\(\int \frac {a+b \text {arcsinh}(c x)}{x^4 (d+c^2 d x^2)^2} \, dx\) [45]

Optimal result

Integrand size = 24, antiderivative size = 239 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\frac {b c^3}{3 d^2 \sqrt {1+c^2 x^2}}-\frac {b c}{6 d^2 x^2 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (1+c^2 x^2\right )}+\frac {5 c^2 (a+b \text {arcsinh}(c x))}{3 d^2 x \left (1+c^2 x^2\right )}+\frac {5 c^4 x (a+b \text {arcsinh}(c x))}{2 d^2 \left (1+c^2 x^2\right )}+\frac {5 c^3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {13 b c^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{6 d^2}-\frac {5 i b c^3 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{2 d^2}+\frac {5 i b c^3 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 d^2} \]

[Out]

1/3*(-a-b*arcsinh(c*x))/d^2/x^3/(c^2*x^2+1)+5/3*c^2*(a+b*arcsinh(c*x))/d^2/x/(c^2*x^2+1)+5/2*c^4*x*(a+b*arcsin
h(c*x))/d^2/(c^2*x^2+1)+5*c^3*(a+b*arcsinh(c*x))*arctan(c*x+(c^2*x^2+1)^(1/2))/d^2+13/6*b*c^3*arctanh((c^2*x^2
+1)^(1/2))/d^2-5/2*I*b*c^3*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/d^2+5/2*I*b*c^3*polylog(2,I*(c*x+(c^2*x^2+1)^
(1/2)))/d^2+1/3*b*c^3/d^2/(c^2*x^2+1)^(1/2)-1/6*b*c/d^2/x^2/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5809, 5788, 5789, 4265, 2317, 2438, 267, 272, 53, 65, 214, 44} \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\frac {5 c^3 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d^2}+\frac {5 c^2 (a+b \text {arcsinh}(c x))}{3 d^2 x \left (c^2 x^2+1\right )}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}+\frac {5 c^4 x (a+b \text {arcsinh}(c x))}{2 d^2 \left (c^2 x^2+1\right )}-\frac {5 i b c^3 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{2 d^2}+\frac {5 i b c^3 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 d^2}+\frac {13 b c^3 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )}{6 d^2}-\frac {b c}{6 d^2 x^2 \sqrt {c^2 x^2+1}}+\frac {b c^3}{3 d^2 \sqrt {c^2 x^2+1}} \]

[In]

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

[Out]

(b*c^3)/(3*d^2*Sqrt[1 + c^2*x^2]) - (b*c)/(6*d^2*x^2*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])/(3*d^2*x^3*(1 +
c^2*x^2)) + (5*c^2*(a + b*ArcSinh[c*x]))/(3*d^2*x*(1 + c^2*x^2)) + (5*c^4*x*(a + b*ArcSinh[c*x]))/(2*d^2*(1 +
c^2*x^2)) + (5*c^3*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/d^2 + (13*b*c^3*ArcTanh[Sqrt[1 + c^2*x^2]])/(
6*d^2) - (((5*I)/2)*b*c^3*PolyLog[2, (-I)*E^ArcSinh[c*x]])/d^2 + (((5*I)/2)*b*c^3*PolyLog[2, I*E^ArcSinh[c*x]]
)/d^2

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 267

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

Rule 272

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 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 5788

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(
p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*(p + 1))), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a
+ b*ArcSinh[c*x])^n, x], x] + Dist[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[x*(1 + c^2*x^2
)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] &
& LtQ[p, -1] && NeQ[p, -3/2]

Rule 5789

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/(c*d), Subst[Int[(a +
b*x)^n*Sech[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]

