Integrand size = 21, antiderivative size = 97 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx=-\frac {2 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{9 d}-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))}{3 d} \] Output:
-2/9*b*e^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-1/9*b*e^2*(d*x+c-1)^(1/2)*(d* x+c)^2*(d*x+c+1)^(1/2)/d+1/3*e^2*(d*x+c)^3*(a+b*arccosh(d*x+c))/d
Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.73 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx=\frac {e^2 \left (-\frac {1}{9} b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (2+c^2+2 c d x+d^2 x^2\right )+\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))\right )}{d} \] Input:
Integrate[(c*e + d*e*x)^2*(a + b*ArcCosh[c + d*x]),x]
Output:
(e^2*(-1/9*(b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(2 + c^2 + 2*c*d*x + d^ 2*x^2)) + ((c + d*x)^3*(a + b*ArcCosh[c + d*x]))/3))/d
Time = 0.40 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6411, 27, 6298, 111, 27, 83}
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.
| \(\displaystyle \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int e^2 (c+d x)^2 (a+b \text {arccosh}(c+d x))d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int (c+d x)^2 (a+b \text {arccosh}(c+d x))d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{3} b \int \frac {(c+d x)^3}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{3} b \left (\frac {1}{3} \int \frac {2 (c+d x)}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{3} b \left (\frac {2}{3} \int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{3} b \left (\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2+\frac {2}{3} \sqrt {c+d x-1} \sqrt {c+d x+1}\right )\right )}{d}\) |
Input:
Int[(c*e + d*e*x)^2*(a + b*ArcCosh[c + d*x]),x]
Output:
(e^2*(-1/3*(b*((2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/3 + (Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/3)) + ((c + d*x)^3*(a + b*ArcCosh[c + d*x]))/3))/d
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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
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 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
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 = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.69
| method | result | size |
| derivativedivides | \(\frac {\frac {e^{2} a \left (d x +c \right )^{3}}{3}+e^{2} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\) | \(67\) |
| default | \(\frac {\frac {e^{2} a \left (d x +c \right )^{3}}{3}+e^{2} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\) | \(67\) |
| parts | \(\frac {e^{2} a \left (d x +c \right )^{3}}{3 d}+\frac {e^{2} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\) | \(69\) |
| orering | \(\frac {\left (5 d^{4} x^{4}+20 c \,d^{3} x^{3}+30 c^{2} d^{2} x^{2}+20 c^{3} d x +5 c^{4}+2 d^{2} x^{2}+4 c d x +2 c^{2}-4\right ) \left (d e x +c e \right )^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}{9 d \left (d x +c \right )^{3}}-\frac {\left (d^{2} x^{2}+2 c d x +c^{2}+2\right ) \left (d x +c -1\right ) \left (d x +c +1\right ) \left (2 \left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) d e +\frac {\left (d e x +c e \right )^{2} b d}{\sqrt {d x +c -1}\, \sqrt {d x +c +1}}\right )}{9 d^{2} \left (d x +c \right )^{2}}\) | \(187\) |
Input:
int((d*e*x+c*e)^2*(a+b*arccosh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(1/3*e^2*a*(d*x+c)^3+e^2*b*(1/3*(d*x+c)^3*arccosh(d*x+c)-1/9*(d*x+c-1) ^(1/2)*(d*x+c+1)^(1/2)*((d*x+c)^2+2)))
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (83) = 166\).
