Integrand size = 21, antiderivative size = 184 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {b e^2 \left (1-c^2 x^2\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}-\frac {2 d e (a+b \text {arccosh}(c x))}{x}+e^2 x (a+b \text {arccosh}(c x))+\frac {b c d \left (c^2 d+12 e\right ) \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}} \]
-1/3*d^2*(a+b*arccosh(c*x))/x^3-2*d*e*(a+b*arccosh(c*x))/x+e^2*x*(a+b*arcc osh(c*x))+b*e^2*(-c^2*x^2+1)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/6*b*c*d^2*(-c ^2*x^2+1)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/6*b*c*d*(c^2*d+12*e)*arctan((c ^2*x^2-1)^(1/2))*(c^2*x^2-1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.16 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.72 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=-\frac {a d^2}{3 x^3}-\frac {2 a d e}{x}+a e^2 x+b \left (-\frac {e^2}{c}+\frac {c d^2}{6 x^2}\right ) \sqrt {-1+c x} \sqrt {1+c x}-\frac {b \left (d^2+6 d e x^2-3 e^2 x^4\right ) \text {arccosh}(c x)}{3 x^3}-\frac {1}{6} b c d \left (c^2 d+12 e\right ) \arctan \left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \]
-1/3*(a*d^2)/x^3 - (2*a*d*e)/x + a*e^2*x + b*(-(e^2/c) + (c*d^2)/(6*x^2))* Sqrt[-1 + c*x]*Sqrt[1 + c*x] - (b*(d^2 + 6*d*e*x^2 - 3*e^2*x^4)*ArcCosh[c* x])/(3*x^3) - (b*c*d*(c^2*d + 12*e)*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x] )])/6
Time = 0.52 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6373, 27, 1905, 1578, 1192, 1471, 25, 299, 216}
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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx\) |
| \(\Big \downarrow \) 6373 |
| \(\displaystyle -b c \int -\frac {-3 e^2 x^4+6 d e x^2+d^2}{3 x^3 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}-\frac {2 d e (a+b \text {arccosh}(c x))}{x}+e^2 x (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} b c \int \frac {-3 e^2 x^4+6 d e x^2+d^2}{x^3 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}-\frac {2 d e (a+b \text {arccosh}(c x))}{x}+e^2 x (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 1905 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \int \frac {-3 e^2 x^4+6 d e x^2+d^2}{x^3 \sqrt {c^2 x^2-1}}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}-\frac {2 d e (a+b \text {arccosh}(c x))}{x}+e^2 x (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \int \frac {-3 e^2 x^4+6 d e x^2+d^2}{x^4 \sqrt {c^2 x^2-1}}dx^2}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}-\frac {2 d e (a+b \text {arccosh}(c x))}{x}+e^2 x (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 1192 |
| \(\displaystyle \frac {b \sqrt {c^2 x^2-1} \int \frac {-3 e^2 x^8+6 \left (c^2 d-e\right ) e x^4+c^4 d^2-3 e^2+6 c^2 d e}{\left (x^4+1\right )^2}d\sqrt {c^2 x^2-1}}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}-\frac {2 d e (a+b \text {arccosh}(c x))}{x}+e^2 x (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {b \sqrt {c^2 x^2-1} \left (\frac {c^4 d^2 \sqrt {c^2 x^2-1}}{2 \left (x^4+1\right )}-\frac {1}{2} \int -\frac {d^2 c^4+12 d e c^2-6 e^2 x^4-6 e^2}{x^4+1}d\sqrt {c^2 x^2-1}\right )}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}-\frac {2 d e (a+b \text {arccosh}(c x))}{x}+e^2 x (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \sqrt {c^2 x^2-1} \left (\frac {1}{2} \int \frac {d^2 c^4+12 d e c^2-6 e^2 x^4-6 e^2}{x^4+1}d\sqrt {c^2 x^2-1}+\frac {c^4 d^2 \sqrt {c^2 x^2-1}}{2 \left (x^4+1\right )}\right )}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}-\frac {2 d e (a+b \text {arccosh}(c x))}{x}+e^2 x (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \sqrt {c^2 x^2-1} \left (\frac {1}{2} \left (c^2 d \left (c^2 d+12 e\right ) \int \frac {1}{x^4+1}d\sqrt {c^2 x^2-1}-6 e^2 \sqrt {c^2 x^2-1}\right )+\frac {c^4 d^2 \sqrt {c^2 x^2-1}}{2 \left (x^4+1\right )}\right )}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}-\frac {2 d e (a+b \text {arccosh}(c x))}{x}+e^2 x (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}-\frac {2 d e (a+b \text {arccosh}(c x))}{x}+e^2 x (a+b \text {arccosh}(c x))+\frac {b \sqrt {c^2 x^2-1} \left (\frac {1}{2} \left (c^2 d \arctan \left (\sqrt {c^2 x^2-1}\right ) \left (c^2 d+12 e\right )-6 e^2 \sqrt {c^2 x^2-1}\right )+\frac {c^4 d^2 \sqrt {c^2 x^2-1}}{2 \left (x^4+1\right )}\right )}{3 c \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/3*(d^2*(a + b*ArcCosh[c*x]))/x^3 - (2*d*e*(a + b*ArcCosh[c*x]))/x + e^2 *x*(a + b*ArcCosh[c*x]) + (b*Sqrt[-1 + c^2*x^2]*((c^4*d^2*Sqrt[-1 + c^2*x^ 2])/(2*(1 + x^4)) + (-6*e^2*Sqrt[-1 + c^2*x^2] + c^2*d*(c^2*d + 12*e)*ArcT an[Sqrt[-1 + c^2*x^2]])/2))/(3*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.5.78.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
