\(\int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx\) [110]

Optimal result

Integrand size = 25, antiderivative size = 552 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=-\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{5 b d (a+b \text {arccosh}(c+d x))^{5/2}}+\frac {16 e^4 (c+d x)^3}{15 b^2 d (a+b \text {arccosh}(c+d x))^{3/2}}-\frac {4 e^4 (c+d x)^5}{3 b^2 d (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {32 e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {40 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b^3 d \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{30 b^{7/2} d}+\frac {9 e^4 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{20 b^{7/2} d}+\frac {5 e^4 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{12 b^{7/2} d}+\frac {e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{30 b^{7/2} d}+\frac {9 e^4 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{20 b^{7/2} d}+\frac {5 e^4 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{12 b^{7/2} d} \] Output:

-2/5*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) 
)^(5/2)+16/15*e^4*(d*x+c)^3/b^2/d/(a+b*arccosh(d*x+c))^(3/2)-4/3*e^4*(d*x+ 
c)^5/b^2/d/(a+b*arccosh(d*x+c))^(3/2)+32/5*e^4*(d*x+c-1)^(1/2)*(d*x+c)^2*( 
d*x+c+1)^(1/2)/b^3/d/(a+b*arccosh(d*x+c))^(1/2)-40/3*e^4*(d*x+c-1)^(1/2)*( 
d*x+c)^4*(d*x+c+1)^(1/2)/b^3/d/(a+b*arccosh(d*x+c))^(1/2)+1/30*e^4*exp(a/b 
)*Pi^(1/2)*erf((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(7/2)/d+9/20*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^ 
(7/2)/d+5/12*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^(7/2)/d+1/30*e^4*Pi^(1/2)*erfi((a+b*arccosh(d*x+c))^( 
1/2)/b^(1/2))/b^(7/2)/d/exp(a/b)+9/20*e^4*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(a 
+b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(7/2)/d/exp(3*a/b)+5/12*e^4*5^(1/2)*Pi 
^(1/2)*erfi(5^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(7/2)/d/exp(5*a/ 
b)
 

Mathematica [A] (warning: unable to verify)

Time = 3.06 (sec) , antiderivative size = 654, normalized size of antiderivative = 1.18 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\frac {e^4 \left (-4 \left (3 b^2 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)+e^{-\text {arccosh}(c+d x)} (a+b \text {arccosh}(c+d x)) \left (-2 a+b-2 b \text {arccosh}(c+d x)+2 e^{\frac {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 )\right )+e^{-\frac {a}{b}} (a+b \text {arccosh}(c+d x)) \left (e^{\frac {a}{b}+\text {arccosh}(c+d x)} (2 a+b+2 b \text {arccosh}(c+d x))+2 b \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 )-9 (a+b \text {arccosh}(c+d x)) \left (12 \sqrt {3} b e^{-\frac {3 a}{b}} \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^{-3 \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 )\right )-5 (a+b \text {arccosh}(c+d x)) \left (2 e^{-5 \text {arccosh}(c+d x)} \left (b+10 a \left (-1+e^{10 \text {arccosh}(c+d x)}\right )-10 b \text {arccosh}(c+d x)+b e^{10 \text {arccosh}(c+d x)} (1+10 \text {arccosh}(c+d x))\right )+20 \sqrt {5} b e^{-\frac {5 a}{b}} \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 )+20 \sqrt {5} e^{\frac {5 a}{b}} \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 )-18 b^2 \sinh (3 \text {arccosh}(c+d x))-6 b^2 \sinh (5 \text {arccosh}(c+d x))\right )}{240 b^3 d (a+b \text {arccosh}(c+d x))^{5/2}} \] Input:

Integrate[(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^(7/2),x]
 

Output:

