Integrand size = 21, antiderivative size = 66 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^3} \, dx=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{2 d e^3 (c+d x)}-\frac {a+b \text {arccosh}(c+d x)}{2 d e^3 (c+d x)^2} \] Output:
1/2*b*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d/e^3/(d*x+c)-1/2*(a+b*arccosh(d*x+c ))/d/e^3/(d*x+c)^2
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^3} \, dx=-\frac {a-b \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}+b \text {arccosh}(c+d x)}{2 d e^3 (c+d x)^2} \] Input:
Integrate[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^3,x]
Output:
-1/2*(a - b*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x] + b*ArcCosh[c + d*x])/(d*e^3*(c + d*x)^2)
Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6411, 27, 6298, 106}
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.
\(\displaystyle \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^3} \, dx\) |
\(\Big \downarrow \) 6411 |
\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c+d x)}{e^3 (c+d x)^3}d(c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c+d x)}{(c+d x)^3}d(c+d x)}{d e^3}\) |
\(\Big \downarrow \) 6298 |
\(\displaystyle \frac {\frac {1}{2} b \int \frac {1}{\sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1}}d(c+d x)-\frac {a+b \text {arccosh}(c+d x)}{2 (c+d x)^2}}{d e^3}\) |
\(\Big \downarrow \) 106 |
\(\displaystyle \frac {\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{2 (c+d x)}-\frac {a+b \text {arccosh}(c+d x)}{2 (c+d x)^2}}{d e^3}\) |
Input:
Int[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^3,x]
Output:
((b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(2*(c + d*x)) - (a + b*ArcCosh[c + d*x])/(2*(c + d*x)^2))/(d*e^3)
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_)*((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] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0] && NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*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, m}, x] && IGtQ[n, 0] & & NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.14 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {-\frac {a}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}+\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 d x +2 c}\right )}{e^{3}}}{d}\) | \(65\) |
default | \(\frac {-\frac {a}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}+\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 d x +2 c}\right )}{e^{3}}}{d}\) | \(65\) |
parts | \(-\frac {a}{2 e^{3} \left (d x +c \right )^{2} d}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}+\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 d x +2 c}\right )}{e^{3} d}\) | \(67\) |
orering | \(\frac {\left (d x +c \right ) \left (3 d^{2} x^{2}+6 c d x +3 c^{2}-4\right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}{2 d \left (d e x +c e \right )^{3}}+\frac {\left (d x +c -1\right ) \left (d x +c +1\right ) \left (d x +c \right )^{2} \left (\frac {b d}{\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d e x +c e \right )^{3}}-\frac {3 \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) d e}{\left (d e x +c e \right )^{4}}\right )}{2 d^{2}}\) | \(130\) |
Input:
int((a+b*arccosh(d*x+c))/(d*e*x+c*e)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/2*a/e^3/(d*x+c)^2+b/e^3*(-1/2/(d*x+c)^2*arccosh(d*x+c)+1/2*(d*x+c- 1)^(1/2)*(d*x+c+1)^(1/2)/(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (58) = 116\).
Time = 0.10 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.77 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^3} \, dx=\frac {a d^{2} x^{2} + 2 \, a c d x - b c^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + {\left (b c^{2} d x + b c^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{2 \, {\left (c^{2} d^{3} e^{3} x^{2} + 2 \, c^{3} d^{2} e^{3} x + c^{4} d e^{3}\right )}} \] Input:
integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^3,x, algorithm="fricas")
Output:
1/2*(a*d^2*x^2 + 2*a*c*d*x - b*c^2*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) + (b*c^2*d*x + b*c^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/(c^2*d^ 3*e^3*x^2 + 2*c^3*d^2*e^3*x + c^4*d*e^3)
\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^3} \, dx=\frac {\int \frac {a}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b \operatorname {acosh}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \] Input:
integrate((a+b*acosh(d*x+c))/(d*e*x+c*e)**3,x)
Output:
(Integral(a/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral (b*acosh(c + d*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x))/e** 3
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (58) = 116\).
Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.79 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^3} \, dx=\frac {1}{2} \, b {\left (\frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d}{d^{3} e^{3} x + c d^{2} e^{3}} - \frac {\operatorname {arcosh}\left (d x + c\right )}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}}\right )} - \frac {a}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \] Input:
integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^3,x, algorithm="maxima")
Output:
1/2*b*(sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d/(d^3*e^3*x + c*d^2*e^3) - arcco sh(d*x + c)/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3)) - 1/2*a/(d^3*e^3*x^ 2 + 2*c*d^2*e^3*x + c^2*d*e^3)
\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^3} \, dx=\int { \frac {b \operatorname {arcosh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{3}} \,d x } \] Input:
integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^3,x, algorithm="giac")
Output:
integrate((b*arccosh(d*x + c) + a)/(d*e*x + c*e)^3, x)
Timed out. \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \] Input:
int((a + b*acosh(c + d*x))/(c*e + d*e*x)^3,x)
Output:
int((a + b*acosh(c + d*x))/(c*e + d*e*x)^3, x)
\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^3} \, dx=\frac {2 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) b \,c^{2} d +4 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) b c \,d^{2} x +2 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) b \,d^{3} x^{2}-a}{2 d \,e^{3} \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:
int((a+b*acosh(d*x+c))/(d*e*x+c*e)^3,x)
Output:
(2*int(acosh(c + d*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*b *c**2*d + 4*int(acosh(c + d*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x **3),x)*b*c*d**2*x + 2*int(acosh(c + d*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x* *2 + d**3*x**3),x)*b*d**3*x**2 - a)/(2*d*e**3*(c**2 + 2*c*d*x + d**2*x**2) )