Integrand size = 19, antiderivative size = 205 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {b c x \sqrt {-1-c^2 x^2}}{8 d \left (c^2 d-e\right ) \sqrt {-c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c x \arctan \left (\sqrt {-1-c^2 x^2}\right )}{4 d^2 e \sqrt {-c^2 x^2}}+\frac {b c \left (3 c^2 d-2 e\right ) x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{\sqrt {c^2 d-e}}\right )}{8 d^2 \left (c^2 d-e\right )^{3/2} \sqrt {e} \sqrt {-c^2 x^2}} \] Output:
1/8*b*c*x*(-c^2*x^2-1)^(1/2)/d/(c^2*d-e)/(-c^2*x^2)^(1/2)/(e*x^2+d)-1/4*(a +b*arccsch(c*x))/e/(e*x^2+d)^2+1/4*b*c*x*arctan((-c^2*x^2-1)^(1/2))/d^2/e/ (-c^2*x^2)^(1/2)+1/8*b*c*(3*c^2*d-2*e)*x*arctanh(e^(1/2)*(-c^2*x^2-1)^(1/2 )/(c^2*d-e)^(1/2))/d^2/(c^2*d-e)^(3/2)/e^(1/2)/(-c^2*x^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.59 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.80 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {1}{16} \left (-\frac {4 a}{e \left (d+e x^2\right )^2}+\frac {2 b c \sqrt {1+\frac {1}{c^2 x^2}} x}{d \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {4 b \text {csch}^{-1}(c x)}{e \left (d+e x^2\right )^2}+\frac {4 b \text {arcsinh}\left (\frac {1}{c x}\right )}{d^2 e}+\frac {b \left (3 c^2 d-2 e\right ) \log \left (\frac {16 d^2 \sqrt {e} \sqrt {-c^2 d+e} \left (\sqrt {e}+c \left (-i c \sqrt {d}+\sqrt {-c^2 d+e} \sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{b \left (-3 c^2 d+2 e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{d^2 \sqrt {e} \left (-c^2 d+e\right )^{3/2}}+\frac {b \left (3 c^2 d-2 e\right ) \log \left (-\frac {16 i d^2 \sqrt {e} \sqrt {-c^2 d+e} \left (\sqrt {e}+c \left (i c \sqrt {d}+\sqrt {-c^2 d+e} \sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{b \left (3 c^2 d-2 e\right ) \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d^2 \sqrt {e} \left (-c^2 d+e\right )^{3/2}}\right ) \] Input:
Integrate[(x*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]
Output:
((-4*a)/(e*(d + e*x^2)^2) + (2*b*c*Sqrt[1 + 1/(c^2*x^2)]*x)/(d*(c^2*d - e) *(d + e*x^2)) - (4*b*ArcCsch[c*x])/(e*(d + e*x^2)^2) + (4*b*ArcSinh[1/(c*x )])/(d^2*e) + (b*(3*c^2*d - 2*e)*Log[(16*d^2*Sqrt[e]*Sqrt[-(c^2*d) + e]*(S qrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x)) /(b*(-3*c^2*d + 2*e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d^2*Sqrt[e]*(-(c^2*d) + e )^(3/2)) + (b*(3*c^2*d - 2*e)*Log[((-16*I)*d^2*Sqrt[e]*Sqrt[-(c^2*d) + e]* (Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/ (b*(3*c^2*d - 2*e)*(Sqrt[d] + I*Sqrt[e]*x))])/(d^2*Sqrt[e]*(-(c^2*d) + e)^ (3/2)))/16
Time = 0.49 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6854, 354, 114, 27, 174, 73, 218, 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.
