Integrand size = 24, antiderivative size = 120 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^2} \, dx=-\frac {5}{3} b c d^2 \sqrt {1+c^2 x^2}-\frac {1}{9} b c d^2 \left (1+c^2 x^2\right )^{3/2}-\frac {d^2 (a+b \text {arcsinh}(c x))}{x}+2 c^2 d^2 x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arcsinh}(c x))-b c d^2 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right ) \]
-1/9*b*c*d^2*(c^2*x^2+1)^(3/2)-d^2*(a+b*arcsinh(c*x))/x+2*c^2*d^2*x*(a+b*a rcsinh(c*x))+1/3*c^4*d^2*x^3*(a+b*arcsinh(c*x))-b*c*d^2*arctanh((c^2*x^2+1 )^(1/2))-5/3*b*c*d^2*(c^2*x^2+1)^(1/2)
Time = 0.09 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.03 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {d^2 \left (-9 a+18 a c^2 x^2+3 a c^4 x^4-16 b c x \sqrt {1+c^2 x^2}-b c^3 x^3 \sqrt {1+c^2 x^2}+3 b \left (-3+6 c^2 x^2+c^4 x^4\right ) \text {arcsinh}(c x)+9 b c x \log (x)-9 b c x \log \left (1+\sqrt {1+c^2 x^2}\right )\right )}{9 x} \]
(d^2*(-9*a + 18*a*c^2*x^2 + 3*a*c^4*x^4 - 16*b*c*x*Sqrt[1 + c^2*x^2] - b*c ^3*x^3*Sqrt[1 + c^2*x^2] + 3*b*(-3 + 6*c^2*x^2 + c^4*x^4)*ArcSinh[c*x] + 9 *b*c*x*Log[x] - 9*b*c*x*Log[1 + Sqrt[1 + c^2*x^2]]))/(9*x)
Time = 0.42 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6218, 27, 1578, 1192, 25, 1467, 2009}
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.
\(\displaystyle \int \frac {\left (c^2 d x^2+d\right )^2 (a+b \text {arcsinh}(c x))}{x^2} \, dx\) |
\(\Big \downarrow \) 6218 |
\(\displaystyle -b c \int -\frac {d^2 \left (-c^4 x^4-6 c^2 x^2+3\right )}{3 x \sqrt {c^2 x^2+1}}dx+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arcsinh}(c x))+2 c^2 d^2 x (a+b \text {arcsinh}(c x))-\frac {d^2 (a+b \text {arcsinh}(c x))}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} b c d^2 \int \frac {-c^4 x^4-6 c^2 x^2+3}{x \sqrt {c^2 x^2+1}}dx+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arcsinh}(c x))+2 c^2 d^2 x (a+b \text {arcsinh}(c x))-\frac {d^2 (a+b \text {arcsinh}(c x))}{x}\) |
\(\Big \downarrow \) 1578 |
\(\displaystyle \frac {1}{6} b c d^2 \int \frac {-c^4 x^4-6 c^2 x^2+3}{x^2 \sqrt {c^2 x^2+1}}dx^2+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arcsinh}(c x))+2 c^2 d^2 x (a+b \text {arcsinh}(c x))-\frac {d^2 (a+b \text {arcsinh}(c x))}{x}\) |
\(\Big \downarrow \) 1192 |
\(\displaystyle \frac {b d^2 \int -\frac {-c^4 x^8-4 c^4 x^4+8 c^4}{1-x^4}d\sqrt {c^2 x^2+1}}{3 c^3}+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arcsinh}(c x))+2 c^2 d^2 x (a+b \text {arcsinh}(c x))-\frac {d^2 (a+b \text {arcsinh}(c x))}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {b d^2 \int \frac {-c^4 x^8-4 c^4 x^4+8 c^4}{1-x^4}d\sqrt {c^2 x^2+1}}{3 c^3}+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arcsinh}(c x))+2 c^2 d^2 x (a+b \text {arcsinh}(c x))-\frac {d^2 (a+b \text {arcsinh}(c x))}{x}\) |
\(\Big \downarrow \) 1467 |
\(\displaystyle -\frac {b d^2 \int \left (x^4 c^4+\frac {3 c^4}{1-x^4}+5 c^4\right )d\sqrt {c^2 x^2+1}}{3 c^3}+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arcsinh}(c x))+2 c^2 d^2 x (a+b \text {arcsinh}(c x))-\frac {d^2 (a+b \text {arcsinh}(c x))}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} c^4 d^2 x^3 (a+b \text {arcsinh}(c x))+2 c^2 d^2 x (a+b \text {arcsinh}(c x))-\frac {d^2 (a+b \text {arcsinh}(c x))}{x}+\frac {b d^2 \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\) |
-((d^2*(a + b*ArcSinh[c*x]))/x) + 2*c^2*d^2*x*(a + b*ArcSinh[c*x]) + (c^4* d^2*x^3*(a + b*ArcSinh[c*x]))/3 + (b*d^2*(-1/3*(c^4*x^6) - 5*c^4*Sqrt[1 + c^2*x^2] - 3*c^4*ArcTanh[Sqrt[1 + c^2*x^2]]))/(3*c^3)
