Integrand size = 22, antiderivative size = 80 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {b c d \sqrt {1+c^2 x^2}}{6 x^2}-\frac {d (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {c^2 d (a+b \text {arcsinh}(c x))}{x}-\frac {5}{6} b c^3 d \text {arctanh}\left (\sqrt {1+c^2 x^2}\right ) \] Output:
-1/6*b*c*d*(c^2*x^2+1)^(1/2)/x^2-1/3*d*(a+b*arcsinh(c*x))/x^3-c^2*d*(a+b*a rcsinh(c*x))/x-5/6*b*c^3*d*arctanh((c^2*x^2+1)^(1/2))
Time = 0.02 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.16 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {a d}{3 x^3}-\frac {a c^2 d}{x}-\frac {b c d \sqrt {1+c^2 x^2}}{6 x^2}-\frac {b d \text {arcsinh}(c x)}{3 x^3}-\frac {b c^2 d \text {arcsinh}(c x)}{x}-\frac {5}{6} b c^3 d \text {arctanh}\left (\sqrt {1+c^2 x^2}\right ) \] Input:
Integrate[((d + c^2*d*x^2)*(a + b*ArcSinh[c*x]))/x^4,x]
Output:
-1/3*(a*d)/x^3 - (a*c^2*d)/x - (b*c*d*Sqrt[1 + c^2*x^2])/(6*x^2) - (b*d*Ar cSinh[c*x])/(3*x^3) - (b*c^2*d*ArcSinh[c*x])/x - (5*b*c^3*d*ArcTanh[Sqrt[1 + c^2*x^2]])/6
Time = 0.33 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6218, 27, 354, 87, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (c^2 d x^2+d\right ) (a+b \text {arcsinh}(c x))}{x^4} \, dx\) |
\(\Big \downarrow \) 6218 |
\(\displaystyle -b c \int -\frac {d \left (3 c^2 x^2+1\right )}{3 x^3 \sqrt {c^2 x^2+1}}dx-\frac {c^2 d (a+b \text {arcsinh}(c x))}{x}-\frac {d (a+b \text {arcsinh}(c x))}{3 x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} b c d \int \frac {3 c^2 x^2+1}{x^3 \sqrt {c^2 x^2+1}}dx-\frac {c^2 d (a+b \text {arcsinh}(c x))}{x}-\frac {d (a+b \text {arcsinh}(c x))}{3 x^3}\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{6} b c d \int \frac {3 c^2 x^2+1}{x^4 \sqrt {c^2 x^2+1}}dx^2-\frac {c^2 d (a+b \text {arcsinh}(c x))}{x}-\frac {d (a+b \text {arcsinh}(c x))}{3 x^3}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{6} b c d \left (\frac {5}{2} c^2 \int \frac {1}{x^2 \sqrt {c^2 x^2+1}}dx^2-\frac {\sqrt {c^2 x^2+1}}{x^2}\right )-\frac {c^2 d (a+b \text {arcsinh}(c x))}{x}-\frac {d (a+b \text {arcsinh}(c x))}{3 x^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{6} b c d \left (5 \int \frac {1}{\frac {x^4}{c^2}-\frac {1}{c^2}}d\sqrt {c^2 x^2+1}-\frac {\sqrt {c^2 x^2+1}}{x^2}\right )-\frac {c^2 d (a+b \text {arcsinh}(c x))}{x}-\frac {d (a+b \text {arcsinh}(c x))}{3 x^3}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {c^2 d (a+b \text {arcsinh}(c x))}{x}-\frac {d (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{6} b c d \left (-5 c^2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {\sqrt {c^2 x^2+1}}{x^2}\right )\) |
Input:
Int[((d + c^2*d*x^2)*(a + b*ArcSinh[c*x]))/x^4,x]
Output:
-1/3*(d*(a + b*ArcSinh[c*x]))/x^3 - (c^2*d*(a + b*ArcSinh[c*x]))/x + (b*c* d*(-(Sqrt[1 + c^2*x^2]/x^2) - 5*c^2*ArcTanh[Sqrt[1 + c^2*x^2]]))/6
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
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]}, Simp [(a + b*ArcSinh[c*x]) u, x] - Simp[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]
Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.04
method | result | size |
parts | \(a d \left (-\frac {1}{3 x^{3}}-\frac {c^{2}}{x}\right )+b d \,c^{3} \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {\operatorname {arcsinh}\left (x c \right )}{3 x^{3} c^{3}}-\frac {\sqrt {c^{2} x^{2}+1}}{6 x^{2} c^{2}}-\frac {5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\) | \(83\) |
derivativedivides | \(c^{3} \left (a d \left (-\frac {1}{x c}-\frac {1}{3 x^{3} c^{3}}\right )+b d \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {\operatorname {arcsinh}\left (x c \right )}{3 x^{3} c^{3}}-\frac {\sqrt {c^{2} x^{2}+1}}{6 x^{2} c^{2}}-\frac {5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(87\) |
default | \(c^{3} \left (a d \left (-\frac {1}{x c}-\frac {1}{3 x^{3} c^{3}}\right )+b d \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {\operatorname {arcsinh}\left (x c \right )}{3 x^{3} c^{3}}-\frac {\sqrt {c^{2} x^{2}+1}}{6 x^{2} c^{2}}-\frac {5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(87\) |
Input:
int((c^2*d*x^2+d)*(a+b*arcsinh(x*c))/x^4,x,method=_RETURNVERBOSE)
Output:
a*d*(-1/3/x^3-c^2/x)+b*d*c^3*(-arcsinh(x*c)/x/c-1/3*arcsinh(x*c)/x^3/c^3-1 /6/x^2/c^2*(c^2*x^2+1)^(1/2)-5/6*arctanh(1/(c^2*x^2+1)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (70) = 140\).
