Integrand size = 19, antiderivative size = 177 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b c x \left (1-c^2 x^2\right )}{8 d \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {a+b \text {arccosh}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{8 d^{3/2} e \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}} \]
1/4*(-a-b*arccosh(c*x))/e/(e*x^2+d)^2+1/8*b*c*x*(-c^2*x^2+1)/d/(c^2*d+e)/( e*x^2+d)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/8*b*c*(2*c^2*d+e)*arctanh(x*(c^2*d+ e)^(1/2)/d^(1/2)/(c^2*x^2-1)^(1/2))*(c^2*x^2-1)^(1/2)/d^(3/2)/e/(c^2*d+e)^ (3/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.83 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.03 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\frac {1}{8} \left (-\frac {\frac {2 a}{e}+\frac {b c x \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}{d \left (c^2 d+e\right )}}{\left (d+e x^2\right )^2}-\frac {2 b \text {arccosh}(c x)}{e \left (d+e x^2\right )^2}-\frac {b c \left (2 c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \arctan \left (\frac {\sqrt {-c^2 d-e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{d^{3/2} \left (-c^2 d-e\right )^{3/2} e \sqrt {-1+c^2 x^2}}\right ) \]
(-(((2*a)/e + (b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2))/(d*(c^2*d + e)))/(d + e*x^2)^2) - (2*b*ArcCosh[c*x])/(e*(d + e*x^2)^2) - (b*c*(2*c^2* d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTan[(Sqrt[-(c^2*d) - e]*x)/(Sqrt[d] *Sqrt[-1 + c^2*x^2])])/(d^(3/2)*(-(c^2*d) - e)^(3/2)*e*Sqrt[-1 + c^2*x^2]) )/8
Time = 0.36 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6372, 648, 296, 291, 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.
| \(\displaystyle \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6372 |
| \(\displaystyle \frac {b c \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1} \left (e x^2+d\right )^2}dx}{4 e}-\frac {a+b \text {arccosh}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 648 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \int \frac {1}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )^2}dx}{4 e \sqrt {c x-1} \sqrt {c x+1}}-\frac {a+b \text {arccosh}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 296 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \left (\frac {\left (2 c^2 d+e\right ) \int \frac {1}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )}dx}{2 d \left (c^2 d+e\right )}-\frac {e x \sqrt {c^2 x^2-1}}{2 d \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{4 e \sqrt {c x-1} \sqrt {c x+1}}-\frac {a+b \text {arccosh}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \left (\frac {\left (2 c^2 d+e\right ) \int \frac {1}{d-\frac {\left (d c^2+e\right ) x^2}{c^2 x^2-1}}d\frac {x}{\sqrt {c^2 x^2-1}}}{2 d \left (c^2 d+e\right )}-\frac {e x \sqrt {c^2 x^2-1}}{2 d \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{4 e \sqrt {c x-1} \sqrt {c x+1}}-\frac {a+b \text {arccosh}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \left (\frac {\left (2 c^2 d+e\right ) \text {arctanh}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{2 d^{3/2} \left (c^2 d+e\right )^{3/2}}-\frac {e x \sqrt {c^2 x^2-1}}{2 d \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{4 e \sqrt {c x-1} \sqrt {c x+1}}-\frac {a+b \text {arccosh}(c x)}{4 e \left (d+e x^2\right )^2}\) |
-1/4*(a + b*ArcCosh[c*x])/(e*(d + e*x^2)^2) + (b*c*Sqrt[-1 + c^2*x^2]*(-1/ 2*(e*x*Sqrt[-1 + c^2*x^2])/(d*(c^2*d + e)*(d + e*x^2)) + ((2*c^2*d + e)*Ar cTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(2*d^(3/2)*(c^2*d + e)^(3/2))))/(4*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.6.8.3.1 Defintions of rubi rules used
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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[(b*c + 2*(p + 1)*(b*c - a*d))/(2*a*(p + 1)*(b*c - a*d)) Int[ (a + b*x^2)^(p + 1)*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, q}, x] && N eQ[b*c - a*d, 0] && EqQ[2*(p + q + 2) + 1, 0] && (LtQ[p, -1] || !LtQ[q, -1 ]) && NeQ[p, -1]
Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) ^2)^(p_), x_Symbol] :> Simp[(c + d*x)^FracPart[m]*((e + f*x)^FracPart[m]/(c *e + d*f*x^2)^FracPart[m]) Int[(c*e + d*f*x^2)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m, n] && EqQ[d*e + c*f, 0] && !(EqQ[p, 2] && LtQ[m, -1])
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])/(2*e*(p + 1))), x] - Simp[b*(c/(2*e*(p + 1))) Int[(d + e*x^2)^(p + 1)/(Sqrt[1 + c*x]*Sqrt [-1 + c*x]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1125\) vs. \(2(154)=308\).
