\(\int \frac {(d+c^2 d x^2)^2 (a+b \text {arcsinh}(c x))}{x^4} \, dx\) [18]

Optimal result

Integrand size = 24, antiderivative size = 126 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^4} \, dx=-b c^3 d^2 \sqrt {1+c^2 x^2}-\frac {b c d^2 \sqrt {1+c^2 x^2}}{6 x^2}-\frac {d^2 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {2 c^2 d^2 (a+b \text {arcsinh}(c x))}{x}+c^4 d^2 x (a+b \text {arcsinh}(c x))-\frac {11}{6} b c^3 d^2 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right ) \]

[Out]

-1/3*d^2*(a+b*arcsinh(c*x))/x^3-2*c^2*d^2*(a+b*arcsinh(c*x))/x+c^4*d^2*x*(a+b*arcsinh(c*x))-11/6*b*c^3*d^2*arc
tanh((c^2*x^2+1)^(1/2))-b*c^3*d^2*(c^2*x^2+1)^(1/2)-1/6*b*c*d^2*(c^2*x^2+1)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {276, 5803, 12, 1265, 911, 1171, 396, 214} \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^4} \, dx=c^4 d^2 x (a+b \text {arcsinh}(c x))-\frac {2 c^2 d^2 (a+b \text {arcsinh}(c x))}{x}-\frac {d^2 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {11}{6} b c^3 d^2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {b c d^2 \sqrt {c^2 x^2+1}}{6 x^2}-b c^3 d^2 \sqrt {c^2 x^2+1} \]

[In]

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

[Out]

-(b*c^3*d^2*Sqrt[1 + c^2*x^2]) - (b*c*d^2*Sqrt[1 + c^2*x^2])/(6*x^2) - (d^2*(a + b*ArcSinh[c*x]))/(3*x^3) - (2
*c^2*d^2*(a + b*ArcSinh[c*x]))/x + c^4*d^2*x*(a + b*ArcSinh[c*x]) - (11*b*c^3*d^2*ArcTanh[Sqrt[1 + c^2*x^2]])/
6

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 396

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

Rule 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], 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] && IntegersQ[n,
p] && FractionQ[m]

Rule 1171

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

Rule 1265

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 5803

Int[((a_.) + ArcSinh[(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*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1
+ c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {2 c^2 d^2 (a+b \text {arcsinh}(c x))}{x}+c^4 d^2 x (a+b \text {arcsinh}(c x))-(b c) \int \frac {d^2 \left (-1-6 c^2 x^2+3 c^4 x^4\right )}{3 x^3 \sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {d^2 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {2 c^2 d^2 (a+b \text {arcsinh}(c x))}{x}+c^4 d^2 x (a+b \text {arcsinh}(c x))-\frac {1}{3} \left (b c d^2\right ) \int \frac {-1-6 c^2 x^2+3 c^4 x^4}{x^3 \sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {d^2 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {2 c^2 d^2 (a+b \text {arcsinh}(c x))}{x}+c^4 d^2 x (a+b \text {arcsinh}(c x))-\frac {1}{6} \left (b c d^2\right ) \text {Subst}\left (\int \frac {-1-6 c^2 x+3 c^4 x^2}{x^2 \sqrt {1+c^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {d^2 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {2 c^2 d^2 (a+b \text {arcsinh}(c x))}{x}+c^4 d^2 x (a+b \text {arcsinh}(c x))-\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {8-12 x^2+3 x^4}{\left (-\frac {1}{c^2}+\frac {x^2}{c^2}\right )^2} \, dx,x,\sqrt {1+c^2 x^2}\right )}{3 c} \\ & = -\frac {b c d^2 \sqrt {1+c^2 x^2}}{6 x^2}-\frac {d^2 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {2 c^2 d^2 (a+b \text {arcsinh}(c x))}{x}+c^4 d^2 x (a+b \text {arcsinh}(c x))-\frac {1}{6} \left (b c d^2\right ) \text {Subst}\left (\int \frac {-17+6 x^2}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right ) \\ & = -b c^3 d^2 \sqrt {1+c^2 x^2}-\frac {b c d^2 \sqrt {1+c^2 x^2}}{6 x^2}-\frac {d^2 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {2 c^2 d^2 (a+b \text {arcsinh}(c x))}{x}+c^4 d^2 x (a+b \text {arcsinh}(c x))+\frac {1}{6} \left (11 b c d^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right ) \\ & = -b c^3 d^2 \sqrt {1+c^2 x^2}-\frac {b c d^2 \sqrt {1+c^2 x^2}}{6 x^2}-\frac {d^2 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {2 c^2 d^2 (a+b \text {arcsinh}(c x))}{x}+c^4 d^2 x (a+b \text {arcsinh}(c x))-\frac {11}{6} b c^3 d^2 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.06 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {d^2 \left (-2 a-12 a c^2 x^2+6 a c^4 x^4-b c x \sqrt {1+c^2 x^2}-6 b c^3 x^3 \sqrt {1+c^2 x^2}+2 b \left (-1-6 c^2 x^2+3 c^4 x^4\right ) \text {arcsinh}(c x)+11 b c^3 x^3 \log (x)-11 b c^3 x^3 \log \left (1+\sqrt {1+c^2 x^2}\right )\right )}{6 x^3} \]

