Integrand size = 14, antiderivative size = 92 \[ \int \frac {1}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d} \] Output:
-1/2*exp(a/b)*Pi^(1/2)*erf((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(1/2)/d+1 /2*Pi^(1/2)*erfi((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(1/2)/d/exp(a/b)
Time = 0.10 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\frac {e^{-\frac {a}{b}} \left (e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )+\sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )\right )}{2 d \sqrt {a+b \text {arccosh}(c+d x)}} \] Input:
Integrate[1/Sqrt[a + b*ArcCosh[c + d*x]],x]
Output:
(E^((2*a)/b)*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[1/2, a/b + ArcCosh[c + d*x ]] + Sqrt[-((a + b*ArcCosh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c + d *x])/b)])/(2*d*E^(a/b)*Sqrt[a + b*ArcCosh[c + d*x]])
Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6410, 6296, 25, 3042, 26, 3789, 2611, 2633, 2634}
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.
\(\displaystyle \int \frac {1}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx\) |
\(\Big \downarrow \) 6410 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 6296 |
\(\displaystyle \frac {\int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle \frac {i \left (\frac {1}{2} i \int \frac {e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))-\frac {1}{2} i \int \frac {e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )}{b d}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}-i \int e^{\frac {a+b \text {arccosh}(c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}\right )}{b d}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b d}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b d}\) |
Input:
Int[1/Sqrt[a + b*ArcCosh[c + d*x]],x]
Output:
(I*((I/2)*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b ]] - ((I/2)*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/E ^(a/b)))/(b*d)
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[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d , n}, x]
\[\int \frac {1}{\sqrt {a +b \,\operatorname {arccosh}\left (d x +c \right )}}d x\]
Input:
int(1/(a+b*arccosh(d*x+c))^(1/2),x)
Output:
int(1/(a+b*arccosh(d*x+c))^(1/2),x)
Exception generated. \[ \int \frac {1}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a+b*arccosh(d*x+c))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\int \frac {1}{\sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}}\, dx \] Input:
integrate(1/(a+b*acosh(d*x+c))**(1/2),x)
Output:
Integral(1/sqrt(a + b*acosh(c + d*x)), x)
\[ \int \frac {1}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \operatorname {arcosh}\left (d x + c\right ) + a}} \,d x } \] Input:
integrate(1/(a+b*arccosh(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(b*arccosh(d*x + c) + a), x)
\[ \int \frac {1}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \operatorname {arcosh}\left (d x + c\right ) + a}} \,d x } \] Input:
integrate(1/(a+b*arccosh(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(b*arccosh(d*x + c) + a), x)
Timed out. \[ \int \frac {1}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\int \frac {1}{\sqrt {a+b\,\mathrm {acosh}\left (c+d\,x\right )}} \,d x \] Input:
int(1/(a + b*acosh(c + d*x))^(1/2),x)
Output:
int(1/(a + b*acosh(c + d*x))^(1/2), x)
\[ \int \frac {1}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\int \frac {\sqrt {\mathit {acosh} \left (d x +c \right ) b +a}}{\mathit {acosh} \left (d x +c \right ) b +a}d x \] Input:
int(1/(a+b*acosh(d*x+c))^(1/2),x)
Output:
int(sqrt(acosh(c + d*x)*b + a)/(acosh(c + d*x)*b + a),x)