 3 \, b c x^{3} \sinh \left (1\right )^{2} + 6 \, {\left (b c d x^{3} - b c x^{3} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) - 3 \, {\left (b c d^{2} x^{3} - b c d^{2} + 3 \, {\left (b c x^{4} - b c x^{3}\right )} \cosh \left (1\right )^{2} + 3 \, {\left (b c x^{4} - b c x^{3}\right )} \sinh \left (1\right )^{2} + 6 \, {\left (b c d x^{3} - b c d x^{2}\right )} \cosh \left (1\right ) + 6 \, {\left (b c d x^{3} - b c d x^{2} + {\left (b c x^{4} - b c x^{3}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 18 \, {\left (b c^{2} d x^{3} + a c x^{4} \cosh \left (1\right ) - a c d x^{2}\right )} \sinh \left (1\right ) + {\left (2 \, b c^{4} d^{2} x^{3} - 18 \, b c^{2} d x^{3} \cosh \left (1\right ) - 18 \, b c^{2} d x^{3} \sinh \left (1\right ) - b c^{2} d^{2} x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{9 \, c x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2*(a+b*arccsch(c*x))/x^4,x, algorithm="fricas")

[Out]

-1/9*(2*b*c^4*d^2*x^3 - 9*a*c*x^4*cosh(1)^2 - 9*a*c*x^4*sinh(1)^2 + 3*a*c*d^2 - 18*(b*c^2*d*x^3 - a*c*d*x^2)*c
osh(1) + 3*(b*c*d^2*x^3 + 6*b*c*d*x^3*cosh(1) - 3*b*c*x^3*cosh(1)^2 - 3*b*c*x^3*sinh(1)^2 + 6*(b*c*d*x^3 - b*c
*x^3*cosh(1))*sinh(1))*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) + 9*(b*x^3*cosh(1)^2 + 2*b*x^3*cosh(1)
*sinh(1) + b*x^3*sinh(1)^2)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x) - 3*(b*c*d^2*x^3 + 6*b*c*d*x^3*cosh(1
) - 3*b*c*x^3*cosh(1)^2 - 3*b*c*x^3*sinh(1)^2 + 6*(b*c*d*x^3 - b*c*x^3*cosh(1))*sinh(1))*log(c*x*sqrt((c^2*x^2
+ 1)/(c^2*x^2)) - c*x - 1) - 3*(b*c*d^2*x^3 - b*c*d^2 + 3*(b*c*x^4 - b*c*x^3)*cosh(1)^2 + 3*(b*c*x^4 - b*c*x^
3)*sinh(1)^2 + 6*(b*c*d*x^3 - b*c*d*x^2)*cosh(1) + 6*(b*c*d*x^3 - b*c*d*x^2 + (b*c*x^4 - b*c*x^3)*cosh(1))*sin
h(1))*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) - 18*(b*c^2*d*x^3 + a*c*x^4*cosh(1) - a*c*d*x^2)*sinh
(1) + (2*b*c^4*d^2*x^3 - 18*b*c^2*d*x^3*cosh(1) - 18*b*c^2*d*x^3*sinh(1) - b*c^2*d^2*x)*sqrt((c^2*x^2 + 1)/(c^
2*x^2)))/(c*x^3)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**2*(a+b*acsch(c*x))/x**4,x)

[Out]

Integral((a + b*acsch(c*x))*(d + e*x**2)**2/x**4, x)

________________________________________________________________________________________

Giac [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((e*x^2+d)^2*(a+b*arccsch(c*x))/x^4,x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^2*(b*arccsch(c*x) + a)/x^4, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^2)^2*(a + b*asinh(1/(c*x))))/x^4,x)

[Out]

int(((d + e*x^2)^2*(a + b*asinh(1/(c*x))))/x^4, x)

________________________________________________________________________________________