[In]

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

[Out]

(d^2*(-2*a - 12*a*c^2*x^2 + 6*a*c^4*x^4 - b*c*x*Sqrt[1 + c^2*x^2] - 6*b*c^3*x^3*Sqrt[1 + c^2*x^2] + 2*b*(-1 -
6*c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x] + 11*b*c^3*x^3*Log[x] - 11*b*c^3*x^3*Log[1 + Sqrt[1 + c^2*x^2]]))/(6*x^3)

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.89

method	result	size
parts	\(d^{2} a \left (c^{4} x -\frac {2 c^{2}}{x}-\frac {1}{3 x^{3}}\right )+d^{2} b \,c^{3} \left (\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c x}-\sqrt {c^{2} x^{2}+1}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {11 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\)	\(112\)
derivativedivides	\(c^{3} \left (d^{2} a \left (c x -\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}\right )+d^{2} b \left (\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c x}-\sqrt {c^{2} x^{2}+1}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {11 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(114\)
default	\(c^{3} \left (d^{2} a \left (c x -\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}\right )+d^{2} b \left (\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c x}-\sqrt {c^{2} x^{2}+1}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {11 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(114\)

[In]

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

[Out]

d^2*a*(c^4*x-2*c^2/x-1/3/x^3)+d^2*b*c^3*(arcsinh(c*x)*c*x-1/3*arcsinh(c*x)/c^3/x^3-2*arcsinh(c*x)/c/x-(c^2*x^2
+1)^(1/2)-1/6/c^2/x^2*(c^2*x^2+1)^(1/2)-11/6*arctanh(1/(c^2*x^2+1)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (114) = 228\).

Time = 0.28 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.93 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {6 \, a c^{4} d^{2} x^{4} - 11 \, b c^{3} d^{2} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} + 1} + 1\right ) + 11 \, b c^{3} d^{2} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} + 1} - 1\right ) - 12 \, a c^{2} d^{2} x^{2} - 2 \, {\left (3 \, b c^{4} - 6 \, b c^{2} - b\right )} d^{2} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, a d^{2} + 2 \, {\left (3 \, b c^{4} d^{2} x^{4} - 6 \, b c^{2} d^{2} x^{2} - {\left (3 \, b c^{4} - 6 \, b c^{2} - b\right )} d^{2} x^{3} - b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (6 \, b c^{3} d^{2} x^{3} + b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}}{6 \, x^{3}} \]

[In]

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

[Out]

1/6*(6*a*c^4*d^2*x^4 - 11*b*c^3*d^2*x^3*log(-c*x + sqrt(c^2*x^2 + 1) + 1) + 11*b*c^3*d^2*x^3*log(-c*x + sqrt(c
^2*x^2 + 1) - 1) - 12*a*c^2*d^2*x^2 - 2*(3*b*c^4 - 6*b*c^2 - b)*d^2*x^3*log(-c*x + sqrt(c^2*x^2 + 1)) - 2*a*d^
2 + 2*(3*b*c^4*d^2*x^4 - 6*b*c^2*d^2*x^2 - (3*b*c^4 - 6*b*c^2 - b)*d^2*x^3 - b*d^2)*log(c*x + sqrt(c^2*x^2 + 1
)) - (6*b*c^3*d^2*x^3 + b*c*d^2*x)*sqrt(c^2*x^2 + 1))/x^3

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^4} \, dx=a c^{4} d^{2} x + {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b c^{3} d^{2} - 2 \, {\left (c \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arsinh}\left (c x\right )}{x}\right )} b c^{2} d^{2} + \frac {1}{6} \, {\left ({\left (c^{2} \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} + 1}}{x^{2}}\right )} c - \frac {2 \, \operatorname {arsinh}\left (c x\right )}{x^{3}}\right )} b d^{2} - \frac {2 \, a c^{2} d^{2}}{x} - \frac {a d^{2}}{3 \, x^{3}} \]

[In]

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

[Out]

a*c^4*d^2*x + (c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*b*c^3*d^2 - 2*(c*arcsinh(1/(c*abs(x))) + arcsinh(c*x)/x)*
b*c^2*d^2 + 1/6*((c^2*arcsinh(1/(c*abs(x))) - sqrt(c^2*x^2 + 1)/x^2)*c - 2*arcsinh(c*x)/x^3)*b*d^2 - 2*a*c^2*d
^2/x - 1/3*a*d^2/x^3

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \]

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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