Integrand size = 25, antiderivative size = 374 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=-\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{b d \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{8 b^{3/2} d}+\frac {3 e^4 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16 b^{3/2} d}+\frac {e^4 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16 b^{3/2} d}+\frac {e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{8 b^{3/2} d}+\frac {3 e^4 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16 b^{3/2} d}+\frac {e^4 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16 b^{3/2} d} \]
1/8*e^4*exp(a/b)*erf((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/ d+1/8*e^4*erfi((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/d/exp( a/b)+3/16*e^4*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)/b^(3/2)/d+3/16*e^4*erfi(3^(1/2)*(a+b*arccosh(d*x+c))^(1/2) /b^(1/2))*3^(1/2)*Pi^(1/2)/b^(3/2)/d/exp(3*a/b)+1/16*e^4*exp(5*a/b)*erf(5^ (1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*5^(1/2)*Pi^(1/2)/b^(3/2)/d+1/16* e^4*erfi(5^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*5^(1/2)*Pi^(1/2)/b^(3 /2)/d/exp(5*a/b)-2*e^4*(d*x+c)^4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b/d/(a+b* arccosh(d*x+c))^(1/2)
Time = 1.16 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.06 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\frac {e^4 e^{-\frac {5 a}{b}} \left (-4 e^{\frac {5 a}{b}} \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)-2 e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )+\sqrt {5} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )+3 \sqrt {3} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+2 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )-3 \sqrt {3} e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )-\sqrt {5} e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )-6 e^{\frac {5 a}{b}} \sinh (3 \text {arccosh}(c+d x))-2 e^{\frac {5 a}{b}} \sinh (5 \text {arccosh}(c+d x))\right )}{16 b d \sqrt {a+b \text {arccosh}(c+d x)}} \]
(e^4*(-4*E^((5*a)/b)*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x) - 2* E^((6*a)/b)*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[1/2, a/b + ArcCosh[c + d*x] ] + Sqrt[5]*Sqrt[-((a + b*ArcCosh[c + d*x])/b)]*Gamma[1/2, (-5*(a + b*ArcC osh[c + d*x]))/b] + 3*Sqrt[3]*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c + d*x])/ b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c + d*x]))/b] + 2*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c + d*x])/b)] - 3*Sq rt[3]*E^((8*a)/b)*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[1/2, (3*(a + b*ArcCos h[c + d*x]))/b] - Sqrt[5]*E^((10*a)/b)*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[ 1/2, (5*(a + b*ArcCosh[c + d*x]))/b] - 6*E^((5*a)/b)*Sinh[3*ArcCosh[c + d* x]] - 2*E^((5*a)/b)*Sinh[5*ArcCosh[c + d*x]]))/(16*b*d*E^((5*a)/b)*Sqrt[a + b*ArcCosh[c + d*x]])
Time = 0.73 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6411, 27, 6300, 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.
\(\displaystyle \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 6411 |
\(\displaystyle \frac {\int \frac {e^4 (c+d x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e^4 \int \frac {(c+d x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 6300 |
\(\displaystyle \frac {e^4 \left (-\frac {2 \int \left (-\frac {5 \cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {9 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{8 \sqrt {a+b \text {arccosh}(c+d x)}}\right )d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^4 \left (-\frac {2 \left (-\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {3}{32} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {5 \pi } \sqrt {b} e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {3}{32} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {5 \pi } \sqrt {b} e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{d}\) |
(e^4*((-2*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(b*Sqrt[a + b* ArcCosh[c + d*x]]) - (2*(-1/16*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*Ar cCosh[c + d*x]]/Sqrt[b]]) - (3*Sqrt[b]*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3] *Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/32 - (Sqrt[b]*E^((5*a)/b)*Sqrt[5* Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/32 - (Sqrt[b]*Sqr t[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(16*E^(a/b)) - (3*Sqrt[b ]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(32*E^( (3*a)/b)) - (Sqrt[b]*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]] )/Sqrt[b]])/(32*E^((5*a)/b))))/b^2))/d
3.2.80.3.1 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[((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]
\[\int \frac {\left (d e x +c e \right )^{4}}{\left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
Exception generated. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=e^{4} \left (\int \frac {c^{4}}{a \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )}}\, dx\right ) \]
e**4*(Integral(c**4/(a*sqrt(a + b*acosh(c + d*x)) + b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)), x) + Integral(d**4*x**4/(a*sqrt(a + b*acosh(c + d* x)) + b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)), x) + Integral(4*c*d**3 *x**3/(a*sqrt(a + b*acosh(c + d*x)) + b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)), x) + Integral(6*c**2*d**2*x**2/(a*sqrt(a + b*acosh(c + d*x)) + b *sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)), x) + Integral(4*c**3*d*x/(a*s qrt(a + b*acosh(c + d*x)) + b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)), x))
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]