Time = 0.62 (sec) , antiderivative size = 1126, normalized size of antiderivative = 6.36
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1126\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1149\) |
| default | \(\text {Expression too large to display}\) | \(1149\) |
-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*arccosh(c*x)+1/ 16*c^4*e^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)/(e*c*x+(-c^2*d*e)^(1/2)))*c^6*x^2*d^2*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)/(e*c*x+(-c^2*d*e)^(1 /2)))*c^6*d^3-2*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)/(e*c*x-(-c^2*d*e)^(1/2)))*c^6*d^2*e*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)/(e*c*x-(-c^2*d*e)^(1/2 )))*c^6*d^3+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)/(e*c*x+(-c^2*d*e)^(1/2)))*c^4*x^2*d*e^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)/(e*c*x+(-c^2*d*e)^(1 /2)))*c^4*d^2*e-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)/(e*c*x-(-c^2*d*e)^(1/2)))*c^4*d*e^2*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)/(e*c*x-(-c^2*d*e)^(1 /2)))*c^4*d^2*e-2*c^3*d*e*(-c^2*d*e)^(1/2)*(c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e )^(1/2)*x+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)/(e*c*x+(-c^2*d*e)^(1/2)))*c^2*x^2*e^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)/(e*c*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)/(e*c*x-(-c^2*d*e)^(1/2)))*e^3*c^2*x^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)/(e*c*x-(-c^2*d*e)^(1/2)))*c^2*d...
Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (150) = 300\).
Time = 0.38 (sec) , antiderivative size = 1233, normalized size of antiderivative = 6.97 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Too large to display} \]
[-1/16*(2*(2*a + b)*c^4*d^4 + 2*(4*a + b)*c^2*d^3*e + 4*a*d^2*e^2 + 2*(b*c ^4*d^2*e^2 + b*c^2*d*e^3)*x^4 + 4*(b*c^4*d^3*e + b*c^2*d^2*e^2)*x^2 - (2*b *c^3*d^3 + b*c*d^2*e + (2*b*c^3*d*e^2 + b*c*e^3)*x^4 + 2*(2*b*c^3*d^2*e + b*c*d*e^2)*x^2)*sqrt(c^2*d^2 + d*e)*log(-(2*c^2*d^2 - (4*c^4*d^2 + 4*c^2*d *e + e^2)*x^2 + d*e - 2*sqrt(c^2*d^2 + d*e)*((2*c^3*d + c*e)*x^2 - c*d) - 2*sqrt(c^2*x^2 - 1)*(sqrt(c^2*d^2 + d*e)*(2*c^2*d + e)*x + 2*(c^3*d^2 + c* d*e)*x))/(e*x^2 + d)) - 4*((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 )) - 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)) + 2*sqrt(c^2*x^2 - 1)*((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))/(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 + b)*c^4*d^4 + (4*a + b)*c^2*d^3*e + 2 *a*d^2*e^2 + (b*c^4*d^2*e^2 + b*c^2*d*e^3)*x^4 + 2*(b*c^4*d^3*e + b*c^2*d^ 2*e^2)*x^2 - (2*b*c^3*d^3 + b*c*d^2*e + (2*b*c^3*d*e^2 + b*c*e^3)*x^4 + 2* (2*b*c^3*d^2*e + b*c*d*e^2)*x^2)*sqrt(-c^2*d^2 - d*e)*arctan((sqrt(-c^2*d^ 2 - d*e)*sqrt(c^2*x^2 - 1)*e*x - sqrt(-c^2*d^2 - d*e)*(c*e*x^2 + c*d))/(c^ 2*d^2 + d*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)) - ...
Timed out. \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
-1/8*(c^4*log(e*x^2 + d)/(c^4*d^2*e + 2*c^2*d*e^2 + e^3) + 8*c*integrate(1 /4/(c^3*e^3*x^7 + (2*c^3*d*e^2 - c*e^3)*x^5 - c*d^2*e*x + (c^3*d^2*e - 2*c *d*e^2)*x^3 + (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)*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1))), x) - (c^4*d^2 + c^2*d*e + (c^4*d*e + c^2*e^2)*x^2 - 2*(c^4*d^2 + 2*c^2*d*e + e^2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + (c^4*e^2*x^4 + 2*c^4*d*e*x^2 + c^4*d^2)*l og(c*x + 1) + (c^4*e^2*x^4 + 2*c^4*d*e*x^2 + c^4*d^2)*log(c*x - 1))/(c^4*d ^4*e + 2*c^2*d^3*e^2 + d^2*e^3 + (c^4*d^2*e^3 + 2*c^2*d*e^4 + e^5)*x^4 + 2 *(c^4*d^3*e^2 + 2*c^2*d^2*e^3 + d*e^4)*x^2))*b - 1/4*a/(e^3*x^4 + 2*d*e^2* x^2 + d^2*e)
\[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]