\(\int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx\) [83]

Optimal result

Integrand size = 25, antiderivative size = 469 \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=-\frac {225 b^2 e^3 \sqrt {a+b \text {arccosh}(c+d x)}}{2048 d}+\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {15 b^{5/2} e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 b^{5/2} e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{512 d}-\frac {15 b^{5/2} e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 b^{5/2} e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{512 d} \] Output:

-225/2048*b^2*e^3*(a+b*arccosh(d*x+c))^(1/2)/d+45/256*b^2*e^3*(d*x+c)^2*(a 
+b*arccosh(d*x+c))^(1/2)/d+15/256*b^2*e^3*(d*x+c)^4*(a+b*arccosh(d*x+c))^( 
1/2)/d-15/64*b*e^3*(d*x+c-1)^(1/2)*(d*x+c)*(d*x+c+1)^(1/2)*(a+b*arccosh(d* 
x+c))^(3/2)/d-5/32*b*e^3*(d*x+c-1)^(1/2)*(d*x+c)^3*(d*x+c+1)^(1/2)*(a+b*ar 
ccosh(d*x+c))^(3/2)/d-3/32*e^3*(a+b*arccosh(d*x+c))^(5/2)/d+1/4*e^3*(d*x+c 
)^4*(a+b*arccosh(d*x+c))^(5/2)/d-15/16384*b^(5/2)*e^3*exp(4*a/b)*Pi^(1/2)* 
erf(2*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/d-15/1024*b^(5/2)*e^3*exp(2*a/b) 
*2^(1/2)*Pi^(1/2)*erf(2^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/d-15/163 
84*b^(5/2)*e^3*Pi^(1/2)*erfi(2*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/d/exp(4 
*a/b)-15/1024*b^(5/2)*e^3*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*(a+b*arccosh(d*x+c 
))^(1/2)/b^(1/2))/d/exp(2*a/b)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(968\) vs. \(2(469)=938\).

Time = 8.39 (sec) , antiderivative size = 968, normalized size of antiderivative = 2.06 \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx =\text {Too large to display} \] Input:

Integrate[(c*e + d*e*x)^3*(a + b*ArcCosh[c + d*x])^(5/2),x]
 

Output:

e^3*((a^2*Sqrt[a + b*ArcCosh[c + d*x]]*(Sqrt[a/b + ArcCosh[c + d*x]]*Gamma 
[3/2, (-4*(a + b*ArcCosh[c + d*x]))/b] + 4*Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + 
ArcCosh[c + d*x]]*Gamma[3/2, (-2*(a + b*ArcCosh[c + d*x]))/b] + E^((6*a)/b 
)*Sqrt[-((a + b*ArcCosh[c + d*x])/b)]*(4*Sqrt[2]*Gamma[3/2, (2*(a + b*ArcC 
osh[c + d*x]))/b] + E^((2*a)/b)*Gamma[3/2, (4*(a + b*ArcCosh[c + d*x]))/b] 
)))/(128*d*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c + d*x])^2/b^2)]) + (a*Sqrt[ 
b]*((8*a + 3*b)*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(C 
osh[(4*a)/b] - Sinh[(4*a)/b]) + (8*a - 3*b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*Arc 
Cosh[c + d*x]])/Sqrt[b]]*(Cosh[(4*a)/b] + Sinh[(4*a)/b]) + 8*((4*a + 3*b)* 
Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a 
)/b] - Sinh[(2*a)/b]) + (4*a - 3*b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*Arc 
Cosh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + 8*Sqrt[b]*Sqrt[ 
a + b*ArcCosh[c + d*x]]*(4*ArcCosh[c + d*x]*Cosh[2*ArcCosh[c + d*x]] - 3*S 
inh[2*ArcCosh[c + d*x]])) + 8*Sqrt[b]*Sqrt[a + b*ArcCosh[c + d*x]]*(8*ArcC 
osh[c + d*x]*Cosh[4*ArcCosh[c + d*x]] - 3*Sinh[4*ArcCosh[c + d*x]])))/(102 
4*d) + (-(Sqrt[b]*(64*a^2 + 48*a*b + 15*b^2)*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*A 
rcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(4*a)/b] - Sinh[(4*a)/b])) - Sqrt[b]*(64* 
a^2 - 48*a*b + 15*b^2)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[ 
b]]*(Cosh[(4*a)/b] + Sinh[(4*a)/b]) - 16*(Sqrt[b]*(16*a^2 + 24*a*b + 15*b^ 
2)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cos...
 

Rubi [A] (verified)

Time = 5.56 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.18, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {6411, 27, 6299, 6354, 6299, 6354, 6299, 6308, 6368, 3042, 3793, 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.

\(\displaystyle \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int e^3 (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^3 \int (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}d(c+d x)}{d}\)

\(\Big \downarrow \) 6299

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 6299

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 6299

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

\(\Big \downarrow \) 6308

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

\(\Big \downarrow \) 6368

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{8} b \left (-\frac {3}{8} b \left (\frac {1}{8} \left (-\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {3}{4} \sqrt {a+b \text {arccosh}(c+d x)}\right )+\frac {1}{4} (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}\right )+\frac {3}{4} \left (-\frac {3}{4} b \left (\frac {1}{4} \left (-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\sqrt {a+b \text {arccosh}(c+d x)}\right )+\frac {1}{2} (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}\right )+\frac {(a+b \text {arccosh}(c+d x))^{5/2}}{5 b}+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}\right )+\frac {1}{4} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}\right )\right )}{d}\)

