\(\int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^{3/2} \, dx\) [79]

Optimal result

Integrand size = 25, antiderivative size = 342 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^{3/2} \, dx=-\frac {b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \sqrt {a+b \text {arccosh}(c+d x)}}{3 d}-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \sqrt {a+b \text {arccosh}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}}{3 d}-\frac {3 b^{3/2} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{32 d}-\frac {b^{3/2} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{96 d}+\frac {3 b^{3/2} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{32 d}+\frac {b^{3/2} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{96 d} \] Output:

-1/3*b*e^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/d-1/ 
6*b*e^2*(d*x+c-1)^(1/2)*(d*x+c)^2*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))^(1/ 
2)/d+1/3*e^2*(d*x+c)^3*(a+b*arccosh(d*x+c))^(3/2)/d-3/32*b^(3/2)*e^2*exp(a 
/b)*Pi^(1/2)*erf((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/d-1/288*b^(3/2)*e^2*e 
xp(3*a/b)*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2)) 
/d+3/32*b^(3/2)*e^2*Pi^(1/2)*erfi((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/d/ex 
p(a/b)+1/288*b^(3/2)*e^2*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(a+b*arccosh(d*x+c) 
)^(1/2)/b^(1/2))/d/exp(3*a/b)
 

Mathematica [A] (warning: unable to verify)

Time = 1.99 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.73 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^{3/2} \, dx=e^2 \left (\frac {a e^{-\frac {3 a}{b}} \sqrt {a+b \text {arccosh}(c+d x)} \left (9 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+9 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )\right )}{72 d \sqrt {-\frac {(a+b \text {arccosh}(c+d x))^2}{b^2}}}+\frac {\sqrt {b} \left (9 \left (-12 \sqrt {b} \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \sqrt {a+b \text {arccosh}(c+d x)}+8 \sqrt {b} (c+d x) \text {arccosh}(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}+(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+(2 a-3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )+(2 a+b) \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )-\sinh \left (\frac {3 a}{b}\right )\right )+(2 a-b) \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )+\sinh \left (\frac {3 a}{b}\right )\right )+12 \sqrt {b} \sqrt {a+b \text {arccosh}(c+d x)} (2 \text {arccosh}(c+d x) \cosh (3 \text {arccosh}(c+d x))-\sinh (3 \text {arccosh}(c+d x)))\right )}{288 d}\right ) \] Input:

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

Output:

e^2*((a*Sqrt[a + b*ArcCosh[c + d*x]]*(9*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[ 
c + d*x])/b)]*Gamma[3/2, a/b + ArcCosh[c + d*x]] + Sqrt[3]*Sqrt[a/b + ArcC 
osh[c + d*x]]*Gamma[3/2, (-3*(a + b*ArcCosh[c + d*x]))/b] + 9*E^((2*a)/b)* 
Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[3/2, -((a + b*ArcCosh[c + d*x])/b)] + S 
qrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcCosh[c + d*x])/b)]*Gamma[3/2, (3*(a + 
b*ArcCosh[c + d*x]))/b]))/(72*d*E^((3*a)/b)*Sqrt[-((a + b*ArcCosh[c + d*x] 
)^2/b^2)]) + (Sqrt[b]*(9*(-12*Sqrt[b]*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*( 
1 + c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]] + 8*Sqrt[b]*(c + d*x)*ArcCosh[c 
+ d*x]*Sqrt[a + b*ArcCosh[c + d*x]] + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b 
*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (2*a - 3*b)*Sqrt[Pi] 
*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + (2*a 
 + b)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cos 
h[(3*a)/b] - Sinh[(3*a)/b]) + (2*a - b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b 
*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(3*a)/b] + Sinh[(3*a)/b]) + 12*Sqrt[b]* 
Sqrt[a + b*ArcCosh[c + d*x]]*(2*ArcCosh[c + d*x]*Cosh[3*ArcCosh[c + d*x]] 
- Sinh[3*ArcCosh[c + d*x]])))/(288*d))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.96 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.21, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {6411, 27, 6299, 6354, 6302, 25, 5971, 2009, 6330, 6296, 25, 3042, 26, 3789, 2611, 2633, 2634}

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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6299

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

\(\Big \downarrow \) 6354

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}-\frac {1}{2} b \left (-\frac {1}{6} b \int \frac {(c+d x)^2}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)+\frac {2}{3} \int \frac {(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 6302

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}-\frac {1}{2} b \left (\frac {2}{3} \int \frac {(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)-\frac {1}{6} \int -\frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}-\frac {1}{2} b \left (\frac {2}{3} \int \frac {(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{6} \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}-\frac {1}{2} b \left (\frac {2}{3} \int \frac {(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{6} \int \left (\frac {\sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c+d x)}}\right )d(a+b \text {arccosh}(c+d x))+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}-\frac {1}{2} b \left (\frac {2}{3} \int \frac {(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 6330

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}-\frac {1}{2} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} b \int \frac {1}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 6296

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}-\frac {1}{2} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} \int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}-\frac {1}{2} b \left (\frac {2}{3} \left (\frac {1}{2} \int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))+\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}-\frac {1}{2} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}+\frac {1}{2} \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}-\frac {1}{2} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 3789

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}-\frac {1}{2} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} i \left (\frac {1}{2} i \int \frac {e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))-\frac {1}{2} i \int \frac {e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}-\frac {1}{2} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}-i \int e^{\frac {a+b \text {arccosh}(c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}\right )\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}-\frac {1}{2} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}-\frac {1}{2} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\)

Input:

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

Output:

(e^2*(((c + d*x)^3*(a + b*ArcCosh[c + d*x])^(3/2))/3 - (b*((Sqrt[-1 + c + 
d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/3 + (2*(S 
qrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]] - (I/2)*( 
(I/2)*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]] - 
 ((I/2)*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/E^(a/ 
b))))/3 + ((Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt 
[b]])/8 + (Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[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[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*Ar 
cCosh[c + d*x]])/Sqrt[b]])/(8*E^((3*a)/b)))/6))/2))/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[2/d   Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d 
*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2634

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 3042

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

rule 3789

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
/2   Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2   Int[(c + d*x)^m*E 
^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
 

rule 5971

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]
 

rule 6296

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c)   S 
ubst[Int[x^n*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, 
b, c, n}, x]
 

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 6302

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]
 

rule 6330

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p 
_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + 
e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 
*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ 
p]   Int[(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, p}, x] && EqQ[e1, c*d1] && E 
qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -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 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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

e**2*(Integral(a*c**2*sqrt(a + b*acosh(c + d*x)), x) + Integral(a*d**2*x** 
2*sqrt(a + b*acosh(c + d*x)), x) + Integral(b*c**2*sqrt(a + b*acosh(c + d* 
x))*acosh(c + d*x), x) + Integral(2*a*c*d*x*sqrt(a + b*acosh(c + d*x)), x) 
 + Integral(b*d**2*x**2*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x), x) + In 
tegral(2*b*c*d*x*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

e**2*(int(sqrt(acosh(c + d*x)*b + a),x)*a*c**2 + int(sqrt(acosh(c + d*x)*b 
 + a)*acosh(c + d*x)*x**2,x)*b*d**2 + 2*int(sqrt(acosh(c + d*x)*b + a)*aco 
sh(c + d*x)*x,x)*b*c*d + int(sqrt(acosh(c + d*x)*b + a)*acosh(c + d*x),x)* 
b*c**2 + int(sqrt(acosh(c + d*x)*b + a)*x**2,x)*a*d**2 + 2*int(sqrt(acosh( 
c + d*x)*b + a)*x,x)*a*c*d)