Rule 5809

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] + (-Dist[c^2*((m + 2*p + 3)/(f^2*
(m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(d +
e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /;
FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (1+c^2 x^2\right )}-\frac {1}{3} \left (5 c^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^2} \, dx+\frac {(b c) \int \frac {1}{x^3 \left (1+c^2 x^2\right )^{3/2}} \, dx}{3 d^2} \\ & = -\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (1+c^2 x^2\right )}+\frac {5 c^2 (a+b \text {arcsinh}(c x))}{3 d^2 x \left (1+c^2 x^2\right )}+\left (5 c^4\right ) \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^2} \, dx+\frac {(b c) \text {Subst}\left (\int \frac {1}{x^2 \left (1+c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{6 d^2}-\frac {\left (5 b c^3\right ) \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2}} \, dx}{3 d^2} \\ & = -\frac {b c}{6 d^2 x^2 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (1+c^2 x^2\right )}+\frac {5 c^2 (a+b \text {arcsinh}(c x))}{3 d^2 x \left (1+c^2 x^2\right )}+\frac {5 c^4 x (a+b \text {arcsinh}(c x))}{2 d^2 \left (1+c^2 x^2\right )}-\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{4 d^2}-\frac {\left (5 b c^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{6 d^2}-\frac {\left (5 b c^5\right ) \int \frac {x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac {\left (5 c^4\right ) \int \frac {a+b \text {arcsinh}(c x)}{d+c^2 d x^2} \, dx}{2 d} \\ & = \frac {b c^3}{3 d^2 \sqrt {1+c^2 x^2}}-\frac {b c}{6 d^2 x^2 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (1+c^2 x^2\right )}+\frac {5 c^2 (a+b \text {arcsinh}(c x))}{3 d^2 x \left (1+c^2 x^2\right )}+\frac {5 c^4 x (a+b \text {arcsinh}(c x))}{2 d^2 \left (1+c^2 x^2\right )}+\frac {\left (5 c^3\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{2 d^2}-\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{4 d^2}-\frac {\left (5 b c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{6 d^2} \\ & = \frac {b c^3}{3 d^2 \sqrt {1+c^2 x^2}}-\frac {b c}{6 d^2 x^2 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (1+c^2 x^2\right )}+\frac {5 c^2 (a+b \text {arcsinh}(c x))}{3 d^2 x \left (1+c^2 x^2\right )}+\frac {5 c^4 x (a+b \text {arcsinh}(c x))}{2 d^2 \left (1+c^2 x^2\right )}+\frac {5 c^3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {(b c) \text {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{2 d^2}-\frac {(5 b c) \text {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{3 d^2}-\frac {\left (5 i b c^3\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{2 d^2}+\frac {\left (5 i b c^3\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{2 d^2} \\ & = \frac {b c^3}{3 d^2 \sqrt {1+c^2 x^2}}-\frac {b c}{6 d^2 x^2 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (1+c^2 x^2\right )}+\frac {5 c^2 (a+b \text {arcsinh}(c x))}{3 d^2 x \left (1+c^2 x^2\right )}+\frac {5 c^4 x (a+b \text {arcsinh}(c x))}{2 d^2 \left (1+c^2 x^2\right )}+\frac {5 c^3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {13 b c^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{6 d^2}-\frac {\left (5 i b c^3\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{2 d^2}+\frac {\left (5 i b c^3\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{2 d^2} \\ & = \frac {b c^3}{3 d^2 \sqrt {1+c^2 x^2}}-\frac {b c}{6 d^2 x^2 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (1+c^2 x^2\right )}+\frac {5 c^2 (a+b \text {arcsinh}(c x))}{3 d^2 x \left (1+c^2 x^2\right )}+\frac {5 c^4 x (a+b \text {arcsinh}(c x))}{2 d^2 \left (1+c^2 x^2\right )}+\frac {5 c^3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {13 b c^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{6 d^2}-\frac {5 i b c^3 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{2 d^2}+\frac {5 i b c^3 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 d^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.42 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.30 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\frac {-\frac {5 a}{3 x^3}+\frac {5 a c^2}{x}-\frac {5 b c \sqrt {1+c^2 x^2}}{6 x^2}+\frac {a}{x^3+c^2 x^5}-\frac {5 b \text {arcsinh}(c x)}{3 x^3}+\frac {5 b c^2 \text {arcsinh}(c x)}{x}+\frac {b \text {arcsinh}(c x)}{x^3+c^2 x^5}+5 a c^3 \arctan (c x)+\frac {35}{6} b c^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )+\frac {b c^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},2,\frac {1}{2},1+c^2 x^2\right )}{\sqrt {1+c^2 x^2}}-5 b \left (-c^2\right )^{3/2} \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+5 b \left (-c^2\right )^{3/2} \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+5 b \left (-c^2\right )^{3/2} \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )-5 b \left (-c^2\right )^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{2 d^2} \]

[In]

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

[Out]

((-5*a)/(3*x^3) + (5*a*c^2)/x - (5*b*c*Sqrt[1 + c^2*x^2])/(6*x^2) + a/(x^3 + c^2*x^5) - (5*b*ArcSinh[c*x])/(3*
x^3) + (5*b*c^2*ArcSinh[c*x])/x + (b*ArcSinh[c*x])/(x^3 + c^2*x^5) + 5*a*c^3*ArcTan[c*x] + (35*b*c^3*ArcTanh[S
qrt[1 + c^2*x^2]])/6 + (b*c^3*Hypergeometric2F1[-1/2, 2, 1/2, 1 + c^2*x^2])/Sqrt[1 + c^2*x^2] - 5*b*(-c^2)^(3/
2)*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 5*b*(-c^2)^(3/2)*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^A
rcSinh[c*x])/c] + 5*b*(-c^2)^(3/2)*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 5*b*(-c^2)^(3/2)*PolyLog[2, (Sq
rt[-c^2]*E^ArcSinh[c*x])/c])/(2*d^2)