| \(\displaystyle \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6854 |
| \(\displaystyle \frac {b c x \int \frac {1}{x \sqrt {-c^2 x^2-1} \left (e x^2+d\right )^2}dx}{4 e \sqrt {-c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {b c x \int \frac {1}{x^2 \sqrt {-c^2 x^2-1} \left (e x^2+d\right )^2}dx^2}{8 e \sqrt {-c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {b c x \left (\frac {\int \frac {-e x^2 c^2+2 d c^2-2 e}{2 x^2 \sqrt {-c^2 x^2-1} \left (e x^2+d\right )}dx^2}{d \left (c^2 d-e\right )}+\frac {e \sqrt {-c^2 x^2-1}}{d \left (c^2 d-e\right ) \left (d+e x^2\right )}\right )}{8 e \sqrt {-c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \left (\frac {\int \frac {2 \left (c^2 d-e\right )-c^2 e x^2}{x^2 \sqrt {-c^2 x^2-1} \left (e x^2+d\right )}dx^2}{2 d \left (c^2 d-e\right )}+\frac {e \sqrt {-c^2 x^2-1}}{d \left (c^2 d-e\right ) \left (d+e x^2\right )}\right )}{8 e \sqrt {-c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {b c x \left (\frac {\frac {2 \left (c^2 d-e\right ) \int \frac {1}{x^2 \sqrt {-c^2 x^2-1}}dx^2}{d}-\frac {e \left (3 c^2 d-2 e\right ) \int \frac {1}{\sqrt {-c^2 x^2-1} \left (e x^2+d\right )}dx^2}{d}}{2 d \left (c^2 d-e\right )}+\frac {e \sqrt {-c^2 x^2-1}}{d \left (c^2 d-e\right ) \left (d+e x^2\right )}\right )}{8 e \sqrt {-c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {b c x \left (\frac {\frac {2 e \left (3 c^2 d-2 e\right ) \int \frac {1}{-\frac {e x^4}{c^2}+d-\frac {e}{c^2}}d\sqrt {-c^2 x^2-1}}{c^2 d}-\frac {4 \left (c^2 d-e\right ) \int \frac {1}{-\frac {x^4}{c^2}-\frac {1}{c^2}}d\sqrt {-c^2 x^2-1}}{c^2 d}}{2 d \left (c^2 d-e\right )}+\frac {e \sqrt {-c^2 x^2-1}}{d \left (c^2 d-e\right ) \left (d+e x^2\right )}\right )}{8 e \sqrt {-c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b c x \left (\frac {\frac {2 e \left (3 c^2 d-2 e\right ) \int \frac {1}{-\frac {e x^4}{c^2}+d-\frac {e}{c^2}}d\sqrt {-c^2 x^2-1}}{c^2 d}+\frac {4 \arctan \left (\sqrt {-c^2 x^2-1}\right ) \left (c^2 d-e\right )}{d}}{2 d \left (c^2 d-e\right )}+\frac {e \sqrt {-c^2 x^2-1}}{d \left (c^2 d-e\right ) \left (d+e x^2\right )}\right )}{8 e \sqrt {-c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b c x \left (\frac {\frac {4 \arctan \left (\sqrt {-c^2 x^2-1}\right ) \left (c^2 d-e\right )}{d}+\frac {2 \sqrt {e} \left (3 c^2 d-2 e\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{\sqrt {c^2 d-e}}\right )}{d \sqrt {c^2 d-e}}}{2 d \left (c^2 d-e\right )}+\frac {e \sqrt {-c^2 x^2-1}}{d \left (c^2 d-e\right ) \left (d+e x^2\right )}\right )}{8 e \sqrt {-c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
Input:
Int[(x*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]
Output:
-1/4*(a + b*ArcCsch[c*x])/(e*(d + e*x^2)^2) + (b*c*x*((e*Sqrt[-1 - c^2*x^2 ])/(d*(c^2*d - e)*(d + e*x^2)) + ((4*(c^2*d - e)*ArcTan[Sqrt[-1 - c^2*x^2] ])/d + (2*(3*c^2*d - 2*e)*Sqrt[e]*ArcTanh[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/Sqr t[c^2*d - e]])/(d*Sqrt[c^2*d - e]))/(2*d*(c^2*d - e))))/(8*e*Sqrt[-(c^2*x^ 2)])
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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_.) + ArcCsch[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCsch[c*x])/(2*e*(p + 1))), x] - Simp[b*c*(x/(2*e*(p + 1)*Sqrt[(-c^2)*x^2])) Int[(d + e*x^2)^(p + 1) /(x*Sqrt[-1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, - 1]
Leaf count of result is larger than twice the leaf count of optimal. \(915\) vs. \(2(177)=354\).
