Integrand size = 25, antiderivative size = 444 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=-\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {16 e^4 (c+d x)^3}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {20 e^4 (c+d x)^5}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{12 b^{5/2} d}-\frac {3 e^4 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{8 b^{5/2} d}-\frac {5 e^4 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{24 b^{5/2} d}+\frac {e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{12 b^{5/2} d}+\frac {3 e^4 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{8 b^{5/2} d}+\frac {5 e^4 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{24 b^{5/2} d} \] Output:
-2/3*e^4*(d*x+c-1)^(1/2)*(d*x+c)^4*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c) )^(3/2)+16/3*e^4*(d*x+c)^3/b^2/d/(a+b*arccosh(d*x+c))^(1/2)-20/3*e^4*(d*x+ c)^5/b^2/d/(a+b*arccosh(d*x+c))^(1/2)-1/12*e^4*exp(a/b)*Pi^(1/2)*erf((a+b* arccosh(d*x+c))^(1/2)/b^(1/2))/b^(5/2)/d-3/8*e^4*exp(3*a/b)*3^(1/2)*Pi^(1/ 2)*erf(3^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(5/2)/d-5/24*e^4*exp( 5*a/b)*5^(1/2)*Pi^(1/2)*erf(5^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^ (5/2)/d+1/12*e^4*Pi^(1/2)*erfi((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(5/2) /d/exp(a/b)+3/8*e^4*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(a+b*arccosh(d*x+c))^(1/ 2)/b^(1/2))/b^(5/2)/d/exp(3*a/b)+5/24*e^4*5^(1/2)*Pi^(1/2)*erfi(5^(1/2)*(a +b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(5/2)/d/exp(5*a/b)
Time = 2.22 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.39 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\frac {e^4 e^{-5 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )} \left (-10 \sqrt {5} b e^{5 \text {arccosh}(c+d x)} \left (-\frac {a+b \text {arccosh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )-18 \sqrt {3} b e^{\frac {2 a}{b}+5 \text {arccosh}(c+d x)} \left (-\frac {a+b \text {arccosh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+2 e^{4 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )} \left (2 e^{\frac {2 a}{b}+\text {arccosh}(c+d x)} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} (a+b \text {arccosh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )-2 \left (e^{a/b} \left (b e^{\text {arccosh}(c+d x)} \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)+\left (1+e^{2 \text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))\right )+b e^{\text {arccosh}(c+d x)} \left (-\frac {a+b \text {arccosh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )\right )\right )+3 e^{\frac {5 a}{b}+2 \text {arccosh}(c+d x)} \left (b-6 a \left (1+e^{6 \text {arccosh}(c+d x)}\right )-6 b \text {arccosh}(c+d x)-b e^{6 \text {arccosh}(c+d x)} (1+6 \text {arccosh}(c+d x))+6 \sqrt {3} e^{3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} (a+b \text {arccosh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )\right )+2 e^{\frac {5 a}{b}} \left (-\frac {1}{2} b \left (-1+e^{10 \text {arccosh}(c+d x)}\right )-5 \left (1+e^{10 \text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))+5 \sqrt {5} e^{5 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} (a+b \text {arccosh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )\right )\right )}{48 b^2 d (a+b \text {arccosh}(c+d x))^{3/2}} \] Input:
Integrate[(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^(5/2),x]
Output:
(e^4*(-10*Sqrt[5]*b*E^(5*ArcCosh[c + d*x])*(-((a + b*ArcCosh[c + d*x])/b)) ^(3/2)*Gamma[1/2, (-5*(a + b*ArcCosh[c + d*x]))/b] - 18*Sqrt[3]*b*E^((2*a) /b + 5*ArcCosh[c + d*x])*(-((a + b*ArcCosh[c + d*x])/b))^(3/2)*Gamma[1/2, (-3*(a + b*ArcCosh[c + d*x]))/b] + 2*E^(4*(a/b + ArcCosh[c + d*x]))*(2*E^( (2*a)/b + ArcCosh[c + d*x])*Sqrt[a/b + ArcCosh[c + d*x]]*(a + b*ArcCosh[c + d*x])*Gamma[1/2, a/b + ArcCosh[c + d*x]] - 2*(E^(a/b)*(b*E^ArcCosh[c + d *x]*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x) + (1 + E^(2*ArcCosh[c + d*x]))*(a + b*ArcCosh[c + d*x])) + b*E^ArcCosh[c + d*x]*(-((a + b*ArcCo sh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcCosh[c + d*x])/b)])) + 3*E^( (5*a)/b + 2*ArcCosh[c + d*x])*(b - 6*a*(1 + E^(6*ArcCosh[c + d*x])) - 6*b* ArcCosh[c + d*x] - b*E^(6*ArcCosh[c + d*x])*(1 + 6*ArcCosh[c + d*x]) + 6*S qrt[3]*E^(3*(a/b + ArcCosh[c + d*x]))*Sqrt[a/b + ArcCosh[c + d*x]]*(a + b* ArcCosh[c + d*x])*Gamma[1/2, (3*(a + b*ArcCosh[c + d*x]))/b]) + 2*E^((5*a) /b)*(-1/2*(b*(-1 + E^(10*ArcCosh[c + d*x]))) - 5*(1 + E^(10*ArcCosh[c + d* x]))*(a + b*ArcCosh[c + d*x]) + 5*Sqrt[5]*E^(5*(a/b + ArcCosh[c + d*x]))*S qrt[a/b + ArcCosh[c + d*x]]*(a + b*ArcCosh[c + d*x])*Gamma[1/2, (5*(a + b* ArcCosh[c + d*x]))/b])))/(48*b^2*d*E^(5*(a/b + ArcCosh[c + d*x]))*(a + b*A rcCosh[c + d*x])^(3/2))
Time = 2.23 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.38, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {6411, 27, 6301, 6366, 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.
