Integrand size = 25, antiderivative size = 217 \[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=-\frac {e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{32 \sqrt {b} d}-\frac {e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d}+\frac {e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{32 \sqrt {b} d}+\frac {e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d} \]
-1/16*e^3*exp(2*a/b)*erf(2^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*2^(1/ 2)*Pi^(1/2)/d/b^(1/2)+1/16*e^3*erfi(2^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^( 1/2))*2^(1/2)*Pi^(1/2)/d/exp(2*a/b)/b^(1/2)-1/32*e^3*exp(4*a/b)*erf(2*(a+b *arccosh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d/b^(1/2)+1/32*e^3*erfi(2*(a+b*ar ccosh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d/exp(4*a/b)/b^(1/2)
Time = 0.30 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.94 \[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\frac {e^3 e^{-\frac {4 a}{b}} \left (\sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )+2 \sqrt {2} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )+e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \left (2 \sqrt {2} \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )+e^{\frac {2 a}{b}} \Gamma \left (\frac {1}{2},\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )\right )\right )}{32 d \sqrt {a+b \text {arccosh}(c+d x)}} \]
(e^3*(Sqrt[-((a + b*ArcCosh[c + d*x])/b)]*Gamma[1/2, (-4*(a + b*ArcCosh[c + d*x]))/b] + 2*Sqrt[2]*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c + d*x])/b)]*Ga mma[1/2, (-2*(a + b*ArcCosh[c + d*x]))/b] + E^((6*a)/b)*Sqrt[a/b + ArcCosh [c + d*x]]*(2*Sqrt[2]*Gamma[1/2, (2*(a + b*ArcCosh[c + d*x]))/b] + E^((2*a )/b)*Gamma[1/2, (4*(a + b*ArcCosh[c + d*x]))/b])))/(32*d*E^((4*a)/b)*Sqrt[ a + b*ArcCosh[c + d*x]])
Time = 0.63 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.94, 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.
\(\displaystyle \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx\) |
\(\Big \downarrow \) 6411 |
\(\displaystyle \frac {\int \frac {e^3 (c+d x)^3}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e^3 \int \frac {(c+d x)^3}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 6302 |
\(\displaystyle \frac {e^3 \int -\frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {e^3 \int \frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle -\frac {e^3 \int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )}{8 \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c+d x)}}\right )d(a+b \text {arccosh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^3 \left (-\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b d}\) |
(e^3*(-1/32*(Sqrt[b]*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c + d* x]])/Sqrt[b]]) - (Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*A rcCosh[c + d*x]])/Sqrt[b]])/8 + (Sqrt[b]*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCo sh[c + d*x]])/Sqrt[b]])/(32*E^((4*a)/b)) + (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[ 2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(8*E^((2*a)/b))))/(b*d)
3.2.75.3.1 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]
\[\int \frac {\left (d e x +c e \right )^{3}}{\sqrt {a +b \,\operatorname {arccosh}\left (d x +c \right )}}d x\]
Exception generated. \[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=e^{3} \left (\int \frac {c^{3}}{\sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}}\, dx + \int \frac {d^{3} x^{3}}{\sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}}\, dx + \int \frac {3 c d^{2} x^{2}}{\sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}}\, dx + \int \frac {3 c^{2} d x}{\sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}}\, dx\right ) \]
e**3*(Integral(c**3/sqrt(a + b*acosh(c + d*x)), x) + Integral(d**3*x**3/sq rt(a + b*acosh(c + d*x)), x) + Integral(3*c*d**2*x**2/sqrt(a + b*acosh(c + d*x)), x) + Integral(3*c**2*d*x/sqrt(a + b*acosh(c + d*x)), x))
\[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{\sqrt {b \operatorname {arcosh}\left (d x + c\right ) + a}} \,d x } \]
\[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{\sqrt {b \operatorname {arcosh}\left (d x + c\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^3}{\sqrt {a+b\,\mathrm {acosh}\left (c+d\,x\right )}} \,d x \]