Integrand size = 21, antiderivative size = 110 \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^2} \, dx=-\frac {e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b d (a+b \text {arccosh}(c+d x))}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2 d}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2 d} \]
e*Chi(2*(a+b*arccosh(d*x+c))/b)*cosh(2*a/b)/b^2/d-e*Shi(2*(a+b*arccosh(d*x +c))/b)*sinh(2*a/b)/b^2/d-e*(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b/d/(a +b*arccosh(d*x+c))
Time = 0.48 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.98 \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^2} \, dx=\frac {e \left (-\frac {b \sqrt {\frac {-1+c+d x}{1+c+d x}} \left (c+c^2+2 c d x+d x (1+d x)\right )}{a+b \text {arccosh}(c+d x)}+\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )}{b^2 d} \]
(e*(-((b*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(c + c^2 + 2*c*d*x + d*x*(1 + d*x)))/(a + b*ArcCosh[c + d*x])) + Cosh[(2*a)/b]*CoshIntegral[2*(a/b + Arc Cosh[c + d*x])] - Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c + d*x])])) /(b^2*d)
Time = 0.55 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {6411, 27, 6300, 25, 3042, 3784, 26, 3042, 26, 3779, 3782}
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.
| \(\displaystyle \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int \frac {e (c+d x)}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int \frac {c+d x}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6300 |
| \(\displaystyle \frac {e \left (-\frac {\int -\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e \left (\frac {\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e \left (-\frac {\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {\int \frac {\sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}\right )}{d}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {e \left (-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}-\frac {-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))+i \sinh \left (\frac {2 a}{b}\right ) \int -\frac {i \sinh \left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e \left (-\frac {\sinh \left (\frac {2 a}{b}\right ) \int \frac {\sinh \left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e \left (-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}-\frac {\sinh \left (\frac {2 a}{b}\right ) \int -\frac {i \sin \left (\frac {2 i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e \left (-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}-\frac {-i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}\right )}{d}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle \frac {e \left (-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}-\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}\right )}{d}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {e \left (-\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )-\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}\right )}{d}\) |
(e*(-((Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(b*(a + b*ArcCosh[c + d*x]))) - (-(Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c + d*x]))/b] ) + Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/b^2))/d
3.2.39.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f , fz}, x] && EqQ[d*e - c*f*fz*I, 0]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz }, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) )), x] + Simp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -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 = 0.08 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.55
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e}{4 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{2 b^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{4 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{2 b^{2}}}{d}\) | \(170\) |
| default | \(\frac {\frac {\left (-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e}{4 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{2 b^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{4 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{2 b^{2}}}{d}\) | \(170\) |
1/d*(1/4*(-2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)+2*(d*x+c)^2-1)*e/b/(a +b*arccosh(d*x+c))-1/2*e/b^2*exp(2*a/b)*Ei(1,2*arccosh(d*x+c)+2*a/b)-1/4/b *e*(2*(d*x+c)^2-1+2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c))/(a+b*arccosh( d*x+c))-1/2/b^2*e*exp(-2*a/b)*Ei(1,-2*arccosh(d*x+c)-2*a/b))
\[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int { \frac {d e x + c e}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^2} \, dx=e \left (\int \frac {c}{a^{2} + 2 a b \operatorname {acosh}{\left (c + d x \right )} + b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a^{2} + 2 a b \operatorname {acosh}{\left (c + d x \right )} + b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx\right ) \]
e*(Integral(c/(a**2 + 2*a*b*acosh(c + d*x) + b**2*acosh(c + d*x)**2), x) + Integral(d*x/(a**2 + 2*a*b*acosh(c + d*x) + b**2*acosh(c + d*x)**2), x))
\[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int { \frac {d e x + c e}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
-(d^4*e*x^4 + 4*c*d^3*e*x^3 + c^4*e - c^2*e + (6*c^2*d^2*e - d^2*e)*x^2 + (d^3*e*x^3 + 3*c*d^2*e*x^2 + c^3*e - c*e + (3*c^2*d*e - d*e)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + 2*(2*c^3*d*e - c*d*e)*x)/(a*b*d^3*x^2 + 2*a*b* c*d^2*x + (c^2*d - d)*a*b + (a*b*d^2*x + a*b*c*d)*sqrt(d*x + c + 1)*sqrt(d *x + c - 1) + (b^2*d^3*x^2 + 2*b^2*c*d^2*x + (c^2*d - d)*b^2 + (b^2*d^2*x + b^2*c*d)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1 )*sqrt(d*x + c - 1) + c)) + integrate((2*d^5*e*x^5 + 10*c*d^4*e*x^4 + 2*c^ 5*e - 4*c^3*e + 4*(5*c^2*d^3*e - d^3*e)*x^3 + 2*(d^3*e*x^3 + 3*c*d^2*e*x^2 + 3*c^2*d*e*x + c^3*e)*(d*x + c + 1)*(d*x + c - 1) + 4*(5*c^3*d^2*e - 3*c *d^2*e)*x^2 + (4*d^4*e*x^4 + 16*c*d^3*e*x^3 + 4*c^4*e - 4*c^2*e + 4*(6*c^2 *d^2*e - d^2*e)*x^2 + 8*(2*c^3*d*e - c*d*e)*x + e)*sqrt(d*x + c + 1)*sqrt( d*x + c - 1) + 2*c*e + 2*(5*c^4*d*e - 6*c^2*d*e + d*e)*x)/(a*b*d^4*x^4 + 4 *a*b*c*d^3*x^3 + 2*(3*c^2*d^2 - d^2)*a*b*x^2 + 4*(c^3*d - c*d)*a*b*x + (a* b*d^2*x^2 + 2*a*b*c*d*x + a*b*c^2)*(d*x + c + 1)*(d*x + c - 1) + (c^4 - 2* c^2 + 1)*a*b + 2*(a*b*d^3*x^3 + 3*a*b*c*d^2*x^2 + (3*c^2*d - d)*a*b*x + (c ^3 - c)*a*b)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (b^2*d^4*x^4 + 4*b^2*c* d^3*x^3 + 2*(3*c^2*d^2 - d^2)*b^2*x^2 + 4*(c^3*d - c*d)*b^2*x + (b^2*d^2*x ^2 + 2*b^2*c*d*x + b^2*c^2)*(d*x + c + 1)*(d*x + c - 1) + (c^4 - 2*c^2 + 1 )*b^2 + 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + (3*c^2*d - d)*b^2*x + (c^3 - c) *b^2)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*...
\[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int { \frac {d e x + c e}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int \frac {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2} \,d x \]