Integrand size = 16, antiderivative size = 138 \[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^3} \, dx=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a+b \text {arccosh}(c x)}{2 e (d+e x)^2}+\frac {b c^3 d \text {arctanh}\left (\frac {\sqrt {c d+e} \sqrt {1+c x}}{\sqrt {c d-e} \sqrt {-1+c x}}\right )}{(c d-e)^{3/2} e (c d+e)^{3/2}} \]
1/2*(-a-b*arccosh(c*x))/e/(e*x+d)^2+b*c^3*d*arctanh((c*d+e)^(1/2)*(c*x+1)^ (1/2)/(c*d-e)^(1/2)/(c*x-1)^(1/2))/(c*d-e)^(3/2)/e/(c*d+e)^(3/2)-1/2*b*c*( c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*d^2-e^2)/(e*x+d)
Time = 0.24 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.33 \[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^3} \, dx=\frac {1}{2} \left (-\frac {a}{e (d+e x)^2}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b \text {arccosh}(c x)}{e (d+e x)^2}+\frac {b c^3 d \log (d+e x)}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c^3 d \log \left (e+c^2 d x-\sqrt {c^2 d^2-e^2} \sqrt {-1+c x} \sqrt {1+c x}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}\right ) \]
(-(a/(e*(d + e*x)^2)) - (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d^2 - e^2 )*(d + e*x)) - (b*ArcCosh[c*x])/(e*(d + e*x)^2) + (b*c^3*d*Log[d + e*x])/( e*(c^2*d^2 - e^2)^(3/2)) - (b*c^3*d*Log[e + c^2*d*x - Sqrt[c^2*d^2 - e^2]* Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(e*(c^2*d^2 - e^2)^(3/2)))/2
Time = 0.31 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6378, 107, 104, 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.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 6378 |
| \(\displaystyle \frac {b c \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}dx}{2 e}-\frac {a+b \text {arccosh}(c x)}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle \frac {b c \left (\frac {c^2 d \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1} (d+e x)}dx}{c^2 d^2-e^2}-\frac {e \sqrt {c x-1} \sqrt {c x+1}}{\left (c^2 d^2-e^2\right ) (d+e x)}\right )}{2 e}-\frac {a+b \text {arccosh}(c x)}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {b c \left (\frac {2 c^2 d \int \frac {1}{c d-e-\frac {(c d+e) (c x+1)}{c x-1}}d\frac {\sqrt {c x+1}}{\sqrt {c x-1}}}{c^2 d^2-e^2}-\frac {e \sqrt {c x-1} \sqrt {c x+1}}{\left (c^2 d^2-e^2\right ) (d+e x)}\right )}{2 e}-\frac {a+b \text {arccosh}(c x)}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b c \left (\frac {2 c^2 d \text {arctanh}\left (\frac {\sqrt {c x+1} \sqrt {c d+e}}{\sqrt {c x-1} \sqrt {c d-e}}\right )}{\sqrt {c d-e} \sqrt {c d+e} \left (c^2 d^2-e^2\right )}-\frac {e \sqrt {c x-1} \sqrt {c x+1}}{\left (c^2 d^2-e^2\right ) (d+e x)}\right )}{2 e}-\frac {a+b \text {arccosh}(c x)}{2 e (d+e x)^2}\) |
-1/2*(a + b*ArcCosh[c*x])/(e*(d + e*x)^2) + (b*c*(-((e*Sqrt[-1 + c*x]*Sqrt [1 + c*x])/((c^2*d^2 - e^2)*(d + e*x))) + (2*c^2*d*ArcTanh[(Sqrt[c*d + e]* Sqrt[1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/(Sqrt[c*d - e]*Sqrt[c*d + e]*(c^2*d^2 - e^2))))/(2*e)
3.1.19.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*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[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
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[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x _Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcCosh[c*x])^( n - 1)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(121)=242\).