Time = 6.28 (sec) , antiderivative size = 916, normalized size of antiderivative = 4.47
| method | result | size |
| parts | \(-\frac {a}{4 e \left (x^{2} e +d \right )^{2}}+\frac {b \left (-\frac {c^{6} \operatorname {arccsch}\left (c x \right )}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}-\frac {c \sqrt {c^{2} x^{2}+1}\, \left (4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{4} d^{2}+4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{4} d e \,x^{2}-3 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-3 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}-3 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-3 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+2 \sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, c^{2} d e -4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{2} d e -4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, e^{2} c^{2} x^{2}+2 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e +2 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}+2 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e +2 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}\right )}{16 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x \,d^{2} \sqrt {-\frac {c^{2} d -e}{e}}\, \left (c^{2} d -e \right ) \left (-c e x +\sqrt {-c^{2} d e}\right ) \left (c e x +\sqrt {-c^{2} d e}\right )}\right )}{c^{2}}\) | \(916\) |
| derivativedivides | \(\frac {-\frac {a \,c^{6}}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}+b \,c^{6} \left (-\frac {\operatorname {arccsch}\left (c x \right )}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{4} d^{2}+4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{4} d e \,x^{2}-3 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-3 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}-3 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-3 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+2 \sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, c^{2} d e -4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{2} d e -4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, e^{2} c^{2} x^{2}+2 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e +2 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}+2 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e +2 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}\right )}{16 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{5} x \,d^{2} \sqrt {-\frac {c^{2} d -e}{e}}\, \left (c^{2} d -e \right ) \left (-c e x +\sqrt {-c^{2} d e}\right ) \left (c e x +\sqrt {-c^{2} d e}\right )}\right )}{c^{2}}\) | \(929\) |
| default | \(\frac {-\frac {a \,c^{6}}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}+b \,c^{6} \left (-\frac {\operatorname {arccsch}\left (c x \right )}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{4} d^{2}+4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{4} d e \,x^{2}-3 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-3 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}-3 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-3 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+2 \sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, c^{2} d e -4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, c^{2} d e -4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}\, e^{2} c^{2} x^{2}+2 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e +2 \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}+2 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e +2 \ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d -e}{e}}\, \sqrt {c^{2} x^{2}+1}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}\right )}{16 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{5} x \,d^{2} \sqrt {-\frac {c^{2} d -e}{e}}\, \left (c^{2} d -e \right ) \left (-c e x +\sqrt {-c^{2} d e}\right ) \left (c e x +\sqrt {-c^{2} d e}\right )}\right )}{c^{2}}\) | \(929\) |