3.1.16.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1) Subst[Int[x^( 2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, 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] && IGtQ[q, -2]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[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] && Int egerQ[(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 = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.93
method | result | size |
parts | \(d^{2} a \left (\frac {c^{4} x^{3}}{3}+2 c^{2} x -\frac {1}{x}\right )+d^{2} b c \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+2 \,\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {16 \sqrt {c^{2} x^{2}+1}}{9}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\) | \(112\) |
derivativedivides | \(c \left (d^{2} a \left (\frac {c^{3} x^{3}}{3}+2 c x -\frac {1}{c x}\right )+d^{2} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+2 \,\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {16 \sqrt {c^{2} x^{2}+1}}{9}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\) | \(114\) |
default | \(c \left (d^{2} a \left (\frac {c^{3} x^{3}}{3}+2 c x -\frac {1}{c x}\right )+d^{2} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+2 \,\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {16 \sqrt {c^{2} x^{2}+1}}{9}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\) | \(114\) |
d^2*a*(1/3*c^4*x^3+2*c^2*x-1/x)+d^2*b*c*(1/3*arcsinh(c*x)*c^3*x^3+2*arcsin h(c*x)*c*x-arcsinh(c*x)/c/x-1/9*c^2*x^2*(c^2*x^2+1)^(1/2)-16/9*(c^2*x^2+1) ^(1/2)-arctanh(1/(c^2*x^2+1)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (108) = 216\).
Time = 0.29 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.90 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {3 \, a c^{4} d^{2} x^{4} + 18 \, a c^{2} d^{2} x^{2} - 9 \, b c d^{2} x \log \left (-c x + \sqrt {c^{2} x^{2} + 1} + 1\right ) + 9 \, b c d^{2} x \log \left (-c x + \sqrt {c^{2} x^{2} + 1} - 1\right ) - 3 \, {\left (b c^{4} + 6 \, b c^{2} - 3 \, b\right )} d^{2} x \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) - 9 \, a d^{2} + 3 \, {\left (b c^{4} d^{2} x^{4} + 6 \, b c^{2} d^{2} x^{2} - {\left (b c^{4} + 6 \, b c^{2} - 3 \, b\right )} d^{2} x - 3 \, b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (b c^{3} d^{2} x^{3} + 16 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}}{9 \, x} \]
1/9*(3*a*c^4*d^2*x^4 + 18*a*c^2*d^2*x^2 - 9*b*c*d^2*x*log(-c*x + sqrt(c^2* x^2 + 1) + 1) + 9*b*c*d^2*x*log(-c*x + sqrt(c^2*x^2 + 1) - 1) - 3*(b*c^4 + 6*b*c^2 - 3*b)*d^2*x*log(-c*x + sqrt(c^2*x^2 + 1)) - 9*a*d^2 + 3*(b*c^4*d ^2*x^4 + 6*b*c^2*d^2*x^2 - (b*c^4 + 6*b*c^2 - 3*b)*d^2*x - 3*b*d^2)*log(c* x + sqrt(c^2*x^2 + 1)) - (b*c^3*d^2*x^3 + 16*b*c*d^2*x)*sqrt(c^2*x^2 + 1)) /x
\[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^2} \, dx=d^{2} \left (\int 2 a c^{2}\, dx + \int \frac {a}{x^{2}}\, dx + \int a c^{4} x^{2}\, dx + \int 2 b c^{2} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx + \int b c^{4} x^{2} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
d**2*(Integral(2*a*c**2, x) + Integral(a/x**2, x) + Integral(a*c**4*x**2, x) + Integral(2*b*c**2*asinh(c*x), x) + Integral(b*asinh(c*x)/x**2, x) + I ntegral(b*c**4*x**2*asinh(c*x), x))
Time = 0.19 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.19 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {1}{3} \, a c^{4} d^{2} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b c^{4} d^{2} + 2 \, a c^{2} d^{2} x + 2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b c d^{2} - {\left (c \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arsinh}\left (c x\right )}{x}\right )} b d^{2} - \frac {a d^{2}}{x} \]
1/3*a*c^4*d^2*x^3 + 1/9*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*b*c^4*d^2 + 2*a*c^2*d^2*x + 2*(c*x*arcsinh(c* x) - sqrt(c^2*x^2 + 1))*b*c*d^2 - (c*arcsinh(1/(c*abs(x))) + arcsinh(c*x)/ x)*b*d^2 - a*d^2/x
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^2}{x^2} \,d x \]