Time = 0.59 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.04
| method | result | size |
| parts | \(-\frac {a}{2 \left (e x +d \right )^{2} e}+\frac {b \left (-\frac {c^{3} \operatorname {arccosh}\left (c x \right )}{2 \left (e c x +c d \right )^{2} e}+\frac {c^{3} \left (-\ln \left (-\frac {2 \left (d \,c^{2} x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d^{2}-\ln \left (-\frac {2 \left (d \,c^{2} x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d e x -e^{2} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\right ) \sqrt {c x +1}\, \sqrt {c x -1}}{2 e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d -e \right ) \left (c d +e \right ) \left (e c x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\) | \(281\) |
| derivativedivides | \(\frac {-\frac {a \,c^{3}}{2 \left (e c x +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\operatorname {arccosh}\left (c x \right )}{2 \left (e c x +c d \right )^{2} e}+\frac {\left (-\ln \left (-\frac {2 \left (d \,c^{2} x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d^{2}-\ln \left (-\frac {2 \left (d \,c^{2} x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d e x -e^{2} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\right ) \sqrt {c x +1}\, \sqrt {c x -1}}{2 e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d -e \right ) \left (c d +e \right ) \left (e c x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\) | \(285\) |
| default | \(\frac {-\frac {a \,c^{3}}{2 \left (e c x +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\operatorname {arccosh}\left (c x \right )}{2 \left (e c x +c d \right )^{2} e}+\frac {\left (-\ln \left (-\frac {2 \left (d \,c^{2} x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d^{2}-\ln \left (-\frac {2 \left (d \,c^{2} x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d e x -e^{2} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\right ) \sqrt {c x +1}\, \sqrt {c x -1}}{2 e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d -e \right ) \left (c d +e \right ) \left (e c x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\) | \(285\) |
-1/2*a/(e*x+d)^2/e+b/c*(-1/2*c^3/(c*e*x+c*d)^2/e*arccosh(c*x)+1/2*c^3/e^2* (-ln(-2*(d*c^2*x-(c^2*x^2-1)^(1/2)*((c^2*d^2-e^2)/e^2)^(1/2)*e+e)/(c*e*x+c *d))*c^2*d^2-ln(-2*(d*c^2*x-(c^2*x^2-1)^(1/2)*((c^2*d^2-e^2)/e^2)^(1/2)*e+ e)/(c*e*x+c*d))*c^2*d*e*x-e^2*(c^2*x^2-1)^(1/2)*((c^2*d^2-e^2)/e^2)^(1/2)) *(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c^2*x^2-1)^(1/2)/(c*d-e)/(c*d+e)/(c*e*x+c*d) /((c^2*d^2-e^2)/e^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (118) = 236\).
Time = 0.36 (sec) , antiderivative size = 1132, normalized size of antiderivative = 8.20 \[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^3} \, dx=\left [-\frac {{\left (a + b\right )} c^{4} d^{6} - {\left (2 \, a + b\right )} c^{2} d^{4} e^{2} + a d^{2} e^{4} + {\left (b c^{4} d^{4} e^{2} - b c^{2} d^{2} e^{4}\right )} x^{2} + {\left (b c^{3} d^{3} e^{2} x^{2} + 2 \, b c^{3} d^{4} e x + b c^{3} d^{5}\right )} \sqrt {c^{2} d^{2} - e^{2}} \log \left (\frac {c^{3} d^{2} x + c d e - \sqrt {c^{2} d^{2} - e^{2}} {\left (c^{2} d x + e\right )} + {\left (c^{2} d^{2} - \sqrt {c^{2} d^{2} - e^{2}} c d - e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{e x + d}\right ) + 2 \, {\left (b c^{4} d^{5} e - b c^{2} d^{3} e^{3}\right )} x - {\left ({\left (b c^{4} d^{4} e^{2} - 2 \, b c^{2} d^{2} e^{4} + b e^{6}\right )} x^{2} + 2 \, {\left (b c^{4} d^{5} e - 2 \, b c^{2} d^{3} e^{3} + b d e^{5}\right )} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{4} d^{6} - 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4} + {\left (b c^{4} d^{4} e^{2} - 2 \, b c^{2} d^{2} e^{4} + b e^{6}\right )} x^{2} + 2 \, {\left (b c^{4} d^{5} e - 2 \, b c^{2} d^{3} e^{3} + b d e^{5}\right )} x\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c^{3} d^{5} e - b c d^{3} e^{3} + {\left (b c^{3} d^{4} e^{2} - b c d^{2} e^{4}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{2 \, {\left (c^{4} d^{8} e - 2 \, c^{2} d^{6} e^{3} + d^{4} e^{5} + {\left (c^{4} d^{6} e^{3} - 2 \, c^{2} d^{4} e^{5} + d^{2} e^{7}\right )} x^{2} + 2 \, {\left (c^{4} d^{7} e^{2} - 2 \, c^{2} d^{5} e^{4} + d^{3} e^{6}\right )} x\right )}}, -\frac {{\left (a + b\right )} c^{4} d^{6} - {\left (2 \, a + b\right )} c^{2} d^{4} e^{2} + a d^{2} e^{4} + {\left (b c^{4} d^{4} e^{2} - b c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (b c^{3} d^{3} e^{2} x^{2} + 2 \, b c^{3} d^{4} e x + b c^{3} d^{5}\right )} \sqrt {-c^{2} d^{2} + e^{2}} \arctan \left (-\frac {\sqrt {-c^{2} d^{2} + e^{2}} \sqrt {c^{2} x^{2} - 1} e - \sqrt {-c^{2} d^{2} + e^{2}} {\left (c e x + c d\right )}}{c^{2} d^{2} - e^{2}}\right ) + 2 \, {\left (b c^{4} d^{5} e - b c^{2} d^{3} e^{3}\right )} x - {\left ({\left (b c^{4} d^{4} e^{2} - 2 \, b c^{2} d^{2} e^{4} + b e^{6}\right )} x^{2} + 2 \, {\left (b c^{4} d^{5} e - 2 \, b c^{2} d^{3} e^{3} + b d e^{5}\right )} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{4} d^{6} - 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4} + {\left (b c^{4} d^{4} e^{2} - 2 \, b c^{2} d^{2} e^{4} + b e^{6}\right )} x^{2} + 2 \, {\left (b c^{4} d^{5} e - 2 \, b c^{2} d^{3} e^{3} + b d e^{5}\right )} x\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c^{3} d^{5} e - b c d^{3} e^{3} + {\left (b c^{3} d^{4} e^{2} - b c d^{2} e^{4}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{2 \, {\left (c^{4} d^{8} e - 2 \, c^{2} d^{6} e^{3} + d^{4} e^{5} + {\left (c^{4} d^{6} e^{3} - 2 \, c^{2} d^{4} e^{5} + d^{2} e^{7}\right )} x^{2} + 2 \, {\left (c^{4} d^{7} e^{2} - 2 \, c^{2} d^{5} e^{4} + d^{3} e^{6}\right )} x\right )}}\right ] \]
[-1/2*((a + b)*c^4*d^6 - (2*a + b)*c^2*d^4*e^2 + a*d^2*e^4 + (b*c^4*d^4*e^ 2 - b*c^2*d^2*e^4)*x^2 + (b*c^3*d^3*e^2*x^2 + 2*b*c^3*d^4*e*x + b*c^3*d^5) *sqrt(c^2*d^2 - e^2)*log((c^3*d^2*x + c*d*e - sqrt(c^2*d^2 - e^2)*(c^2*d*x + e) + (c^2*d^2 - sqrt(c^2*d^2 - e^2)*c*d - e^2)*sqrt(c^2*x^2 - 1))/(e*x + d)) + 2*(b*c^4*d^5*e - b*c^2*d^3*e^3)*x - ((b*c^4*d^4*e^2 - 2*b*c^2*d^2* e^4 + b*e^6)*x^2 + 2*(b*c^4*d^5*e - 2*b*c^2*d^3*e^3 + b*d*e^5)*x)*log(c*x + sqrt(c^2*x^2 - 1)) - (b*c^4*d^6 - 2*b*c^2*d^4*e^2 + b*d^2*e^4 + (b*c^4*d ^4*e^2 - 2*b*c^2*d^2*e^4 + b*e^6)*x^2 + 2*(b*c^4*d^5*e - 2*b*c^2*d^3*e^3 + b*d*e^5)*x)*log(-c*x + sqrt(c^2*x^2 - 1)) + (b*c^3*d^5*e - b*c*d^3*e^3 + (b*c^3*d^4*e^2 - b*c*d^2*e^4)*x)*sqrt(c^2*x^2 - 1))/(c^4*d^8*e - 2*c^2*d^6 *e^3 + d^4*e^5 + (c^4*d^6*e^3 - 2*c^2*d^4*e^5 + d^2*e^7)*x^2 + 2*(c^4*d^7* e^2 - 2*c^2*d^5*e^4 + d^3*e^6)*x), -1/2*((a + b)*c^4*d^6 - (2*a + b)*c^2*d ^4*e^2 + a*d^2*e^4 + (b*c^4*d^4*e^2 - b*c^2*d^2*e^4)*x^2 + 2*(b*c^3*d^3*e^ 2*x^2 + 2*b*c^3*d^4*e*x + b*c^3*d^5)*sqrt(-c^2*d^2 + e^2)*arctan(-(sqrt(-c ^2*d^2 + e^2)*sqrt(c^2*x^2 - 1)*e - sqrt(-c^2*d^2 + e^2)*(c*e*x + c*d))/(c ^2*d^2 - e^2)) + 2*(b*c^4*d^5*e - b*c^2*d^3*e^3)*x - ((b*c^4*d^4*e^2 - 2*b *c^2*d^2*e^4 + b*e^6)*x^2 + 2*(b*c^4*d^5*e - 2*b*c^2*d^3*e^3 + b*d*e^5)*x) *log(c*x + sqrt(c^2*x^2 - 1)) - (b*c^4*d^6 - 2*b*c^2*d^4*e^2 + b*d^2*e^4 + (b*c^4*d^4*e^2 - 2*b*c^2*d^2*e^4 + b*e^6)*x^2 + 2*(b*c^4*d^5*e - 2*b*c^2* d^3*e^3 + b*d*e^5)*x)*log(-c*x + sqrt(c^2*x^2 - 1)) + (b*c^3*d^5*e - b*...
\[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^3} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (d + e x\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((e-c*d)*(e+c*d)>0)', see `assume ?` for mor
\[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^3} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (d+e\,x\right )}^3} \,d x \]