Input:
int(x*(a+b*arccsch(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
-1/4*a/e/(e*x^2+d)^2+b/c^2*(-1/4*c^6/e/(c^2*e*x^2+c^2*d)^2*arccsch(c*x)-1/ 16*c*(c^2*x^2+1)^(1/2)*(4*arctanh(1/(c^2*x^2+1)^(1/2))*(-(c^2*d-e)/e)^(1/2 )*c^4*d^2+4*arctanh(1/(c^2*x^2+1)^(1/2))*(-(c^2*d-e)/e)^(1/2)*c^4*d*e*x^2- 3*ln(-2*((-(c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/ (-c*e*x+(-c^2*d*e)^(1/2)))*c^4*d^2-3*ln(-2*((-(c^2*d-e)/e)^(1/2)*(c^2*x^2+ 1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(-c*e*x+(-c^2*d*e)^(1/2)))*c^4*d*e*x^2- 3*ln(-2*(-(-(c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e) /(c*e*x+(-c^2*d*e)^(1/2)))*c^4*d^2-3*ln(-2*(-(-(c^2*d-e)/e)^(1/2)*(c^2*x^2 +1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(c*e*x+(-c^2*d*e)^(1/2)))*c^4*d*e*x^2+ 2*(c^2*x^2+1)^(1/2)*(-(c^2*d-e)/e)^(1/2)*c^2*d*e-4*arctanh(1/(c^2*x^2+1)^( 1/2))*(-(c^2*d-e)/e)^(1/2)*c^2*d*e-4*arctanh(1/(c^2*x^2+1)^(1/2))*(-(c^2*d -e)/e)^(1/2)*e^2*c^2*x^2+2*ln(-2*((-(c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)*e +(-c^2*d*e)^(1/2)*c*x+e)/(-c*e*x+(-c^2*d*e)^(1/2)))*c^2*d*e+2*ln(-2*((-(c^ 2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(-c*e*x+(-c^2* d*e)^(1/2)))*e^2*c^2*x^2+2*ln(-2*(-(-(c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)* e+(-c^2*d*e)^(1/2)*c*x-e)/(c*e*x+(-c^2*d*e)^(1/2)))*c^2*d*e+2*ln(-2*(-(-(c ^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(c*e*x+(-c^2* d*e)^(1/2)))*e^2*c^2*x^2)/((c^2*x^2+1)/c^2/x^2)^(1/2)/x/d^2/(-(c^2*d-e)/e) ^(1/2)/(c^2*d-e)/(-c*e*x+(-c^2*d*e)^(1/2))/(c*e*x+(-c^2*d*e)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (177) = 354\).
Time = 0.25 (sec) , antiderivative size = 1256, normalized size of antiderivative = 6.13 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x*(a+b*arccsch(c*x))/(e*x^2+d)^3,x, algorithm="fricas")
Output:
[-1/16*(4*a*c^4*d^4 - 8*a*c^2*d^3*e + 4*a*d^2*e^2 + (3*b*c^2*d^3 + (3*b*c^ 2*d*e^2 - 2*b*e^3)*x^4 - 2*b*d^2*e + 2*(3*b*c^2*d^2*e - 2*b*d*e^2)*x^2)*sq rt(-c^2*d*e + e^2)*log((c^2*e*x^2 - c^2*d - 2*sqrt(-c^2*d*e + e^2)*c*x*sqr t((c^2*x^2 + 1)/(c^2*x^2)) + 2*e)/(e*x^2 + d)) - 4*(b*c^4*d^4 - 2*b*c^2*d^ 3*e + b*d^2*e^2 + (b*c^4*d^2*e^2 - 2*b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d ^3*e - 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2 )) - c*x + 1) + 4*(b*c^4*d^4 - 2*b*c^2*d^3*e + b*d^2*e^2 + (b*c^4*d^2*e^2 - 2*b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e - 2*b*c^2*d^2*e^2 + b*d*e^3) *x^2)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x - 1) + 4*(b*c^4*d^4 - 2* b*c^2*d^3*e + b*d^2*e^2)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x) ) - 2*((b*c^3*d^2*e^2 - b*c*d*e^3)*x^3 + (b*c^3*d^3*e - b*c*d^2*e^2)*x)*sq rt((c^2*x^2 + 1)/(c^2*x^2)))/(c^4*d^6*e - 2*c^2*d^5*e^2 + d^4*e^3 + (c^4*d ^4*e^3 - 2*c^2*d^3*e^4 + d^2*e^5)*x^4 + 2*(c^4*d^5*e^2 - 2*c^2*d^4*e^3 + d ^3*e^4)*x^2), -1/8*(2*a*c^4*d^4 - 4*a*c^2*d^3*e + 2*a*d^2*e^2 + (3*b*c^2*d ^3 + (3*b*c^2*d*e^2 - 2*b*e^3)*x^4 - 2*b*d^2*e + 2*(3*b*c^2*d^2*e - 2*b*d* e^2)*x^2)*sqrt(c^2*d*e - e^2)*arctan(sqrt(c^2*d*e - e^2)*c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^2*e*x^2 + e)) - 2*(b*c^4*d^4 - 2*b*c^2*d^3*e + b*d^2*e ^2 + (b*c^4*d^2*e^2 - 2*b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e - 2*b*c^ 2*d^2*e^2 + b*d*e^3)*x^2)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) + 2*(b*c^4*d^4 - 2*b*c^2*d^3*e + b*d^2*e^2 + (b*c^4*d^2*e^2 - 2*b*c^2*...