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.12

method	result	size
derivativedivides	\(c^{3} \left (\frac {a \left (-\frac {1}{3 c^{3} x^{3}}+\frac {2}{c x}+\frac {c x}{2 c^{2} x^{2}+2}+\frac {5 \arctan \left (c x \right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c x}+\frac {\operatorname {arcsinh}\left (c x \right ) c x}{2 c^{2} x^{2}+2}+\frac {5 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{2}+\frac {1}{3 \sqrt {c^{2} x^{2}+1}}-\frac {1}{6 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}+\frac {13 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}+\frac {5 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {5 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {5 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {5 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}\right )}{d^{2}}\right )\)	\(268\)
default	\(c^{3} \left (\frac {a \left (-\frac {1}{3 c^{3} x^{3}}+\frac {2}{c x}+\frac {c x}{2 c^{2} x^{2}+2}+\frac {5 \arctan \left (c x \right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c x}+\frac {\operatorname {arcsinh}\left (c x \right ) c x}{2 c^{2} x^{2}+2}+\frac {5 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{2}+\frac {1}{3 \sqrt {c^{2} x^{2}+1}}-\frac {1}{6 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}+\frac {13 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}+\frac {5 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {5 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {5 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {5 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}\right )}{d^{2}}\right )\)	\(268\)
parts	\(\frac {a \left (c^{4} \left (\frac {x}{2 c^{2} x^{2}+2}+\frac {5 \arctan \left (c x \right )}{2 c}\right )-\frac {1}{3 x^{3}}+\frac {2 c^{2}}{x}\right )}{d^{2}}+\frac {b \,c^{3} \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c x}+\frac {\operatorname {arcsinh}\left (c x \right ) c x}{2 c^{2} x^{2}+2}+\frac {5 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{2}+\frac {1}{3 \sqrt {c^{2} x^{2}+1}}-\frac {1}{6 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}+\frac {13 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}+\frac {5 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {5 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {5 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {5 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}\right )}{d^{2}}\)	\(271\)

[In]

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

[Out]

c^3*(a/d^2*(-1/3/c^3/x^3+2/c/x+1/2*c*x/(c^2*x^2+1)+5/2*arctan(c*x))+b/d^2*(-1/3*arcsinh(c*x)/c^3/x^3+2*arcsinh
(c*x)/c/x+1/2*c*x/(c^2*x^2+1)*arcsinh(c*x)+5/2*arcsinh(c*x)*arctan(c*x)+1/3/(c^2*x^2+1)^(1/2)-1/6/c^2/x^2/(c^2
*x^2+1)^(1/2)+13/6*arctanh(1/(c^2*x^2+1)^(1/2))+5/2*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-5/2*arctan
(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-5/2*I*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+5/2*I*dilog(1-I*(1+I*c*
x)/(c^2*x^2+1)^(1/2))))

Fricas [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{4}} \,d x } \]

[In]

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

[Out]

integral((b*arcsinh(c*x) + a)/(c^4*d^2*x^8 + 2*c^2*d^2*x^6 + d^2*x^4), x)

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{8} + 2 c^{2} x^{6} + x^{4}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{8} + 2 c^{2} x^{6} + x^{4}}\, dx}{d^{2}} \]

[In]

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

[Out]

(Integral(a/(c**4*x**8 + 2*c**2*x**6 + x**4), x) + Integral(b*asinh(c*x)/(c**4*x**8 + 2*c**2*x**6 + x**4), x))
/d**2

Maxima [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{4}} \,d x } \]

[In]

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

[Out]

1/6*(15*c^3*arctan(c*x)/d^2 + (15*c^4*x^4 + 10*c^2*x^2 - 2)/(c^2*d^2*x^5 + d^2*x^3))*a + b*integrate(log(c*x +
sqrt(c^2*x^2 + 1))/(c^4*d^2*x^8 + 2*c^2*d^2*x^6 + d^2*x^4), x)

Giac [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{4}} \,d x } \]

[In]

integrate((a+b*arcsinh(c*x))/x^4/(c^2*d*x^2+d)^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^4\,{\left (d\,c^2\,x^2+d\right )}^2} \,d x \]

[In]

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

[Out]

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