Time = 0.11 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.73 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx=\frac {3 \, a d^{3} e^{2} x^{3} + 9 \, a c d^{2} e^{2} x^{2} + 9 \, a c^{2} d e^{2} x + 3 \, {\left (b d^{3} e^{2} x^{3} + 3 \, b c d^{2} e^{2} x^{2} + 3 \, b c^{2} d e^{2} x + b c^{3} e^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (b d^{2} e^{2} x^{2} + 2 \, b c d e^{2} x + {\left (b c^{2} + 2 \, b\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{9 \, d} \] Input:
integrate((d*e*x+c*e)^2*(a+b*arccosh(d*x+c)),x, algorithm="fricas")
Output:
1/9*(3*a*d^3*e^2*x^3 + 9*a*c*d^2*e^2*x^2 + 9*a*c^2*d*e^2*x + 3*(b*d^3*e^2* x^3 + 3*b*c*d^2*e^2*x^2 + 3*b*c^2*d*e^2*x + b*c^3*e^2)*log(d*x + c + sqrt( d^2*x^2 + 2*c*d*x + c^2 - 1)) - (b*d^2*e^2*x^2 + 2*b*c*d*e^2*x + (b*c^2 + 2*b)*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/d
\[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx=e^{2} \left (\int a c^{2}\, dx + \int a d^{2} x^{2}\, dx + \int b c^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 2 a c d x\, dx + \int b d^{2} x^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 2 b c d x \operatorname {acosh}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate((d*e*x+c*e)**2*(a+b*acosh(d*x+c)),x)
Output:
e**2*(Integral(a*c**2, x) + Integral(a*d**2*x**2, x) + Integral(b*c**2*aco sh(c + d*x), x) + Integral(2*a*c*d*x, x) + Integral(b*d**2*x**2*acosh(c + d*x), x) + Integral(2*b*c*d*x*acosh(c + d*x), x))
Leaf count of result is larger than twice the leaf count of optimal. 449 vs. \(2 (83) = 166\).
Time = 0.04 (sec) , antiderivative size = 449, normalized size of antiderivative = 4.63 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx=\frac {1}{3} \, a d^{2} e^{2} x^{3} + a c d e^{2} x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (d x + c\right ) - d {\left (\frac {3 \, c^{2} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x}{d^{2}} - \frac {{\left (c^{2} - 1\right )} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} - \frac {3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c}{d^{3}}\right )}\right )} b c d e^{2} + \frac {1}{18} \, {\left (6 \, x^{3} \operatorname {arcosh}\left (d x + c\right ) - d {\left (\frac {2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x^{2}}{d^{2}} - \frac {15 \, c^{3} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{4}} - \frac {5 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c x}{d^{3}} + \frac {9 \, {\left (c^{2} - 1\right )} c \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{4}} + \frac {15 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c^{2}}{d^{4}} - \frac {4 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (c^{2} - 1\right )}}{d^{4}}\right )}\right )} b d^{2} e^{2} + a c^{2} e^{2} x + \frac {{\left ({\left (d x + c\right )} \operatorname {arcosh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} - 1}\right )} b c^{2} e^{2}}{d} \] Input:
integrate((d*e*x+c*e)^2*(a+b*arccosh(d*x+c)),x, algorithm="maxima")
Output:
1/3*a*d^2*e^2*x^3 + a*c*d*e^2*x^2 + 1/2*(2*x^2*arccosh(d*x + c) - d*(3*c^2 *log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 + sqrt(d ^2*x^2 + 2*c*d*x + c^2 - 1)*x/d^2 - (c^2 - 1)*log(2*d^2*x + 2*c*d + 2*sqrt (d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1) *c/d^3))*b*c*d*e^2 + 1/18*(6*x^3*arccosh(d*x + c) - d*(2*sqrt(d^2*x^2 + 2* c*d*x + c^2 - 1)*x^2/d^2 - 15*c^3*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2 *c*d*x + c^2 - 1)*d)/d^4 - 5*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c*x/d^3 + 9 *(c^2 - 1)*c*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/ d^4 + 15*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c^2/d^4 - 4*sqrt(d^2*x^2 + 2*c* d*x + c^2 - 1)*(c^2 - 1)/d^4))*b*d^2*e^2 + a*c^2*e^2*x + ((d*x + c)*arccos h(d*x + c) - sqrt((d*x + c)^2 - 1))*b*c^2*e^2/d
Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (83) = 166\).