(e^4*(-4*(3*b^2*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x) + ((a + b 
*ArcCosh[c + d*x])*(-2*a + b - 2*b*ArcCosh[c + d*x] + 2*E^(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]]))/E^ArcCosh[c + d*x] + ((a + b*ArcCosh[c + d*x])*( 
E^(a/b + ArcCosh[c + d*x])*(2*a + b + 2*b*ArcCosh[c + d*x]) + 2*b*(-((a + 
b*ArcCosh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcCosh[c + d*x])/b)]))/ 
E^(a/b)) - 9*(a + b*ArcCosh[c + d*x])*((12*Sqrt[3]*b*(-((a + b*ArcCosh[c + 
 d*x])/b))^(3/2)*Gamma[1/2, (-3*(a + b*ArcCosh[c + d*x]))/b])/E^((3*a)/b) 
+ (2*(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*Sqrt[3]*E^(3*(a/b + ArcCo 
sh[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]))/E^(3*ArcCosh[c + d*x])) - 5*(a + b* 
ArcCosh[c + d*x])*((2*(b + 10*a*(-1 + E^(10*ArcCosh[c + d*x])) - 10*b*ArcC 
osh[c + d*x] + b*E^(10*ArcCosh[c + d*x])*(1 + 10*ArcCosh[c + d*x])))/E^(5* 
ArcCosh[c + d*x]) + (20*Sqrt[5]*b*(-((a + b*ArcCosh[c + d*x])/b))^(3/2)*Ga 
mma[1/2, (-5*(a + b*ArcCosh[c + d*x]))/b])/E^((5*a)/b) + 20*Sqrt[5]*E^((5* 
a)/b)*Sqrt[a/b + ArcCosh[c + d*x]]*(a + b*ArcCosh[c + d*x])*Gamma[1/2, (5* 
(a + b*ArcCosh[c + d*x]))/b]) - 18*b^2*Sinh[3*ArcCosh[c + d*x]] - 6*b^2*Si 
nh[5*ArcCosh[c + d*x]]))/(240*b^3*d*(a + b*ArcCosh[c + d*x])^(5/2))
 

Rubi [A] (verified)

Time = 2.61 (sec) , antiderivative size = 714, normalized size of antiderivative = 1.29, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6411, 27, 6301, 6366, 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.

\(\displaystyle \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int \frac {e^4 (c+d x)^4}{(a+b \text {arccosh}(c+d x))^{7/2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^4 \int \frac {(c+d x)^4}{(a+b \text {arccosh}(c+d x))^{7/2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 6301

\(\displaystyle \frac {e^4 \left (-\frac {8 \int \frac {(c+d x)^3}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}}d(c+d x)}{5 b}+\frac {2 \int \frac {(c+d x)^5}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}}d(c+d x)}{b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\)

\(\Big \downarrow \) 6366

\(\displaystyle \frac {e^4 \left (-\frac {8 \left (\frac {2 \int \frac {(c+d x)^2}{(a+b \text {arccosh}(c+d x))^{3/2}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {2 \left (\frac {10 \int \frac {(c+d x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}}d(c+d x)}{3 b}-\frac {2 (c+d x)^5}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\)

\(\Big \downarrow \) 6300

\(\displaystyle \frac {e^4 \left (-\frac {8 \left (\frac {2 \left (-\frac {2 \int \left (-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c+d x)}}\right )d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {2 \left (\frac {10 \left (-\frac {2 \int \left (-\frac {5 \cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {9 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{8 \sqrt {a+b \text {arccosh}(c+d x)}}\right )d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^5}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^4 \left (-\frac {8 \left (\frac {2 \left (-\frac {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 {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}{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 {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 )\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {2 \left (\frac {10 \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 )}{3 b}-\frac {2 (c+d x)^5}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\)

Input:

Int[(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^(7/2),x]
 

Output:

(e^4*((-2*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(5*b*(a + b*Ar 
cCosh[c + d*x])^(5/2)) - (8*((-2*(c + d*x)^3)/(3*b*(a + b*ArcCosh[c + d*x] 
)^(3/2)) + (2*((-2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b*Sq 
rt[a + b*ArcCosh[c + d*x]]) - (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[3*Pi]*Erf[(S 
qrt[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[3*Pi]*E 
rfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(8*E^((3*a)/b))))/b^2 
))/b))/(5*b) + (2*((-2*(c + d*x)^5)/(3*b*(a + b*ArcCosh[c + d*x])^(3/2)) + 
 (10*((-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))/(3*b)))/b))/d
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6300

