Integrand size = 25, antiderivative size = 214 \[ \int \frac {(c e+d e x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=-\frac {e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d}-\frac {e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d}+\frac {e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d}+\frac {e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d} \]
-1/24*e^2*exp(3*a/b)*erf(3^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*3^(1/ 2)*Pi^(1/2)/d/b^(1/2)+1/24*e^2*erfi(3^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^( 1/2))*3^(1/2)*Pi^(1/2)/d/exp(3*a/b)/b^(1/2)-1/8*e^2*exp(a/b)*erf((a+b*arcc osh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d/b^(1/2)+1/8*e^2*erfi((a+b*arccosh(d* x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d/exp(a/b)/b^(1/2)
Time = 0.27 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.01 \[ \int \frac {(c e+d e x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\frac {e^2 e^{-\frac {3 a}{b}} \left (3 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )+\sqrt {3} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+3 e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )\right )}{24 d \sqrt {a+b \text {arccosh}(c+d x)}} \]
(e^2*(3*E^((4*a)/b)*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[1/2, a/b + ArcCosh[ c + d*x]] + Sqrt[3]*Sqrt[-((a + b*ArcCosh[c + d*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c + d*x]))/b] + 3*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c + d*x])/ b)]*Gamma[1/2, -((a + b*ArcCosh[c + d*x])/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt[a /b + ArcCosh[c + d*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c + d*x]))/b]))/(24*d* E^((3*a)/b)*Sqrt[a + b*ArcCosh[c + d*x]])
Time = 0.63 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6411, 27, 6302, 25, 5971, 2009}
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.
\(\displaystyle \int \frac {(c e+d e x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx\) |
\(\Big \downarrow \) 6411 |
\(\displaystyle \frac {\int \frac {e^2 (c+d x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e^2 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 6302 |
\(\displaystyle \frac {e^2 \int -\frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {e^2 \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle -\frac {e^2 \int \left (\frac {\sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c+d x)}}\right )d(a+b \text {arccosh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b d}\) |
(e^2*(-1/8*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt [b]]) - (Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/8 + (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]] /Sqrt[b]])/(8*E^(a/b)) + (Sqrt[b]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcC osh[c + d*x]])/Sqrt[b]])/(8*E^((3*a)/b))))/(b*d)
3.2.76.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
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]
\[\int \frac {\left (d e x +c e \right )^{2}}{\sqrt {a +b \,\operatorname {arccosh}\left (d x +c \right )}}d x\]
Exception generated. \[ \int \frac {(c e+d e x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(c e+d e x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=e^{2} \left (\int \frac {c^{2}}{\sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}}\, dx + \int \frac {d^{2} x^{2}}{\sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}}\, dx + \int \frac {2 c d x}{\sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}}\, dx\right ) \]
e**2*(Integral(c**2/sqrt(a + b*acosh(c + d*x)), x) + Integral(d**2*x**2/sq rt(a + b*acosh(c + d*x)), x) + Integral(2*c*d*x/sqrt(a + b*acosh(c + d*x)) , x))
\[ \int \frac {(c e+d e x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{\sqrt {b \operatorname {arcosh}\left (d x + c\right ) + a}} \,d x } \]
\[ \int \frac {(c e+d e x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{\sqrt {b \operatorname {arcosh}\left (d x + c\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^2}{\sqrt {a+b\,\mathrm {acosh}\left (c+d\,x\right )}} \,d x \]