Time = 0.49 (sec) , antiderivative size = 419, normalized size of antiderivative = 4.32 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx=\frac {1}{3} \, a d^{2} e^{2} x^{3} + a c d e^{2} x^{2} - {\left (d {\left (\frac {c \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d {\left | d \right |}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )\right )} b c^{2} e^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (\frac {x}{d^{2}} - \frac {3 \, c}{d^{3}}\right )} - \frac {{\left (2 \, c^{2} + 1\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{2} {\left | d \right |}}\right )} d\right )} b c d e^{2} + \frac {1}{18} \, {\left (6 \, x^{3} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (x {\left (\frac {2 \, x}{d^{2}} - \frac {5 \, c}{d^{3}}\right )} + \frac {11 \, c^{2} d + 4 \, d}{d^{5}}\right )} + \frac {3 \, {\left (2 \, c^{3} + 3 \, c\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{3} {\left | d \right |}}\right )} d\right )} b d^{2} e^{2} + a c^{2} e^{2} x \] Input:
integrate((d*e*x+c*e)^2*(a+b*arccosh(d*x+c)),x, algorithm="giac")
Output:
1/3*a*d^2*e^2*x^3 + a*c*d*e^2*x^2 - (d*(c*log(abs(-c*d - (x*abs(d) - sqrt( d^2*x^2 + 2*c*d*x + c^2 - 1))*abs(d)))/(d*abs(d)) + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)/d^2) - x*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)))*b*c ^2*e^2 + 1/2*(2*x^2*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - (sq rt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(x/d^2 - 3*c/d^3) - (2*c^2 + 1)*log(abs(-c *d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*abs(d)))/(d^2*abs(d))) *d)*b*c*d*e^2 + 1/18*(6*x^3*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1 )) - (sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(x*(2*x/d^2 - 5*c/d^3) + (11*c^2*d + 4*d)/d^5) + 3*(2*c^3 + 3*c)*log(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2 *c*d*x + c^2 - 1))*abs(d)))/(d^3*abs(d)))*d)*b*d^2*e^2 + a*c^2*e^2*x
Timed out. \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,d x \] Input:
int((c*e + d*e*x)^2*(a + b*acosh(c + d*x)),x)
Output:
int((c*e + d*e*x)^2*(a + b*acosh(c + d*x)), x)
Time = 0.19 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.46 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx=\frac {e^{2} \left (9 \mathit {acosh} \left (d x +c \right ) b \,c^{3}+9 \mathit {acosh} \left (d x +c \right ) b \,c^{2} d x +9 \mathit {acosh} \left (d x +c \right ) b c \,d^{2} x^{2}+3 \mathit {acosh} \left (d x +c \right ) b \,d^{3} x^{3}+8 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, b \,c^{2}-2 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, b c d x -\sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, b \,d^{2} x^{2}-2 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, b -9 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, b \,c^{2}-6 \,\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}+c +d x \right ) b \,c^{3}+9 a \,c^{2} d x +9 a c \,d^{2} x^{2}+3 a \,d^{3} x^{3}\right )}{9 d} \] Input:
int((d*e*x+c*e)^2*(a+b*acosh(d*x+c)),x)
Output:
(e**2*(9*acosh(c + d*x)*b*c**3 + 9*acosh(c + d*x)*b*c**2*d*x + 9*acosh(c + d*x)*b*c*d**2*x**2 + 3*acosh(c + d*x)*b*d**3*x**3 + 8*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*b*c**2 - 2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*b*c*d*x - sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*b*d**2*x**2 - 2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*b - 9*sqrt(c + d*x + 1)*sqrt(c + d*x - 1)*b*c**2 - 6*log (sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1) + c + d*x)*b*c**3 + 9*a*c**2*d*x + 9 *a*c*d**2*x**2 + 3*a*d**3*x**3))/(9*d)