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]
 

rule 6301

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]
 

rule 6366

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]
 

rule 6411

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]
 

Maple [F]

\[\int \frac {\left (d e x +c e \right )^{4}}{\left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{\frac {7}{2}}}d x\]

Input:

int((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(7/2),x)
 

Output:

int((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(7/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(7/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((d*e*x+c*e)**4/(a+b*acosh(d*x+c))**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(7/2),x, algorithm="maxima")
 

Output:

integrate((d*e*x + c*e)^4/(b*arccosh(d*x + c) + a)^(7/2), x)
 

Giac [F]

\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(7/2),x, algorithm="giac")
 

Output:

integrate((d*e*x + c*e)^4/(b*arccosh(d*x + c) + a)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \] Input:

int((c*e + d*e*x)^4/(a + b*acosh(c + d*x))^(7/2),x)
 

Output:

int((c*e + d*e*x)^4/(a + b*acosh(c + d*x))^(7/2), x)
 

Reduce [F]

\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\text {too large to display} \] Input:

int((d*e*x+c*e)^4/(a+b*acosh(d*x+c))^(7/2),x)
                                                                                    
                                                                                    
 

Output:

(e**4*(5*acosh(c + d*x)**3*int((sqrt(acosh(c + d*x)*b + a)*x**6)/(acosh(c 
+ d*x)**4*b**4*c**2 + 2*acosh(c + d*x)**4*b**4*c*d*x + acosh(c + d*x)**4*b 
**4*d**2*x**2 - acosh(c + d*x)**4*b**4 + 4*acosh(c + d*x)**3*a*b**3*c**2 + 
 8*acosh(c + d*x)**3*a*b**3*c*d*x + 4*acosh(c + d*x)**3*a*b**3*d**2*x**2 - 
 4*acosh(c + d*x)**3*a*b**3 + 6*acosh(c + d*x)**2*a**2*b**2*c**2 + 12*acos 
h(c + d*x)**2*a**2*b**2*c*d*x + 6*acosh(c + d*x)**2*a**2*b**2*d**2*x**2 - 
6*acosh(c + d*x)**2*a**2*b**2 + 4*acosh(c + d*x)*a**3*b*c**2 + 8*acosh(c + 
 d*x)*a**3*b*c*d*x + 4*acosh(c + d*x)*a**3*b*d**2*x**2 - 4*acosh(c + d*x)* 
a**3*b + a**4*c**2 + 2*a**4*c*d*x + a**4*d**2*x**2 - a**4),x)*b**4*d**7 + 
30*acosh(c + d*x)**3*int((sqrt(acosh(c + d*x)*b + a)*x**5)/(acosh(c + d*x) 
**4*b**4*c**2 + 2*acosh(c + d*x)**4*b**4*c*d*x + acosh(c + d*x)**4*b**4*d* 
*2*x**2 - acosh(c + d*x)**4*b**4 + 4*acosh(c + d*x)**3*a*b**3*c**2 + 8*aco 
sh(c + d*x)**3*a*b**3*c*d*x + 4*acosh(c + d*x)**3*a*b**3*d**2*x**2 - 4*aco 
sh(c + d*x)**3*a*b**3 + 6*acosh(c + d*x)**2*a**2*b**2*c**2 + 12*acosh(c + 
d*x)**2*a**2*b**2*c*d*x + 6*acosh(c + d*x)**2*a**2*b**2*d**2*x**2 - 6*acos 
h(c + d*x)**2*a**2*b**2 + 4*acosh(c + d*x)*a**3*b*c**2 + 8*acosh(c + d*x)* 
a**3*b*c*d*x + 4*acosh(c + d*x)*a**3*b*d**2*x**2 - 4*acosh(c + d*x)*a**3*b 
 + a**4*c**2 + 2*a**4*c*d*x + a**4*d**2*x**2 - a**4),x)*b**4*c*d**6 + 75*a 
cosh(c + d*x)**3*int((sqrt(acosh(c + d*x)*b + a)*x**4)/(acosh(c + d*x)**4* 
b**4*c**2 + 2*acosh(c + d*x)**4*b**4*c*d*x + acosh(c + d*x)**4*b**4*d**...