Time = 0.13 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.11 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {5 \, b c^{3} d x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} + 1} + 1\right ) - 5 \, b c^{3} d x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} + 1} - 1\right ) + 6 \, a c^{2} d x^{2} - 2 \, {\left (3 \, b c^{2} + b\right )} d x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {c^{2} x^{2} + 1} b c d x + 2 \, a d + 2 \, {\left (3 \, b c^{2} d x^{2} - {\left (3 \, b c^{2} + b\right )} d x^{3} + b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{6 \, x^{3}} \] Input:
integrate((c^2*d*x^2+d)*(a+b*arcsinh(c*x))/x^4,x, algorithm="fricas")
Output:
-1/6*(5*b*c^3*d*x^3*log(-c*x + sqrt(c^2*x^2 + 1) + 1) - 5*b*c^3*d*x^3*log( -c*x + sqrt(c^2*x^2 + 1) - 1) + 6*a*c^2*d*x^2 - 2*(3*b*c^2 + b)*d*x^3*log( -c*x + sqrt(c^2*x^2 + 1)) + sqrt(c^2*x^2 + 1)*b*c*d*x + 2*a*d + 2*(3*b*c^2 *d*x^2 - (3*b*c^2 + b)*d*x^3 + b*d)*log(c*x + sqrt(c^2*x^2 + 1)))/x^3
\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x^4} \, dx=d \left (\int \frac {a}{x^{4}}\, dx + \int \frac {a c^{2}}{x^{2}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {b c^{2} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx\right ) \] Input:
integrate((c**2*d*x**2+d)*(a+b*asinh(c*x))/x**4,x)
Output:
d*(Integral(a/x**4, x) + Integral(a*c**2/x**2, x) + Integral(b*asinh(c*x)/ x**4, x) + Integral(b*c**2*asinh(c*x)/x**2, x))
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.14 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x^4} \, dx=-{\left (c \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arsinh}\left (c x\right )}{x}\right )} b c^{2} d + \frac {1}{6} \, {\left ({\left (c^{2} \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} + 1}}{x^{2}}\right )} c - \frac {2 \, \operatorname {arsinh}\left (c x\right )}{x^{3}}\right )} b d - \frac {a c^{2} d}{x} - \frac {a d}{3 \, x^{3}} \] Input:
integrate((c^2*d*x^2+d)*(a+b*arcsinh(c*x))/x^4,x, algorithm="maxima")
Output:
-(c*arcsinh(1/(c*abs(x))) + arcsinh(c*x)/x)*b*c^2*d + 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 - a*c^2*d /x - 1/3*a*d/x^3
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c^2*d*x^2+d)*(a+b*arcsinh(c*x))/x^4,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right )}{x^4} \,d x \] Input:
int(((a + b*asinh(c*x))*(d + c^2*d*x^2))/x^4,x)
Output:
int(((a + b*asinh(c*x))*(d + c^2*d*x^2))/x^4, x)
Time = 0.18 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.30 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {d \left (-6 \mathit {asinh} \left (c x \right ) b \,c^{2} x^{2}-2 \mathit {asinh} \left (c x \right ) b -\sqrt {c^{2} x^{2}+1}\, b c x +5 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) b \,c^{3} x^{3}-5 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) b \,c^{3} x^{3}-6 a \,c^{2} x^{2}-2 a \right )}{6 x^{3}} \] Input:
int((c^2*d*x^2+d)*(a+b*asinh(c*x))/x^4,x)
Output:
(d*( - 6*asinh(c*x)*b*c**2*x**2 - 2*asinh(c*x)*b - sqrt(c**2*x**2 + 1)*b*c *x + 5*log(sqrt(c**2*x**2 + 1) + c*x - 1)*b*c**3*x**3 - 5*log(sqrt(c**2*x* *2 + 1) + c*x + 1)*b*c**3*x**3 - 6*a*c**2*x**2 - 2*a))/(6*x**3)