\(\displaystyle \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx\) |
\(\Big \downarrow \) 6411 |
\(\displaystyle \frac {\int \frac {e^4 (c+d x)^4}{(a+b \text {arccosh}(c+d x))^{5/2}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e^4 \int \frac {(c+d x)^4}{(a+b \text {arccosh}(c+d x))^{5/2}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 6301 |
\(\displaystyle \frac {e^4 \left (-\frac {8 \int \frac {(c+d x)^3}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}}d(c+d x)}{3 b}+\frac {10 \int \frac {(c+d x)^5}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}}d(c+d x)}{3 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\) |
\(\Big \downarrow \) 6366 |
\(\displaystyle \frac {e^4 \left (-\frac {8 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}+\frac {10 \left (\frac {10 \int \frac {(c+d x)^4}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^5}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\) |
\(\Big \downarrow \) 6302 |
\(\displaystyle \frac {e^4 \left (\frac {10 \left (\frac {10 \int -\frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 (c+d x)^5}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {8 \left (\frac {6 \int -\frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {e^4 \left (\frac {10 \left (-\frac {10 \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 (c+d x)^5}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {8 \left (-\frac {6 \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \frac {e^4 \left (\frac {10 \left (-\frac {10 \int \left (\frac {\sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{8 \sqrt {a+b \text {arccosh}(c+d x)}}\right )d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 (c+d x)^5}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {8 \left (-\frac {6 \int \left (\frac {\sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c+d x)}}\right )d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^4 \left (-\frac {8 \left (\frac {6 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}+\frac {10 \left (\frac {10 \left (-\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^5}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{d}\) |
Input:
Int[(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^(5/2),x]
Output:
(e^4*((-2*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(3*b*(a + b*Ar cCosh[c + d*x])^(3/2)) - (8*((-2*(c + d*x)^3)/(b*Sqrt[a + b*ArcCosh[c + d* x]]) + (6*(-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*ArcCo sh[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 *ArcCosh[c + d*x]])/Sqrt[b]])/(8*E^((3*a)/b))))/b^2))/(3*b) + (10*((-2*(c + d*x)^5)/(b*Sqrt[a + b*ArcCosh[c + d*x]]) + (10*(-1/16*(Sqrt[b]*E^(a/b)*S qrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]) - (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[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt [b]])/32 + (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/( 16*E^(a/b)) + (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[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(32*E^((5*a)/b))))/b^2))/(3*b)))/d
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[ x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) )), x] + (-Simp[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcCosh[c*x ])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]) ), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
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_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1 _) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x ]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Simp[f*(m/(b*c*(n + 1)))*Simp [Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Int[ (f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e 1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && 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]