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] :> With[{Qx = PolynomialQuotient[(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] + Simp[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]
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[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) *(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x _Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ q]/(d1*d2 + e1*e2*x^n)^FracPart[q]) Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
Int[((a_.) + ArcCosh[(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]}, Sim p[(a + b*ArcCosh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && Le Q[m + p, 0]))
Time = 0.58 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.04
| method | result | size |
| parts | \(a \left (e^{2} x -\frac {2 d e}{x}-\frac {d^{2}}{3 x^{3}}\right )+b \,c^{3} \left (\frac {\operatorname {arccosh}\left (c x \right ) x \,e^{2}}{c^{3}}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) d e}{c^{3} x}-\frac {\operatorname {arccosh}\left (c x \right ) d^{2}}{3 c^{3} x^{3}}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{6} d^{2} x^{2}+12 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{4} d e \,x^{2}-\sqrt {c^{2} x^{2}-1}\, c^{4} d^{2}+6 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{2} x^{2}\right )}{6 c^{6} \sqrt {c^{2} x^{2}-1}\, x^{2}}\right )\) | \(191\) |
| derivativedivides | \(c^{3} \left (\frac {a \left (c x \,e^{2}-\frac {c \,d^{2}}{3 x^{3}}-\frac {2 d c e}{x}\right )}{c^{4}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) e^{2} c x -\frac {\operatorname {arccosh}\left (c x \right ) c \,d^{2}}{3 x^{3}}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) d c e}{x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{6} d^{2} x^{2}+12 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{4} d e \,x^{2}-\sqrt {c^{2} x^{2}-1}\, c^{4} d^{2}+6 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{2} x^{2}\right )}{6 \sqrt {c^{2} x^{2}-1}\, c^{2} x^{2}}\right )}{c^{4}}\right )\) | \(195\) |
| default | \(c^{3} \left (\frac {a \left (c x \,e^{2}-\frac {c \,d^{2}}{3 x^{3}}-\frac {2 d c e}{x}\right )}{c^{4}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) e^{2} c x -\frac {\operatorname {arccosh}\left (c x \right ) c \,d^{2}}{3 x^{3}}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) d c e}{x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{6} d^{2} x^{2}+12 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{4} d e \,x^{2}-\sqrt {c^{2} x^{2}-1}\, c^{4} d^{2}+6 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{2} x^{2}\right )}{6 \sqrt {c^{2} x^{2}-1}\, c^{2} x^{2}}\right )}{c^{4}}\right )\) | \(195\) |
a*(e^2*x-2*d*e/x-1/3*d^2/x^3)+b*c^3*(1/c^3*arccosh(c*x)*x*e^2-2/c^3*arccos h(c*x)*d*e/x-1/3*arccosh(c*x)*d^2/c^3/x^3-1/6/c^6*(c*x-1)^(1/2)*(c*x+1)^(1 /2)*(arctan(1/(c^2*x^2-1)^(1/2))*c^6*d^2*x^2+12*arctan(1/(c^2*x^2-1)^(1/2) )*c^4*d*e*x^2-(c^2*x^2-1)^(1/2)*c^4*d^2+6*e^2*(c^2*x^2-1)^(1/2)*c^2*x^2)/( c^2*x^2-1)^(1/2)/x^2)
Time = 0.31 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.17 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {6 \, a c e^{2} x^{4} - 12 \, a c d e x^{2} + 2 \, {\left (b c^{4} d^{2} + 12 \, b c^{2} d e\right )} x^{3} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, a c d^{2} + 2 \, {\left (3 \, b c e^{2} x^{4} - 6 \, b c d e x^{2} - b c d^{2} + {\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c^{2} d^{2} x - 6 \, b e^{2} x^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{6 \, c x^{3}} \]
1/6*(6*a*c*e^2*x^4 - 12*a*c*d*e*x^2 + 2*(b*c^4*d^2 + 12*b*c^2*d*e)*x^3*arc tan(-c*x + sqrt(c^2*x^2 - 1)) + 2*(b*c*d^2 + 6*b*c*d*e - 3*b*c*e^2)*x^3*lo g(-c*x + sqrt(c^2*x^2 - 1)) - 2*a*c*d^2 + 2*(3*b*c*e^2*x^4 - 6*b*c*d*e*x^2 - b*c*d^2 + (b*c*d^2 + 6*b*c*d*e - 3*b*c*e^2)*x^3)*log(c*x + sqrt(c^2*x^2 - 1)) + (b*c^2*d^2*x - 6*b*e^2*x^3)*sqrt(c^2*x^2 - 1))/(c*x^3)
\[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{4}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.68 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=-\frac {1}{6} \, {\left ({\left (c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} - 1}}{x^{2}}\right )} c + \frac {2 \, \operatorname {arcosh}\left (c x\right )}{x^{3}}\right )} b d^{2} - 2 \, {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d e + a e^{2} x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b e^{2}}{c} - \frac {2 \, a d e}{x} - \frac {a d^{2}}{3 \, x^{3}} \]
-1/6*((c^2*arcsin(1/(c*abs(x))) - sqrt(c^2*x^2 - 1)/x^2)*c + 2*arccosh(c*x )/x^3)*b*d^2 - 2*(c*arcsin(1/(c*abs(x))) + arccosh(c*x)/x)*b*d*e + a*e^2*x + (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*e^2/c - 2*a*d*e/x - 1/3*a*d^2/ x^3
\[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2}{x^4} \,d x \]