Input:

Int[(c*e + d*e*x)^3*(a + b*ArcCosh[c + d*x])^(5/2),x]
 

Output:

(e^3*(((c + d*x)^4*(a + b*ArcCosh[c + d*x])^(5/2))/4 - (5*b*((Sqrt[-1 + c 
+ d*x]*(c + d*x)^3*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2))/4 - ( 
3*b*(((c + d*x)^4*Sqrt[a + b*ArcCosh[c + d*x]])/4 + ((-3*Sqrt[a + b*ArcCos 
h[c + d*x]])/4 - (Sqrt[b]*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c 
 + d*x]])/Sqrt[b]])/32 - (Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt 
[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/4 - (Sqrt[b]*Sqrt[Pi]*Erfi[(2*Sqrt[a + 
 b*ArcCosh[c + d*x]])/Sqrt[b]])/(32*E^((4*a)/b)) - (Sqrt[b]*Sqrt[Pi/2]*Erf 
i[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(4*E^((2*a)/b)))/8))/8 
+ (3*((Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d 
*x])^(3/2))/2 + (a + b*ArcCosh[c + d*x])^(5/2)/(5*b) - (3*b*(((c + d*x)^2* 
Sqrt[a + b*ArcCosh[c + d*x]])/2 + (-Sqrt[a + b*ArcCosh[c + d*x]] - (Sqrt[b 
]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b 
]])/4 - (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sq 
rt[b]])/(4*E^((2*a)/b)))/4))/4))/4))/8))/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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6299

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^(m + 1)*((a + b*ArcCosh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 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] && GtQ[n, 0]
 

rule 6308

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq 
rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + 
 c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ 
c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 
] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
 

rule 6354

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 
1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 
1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( 
m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1)))   Int[(f*x)^(m 
 - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f 
*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/( 
-1 + c*x)^p]   Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*( 
a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, 
p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N 
eQ[m + 2*p + 1, 0]
 

rule 6368

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x 
_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* 
Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p]   Subst[In 
t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c 
*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ 
e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
 

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]

\[\int \left (d e x +c e \right )^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{\frac {5}{2}}d x\]

Input:

int((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^(5/2),x)
 

Output:

int((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^(5/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d*e*x+c*e)^3*(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)
 

Sympy [F(-1)]

Timed out. \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\text {Timed out} \] Input:

integrate((d*e*x+c*e)**3*(a+b*acosh(d*x+c))**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^(5/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate((d*e*x + c*e)^3*(b*arccosh(d*x + c) + a)^(5/2), x)
 

Giac [F]

\[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^(5/2),x, algorithm="giac")
 

Output:

integrate((d*e*x + c*e)^3*(b*arccosh(d*x + c) + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{5/2} \,d x \] Input:

int((c*e + d*e*x)^3*(a + b*acosh(c + d*x))^(5/2),x)
 

Output:

int((c*e + d*e*x)^3*(a + b*acosh(c + d*x))^(5/2), x)
 

Reduce [F]

\[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=e^{3} \left (\left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}d x \right ) a^{2} c^{3}+2 \left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right ) x^{3}d x \right ) a b \,d^{3}+6 \left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right ) x^{2}d x \right ) a b c \,d^{2}+6 \left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right ) x d x \right ) a b \,c^{2} d +2 \left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right )d x \right ) a b \,c^{3}+\left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right )^{2} x^{3}d x \right ) b^{2} d^{3}+3 \left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right )^{2} x^{2}d x \right ) b^{2} c \,d^{2}+3 \left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right )^{2} x d x \right ) b^{2} c^{2} d +\left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right )^{2}d x \right ) b^{2} c^{3}+\left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, x^{3}d x \right ) a^{2} d^{3}+3 \left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, x^{2}d x \right ) a^{2} c \,d^{2}+3 \left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, x d x \right ) a^{2} c^{2} d \right ) \] Input:

int((d*e*x+c*e)^3*(a+b*acosh(d*x+c))^(5/2),x)
 

Output:

e**3*(int(sqrt(acosh(c + d*x)*b + a),x)*a**2*c**3 + 2*int(sqrt(acosh(c + d 
*x)*b + a)*acosh(c + d*x)*x**3,x)*a*b*d**3 + 6*int(sqrt(acosh(c + d*x)*b + 
 a)*acosh(c + d*x)*x**2,x)*a*b*c*d**2 + 6*int(sqrt(acosh(c + d*x)*b + a)*a 
cosh(c + d*x)*x,x)*a*b*c**2*d + 2*int(sqrt(acosh(c + d*x)*b + a)*acosh(c + 
 d*x),x)*a*b*c**3 + int(sqrt(acosh(c + d*x)*b + a)*acosh(c + d*x)**2*x**3, 
x)*b**2*d**3 + 3*int(sqrt(acosh(c + d*x)*b + a)*acosh(c + d*x)**2*x**2,x)* 
b**2*c*d**2 + 3*int(sqrt(acosh(c + d*x)*b + a)*acosh(c + d*x)**2*x,x)*b**2 
*c**2*d + int(sqrt(acosh(c + d*x)*b + a)*acosh(c + d*x)**2,x)*b**2*c**3 + 
int(sqrt(acosh(c + d*x)*b + a)*x**3,x)*a**2*d**3 + 3*int(sqrt(acosh(c + d* 
x)*b + a)*x**2,x)*a**2*c*d**2 + 3*int(sqrt(acosh(c + d*x)*b + a)*x,x)*a**2 
*c**2*d)