\[\int \frac {\left (d e x +c e \right )^{4}}{\left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{\frac {5}{2}}}d x\]
Input:
int((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(5/2),x)
Output:
int((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(5/2),x)
Exception generated. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(5/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=e^{4} \left (\int \frac {c^{4}}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx\right ) \] Input:
integrate((d*e*x+c*e)**4/(a+b*acosh(d*x+c))**(5/2),x)
Output:
e**4*(Integral(c**4/(a**2*sqrt(a + b*acosh(c + d*x)) + 2*a*b*sqrt(a + b*ac osh(c + d*x))*acosh(c + d*x) + b**2*sqrt(a + b*acosh(c + d*x))*acosh(c + d *x)**2), x) + Integral(d**4*x**4/(a**2*sqrt(a + b*acosh(c + d*x)) + 2*a*b* sqrt(a + b*acosh(c + d*x))*acosh(c + d*x) + b**2*sqrt(a + b*acosh(c + d*x) )*acosh(c + d*x)**2), x) + Integral(4*c*d**3*x**3/(a**2*sqrt(a + b*acosh(c + d*x)) + 2*a*b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x) + b**2*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)**2), x) + Integral(6*c**2*d**2*x**2/(a** 2*sqrt(a + b*acosh(c + d*x)) + 2*a*b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x) + b**2*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)**2), x) + Integral(4 *c**3*d*x/(a**2*sqrt(a + b*acosh(c + d*x)) + 2*a*b*sqrt(a + b*acosh(c + d* x))*acosh(c + d*x) + b**2*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)**2), x ))
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(5/2),x, algorithm="maxima")
Output:
integrate((d*e*x + c*e)^4/(b*arccosh(d*x + c) + a)^(5/2), x)
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(5/2),x, algorithm="giac")
Output:
integrate((d*e*x + c*e)^4/(b*arccosh(d*x + c) + a)^(5/2), x)
Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{5/2}} \,d x \] Input:
int((c*e + d*e*x)^4/(a + b*acosh(c + d*x))^(5/2),x)
Output:
int((c*e + d*e*x)^4/(a + b*acosh(c + d*x))^(5/2), x)
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\text {too large to display} \] Input:
int((d*e*x+c*e)^4/(a+b*acosh(d*x+c))^(5/2),x)
Output:
(e**4*(3*acosh(c + d*x)**2*int((sqrt(acosh(c + d*x)*b + a)*x**6)/(acosh(c + d*x)**3*b**3*c**2 + 2*acosh(c + d*x)**3*b**3*c*d*x + acosh(c + d*x)**3*b **3*d**2*x**2 - acosh(c + d*x)**3*b**3 + 3*acosh(c + d*x)**2*a*b**2*c**2 + 6*acosh(c + d*x)**2*a*b**2*c*d*x + 3*acosh(c + d*x)**2*a*b**2*d**2*x**2 - 3*acosh(c + d*x)**2*a*b**2 + 3*acosh(c + d*x)*a**2*b*c**2 + 6*acosh(c + d *x)*a**2*b*c*d*x + 3*acosh(c + d*x)*a**2*b*d**2*x**2 - 3*acosh(c + d*x)*a* *2*b + a**3*c**2 + 2*a**3*c*d*x + a**3*d**2*x**2 - a**3),x)*b**3*d**7 + 18 *acosh(c + d*x)**2*int((sqrt(acosh(c + d*x)*b + a)*x**5)/(acosh(c + d*x)** 3*b**3*c**2 + 2*acosh(c + d*x)**3*b**3*c*d*x + acosh(c + d*x)**3*b**3*d**2 *x**2 - acosh(c + d*x)**3*b**3 + 3*acosh(c + d*x)**2*a*b**2*c**2 + 6*acosh (c + d*x)**2*a*b**2*c*d*x + 3*acosh(c + d*x)**2*a*b**2*d**2*x**2 - 3*acosh (c + d*x)**2*a*b**2 + 3*acosh(c + d*x)*a**2*b*c**2 + 6*acosh(c + d*x)*a**2 *b*c*d*x + 3*acosh(c + d*x)*a**2*b*d**2*x**2 - 3*acosh(c + d*x)*a**2*b + a **3*c**2 + 2*a**3*c*d*x + a**3*d**2*x**2 - a**3),x)*b**3*c*d**6 + 45*acosh (c + d*x)**2*int((sqrt(acosh(c + d*x)*b + a)*x**4)/(acosh(c + d*x)**3*b**3 *c**2 + 2*acosh(c + d*x)**3*b**3*c*d*x + acosh(c + d*x)**3*b**3*d**2*x**2 - acosh(c + d*x)**3*b**3 + 3*acosh(c + d*x)**2*a*b**2*c**2 + 6*acosh(c + d *x)**2*a*b**2*c*d*x + 3*acosh(c + d*x)**2*a*b**2*d**2*x**2 - 3*acosh(c + d *x)**2*a*b**2 + 3*acosh(c + d*x)*a**2*b*c**2 + 6*acosh(c + d*x)*a**2*b*c*d *x + 3*acosh(c + d*x)*a**2*b*d**2*x**2 - 3*acosh(c + d*x)*a**2*b + a**3...