Timed out. \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x*(a+b*acsch(c*x))/(e*x**2+d)**3,x)
Output:
Timed out
\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate(x*(a+b*arccsch(c*x))/(e*x^2+d)^3,x, algorithm="maxima")
Output:
-1/8*(8*c^2*integrate(1/4*x/(c^2*e^3*x^6 + (2*c^2*d*e^2 + e^3)*x^4 + d^2*e + (c^2*d^2*e + 2*d*e^2)*x^2 + (c^2*e^3*x^6 + (2*c^2*d*e^2 + e^3)*x^4 + d^ 2*e + (c^2*d^2*e + 2*d*e^2)*x^2)*sqrt(c^2*x^2 + 1)), x) + (2*c^2*d - e)*lo g(e*x^2 + d)/(c^4*d^4 - 2*c^2*d^3*e + d^2*e^2) - (2*c^4*d^4*log(c) + 2*d^2 *e^2*log(c) - d^2*e^2 - (4*d^3*e*log(c) - d^3*e)*c^2 + (c^2*d^2*e^2 - d*e^ 3)*x^2 + (c^4*d^2*e^2*x^4 + 2*c^4*d^3*e*x^2 + c^4*d^4)*log(c^2*x^2 + 1) - 2*((c^4*d^2*e^2 - 2*c^2*d*e^3 + e^4)*x^4 + 2*(c^4*d^3*e - 2*c^2*d^2*e^2 + d*e^3)*x^2)*log(x) - 2*(c^4*d^4 - 2*c^2*d^3*e + d^2*e^2)*log(sqrt(c^2*x^2 + 1) + 1))/(c^4*d^6*e - 2*c^2*d^5*e^2 + d^4*e^3 + (c^4*d^4*e^3 - 2*c^2*d^3 *e^4 + d^2*e^5)*x^4 + 2*(c^4*d^5*e^2 - 2*c^2*d^4*e^3 + d^3*e^4)*x^2))*b - 1/4*a/(e^3*x^4 + 2*d*e^2*x^2 + d^2*e)
\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate(x*(a+b*arccsch(c*x))/(e*x^2+d)^3,x, algorithm="giac")
Output:
integrate((b*arccsch(c*x) + a)*x/(e*x^2 + d)^3, x)
Timed out. \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \] Input:
int((x*(a + b*asinh(1/(c*x))))/(d + e*x^2)^3,x)
Output:
int((x*(a + b*asinh(1/(c*x))))/(d + e*x^2)^3, x)
\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {4 \left (\int \frac {\mathit {acsch} \left (c x \right ) x}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{2} e +8 \left (\int \frac {\mathit {acsch} \left (c x \right ) x}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b d \,e^{2} x^{2}+4 \left (\int \frac {\mathit {acsch} \left (c x \right ) x}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,e^{3} x^{4}-a}{4 e \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:
int(x*(a+b*acsch(c*x))/(e*x^2+d)^3,x)
Output:
(4*int((acsch(c*x)*x)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x )*b*d**2*e + 8*int((acsch(c*x)*x)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b*d*e**2*x**2 + 4*int((acsch(c*x)*x)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b*e**3*x**4 - a)/(4*e*(d**2 + 2*d*e*x**2 + e**2*x**4))