3.2.62 \(\int (c e+d e x) (a+b \text {arccosh}(c+d x))^{3/2} \, dx\) [162]

3.2.62.1 Optimal result

Integrand size = 23, antiderivative size = 212 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{3/2} \, dx=-\frac {3 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \sqrt {a+b \text {arccosh}(c+d x)}}{8 d}-\frac {e (a+b \text {arccosh}(c+d x))^{3/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^{3/2}}{2 d}-\frac {3 b^{3/2} e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{64 d}+\frac {3 b^{3/2} e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{64 d} \]

output

-1/4*e*(a+b*arccosh(d*x+c))^(3/2)/d+1/2*e*(d*x+c)^2*(a+b*arccosh(d*x+c))^( 
3/2)/d-3/128*b^(3/2)*e*exp(2*a/b)*erf(2^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b 
^(1/2))*2^(1/2)*Pi^(1/2)/d+3/128*b^(3/2)*e*erfi(2^(1/2)*(a+b*arccosh(d*x+c 
))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d/exp(2*a/b)-3/8*b*e*(d*x+c)*(d*x+c-1)^ 
(1/2)*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/d
 

3.2.62.2 Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.85 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{3/2} \, dx=\frac {e \left (3 b^{3/2} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right )-3 b^{3/2} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+8 \sqrt {a+b \text {arccosh}(c+d x)} (4 a \cosh (2 \text {arccosh}(c+d x))+4 b \text {arccosh}(c+d x) \cosh (2 \text {arccosh}(c+d x))-3 b \sinh (2 \text {arccosh}(c+d x)))\right )}{128 d} \]

input

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

output

(e*(3*b^(3/2)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[ 
b]]*(Cosh[(2*a)/b] - Sinh[(2*a)/b]) - 3*b^(3/2)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sq 
rt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + 8*S 
qrt[a + b*ArcCosh[c + d*x]]*(4*a*Cosh[2*ArcCosh[c + d*x]] + 4*b*ArcCosh[c 
+ d*x]*Cosh[2*ArcCosh[c + d*x]] - 3*b*Sinh[2*ArcCosh[c + d*x]])))/(128*d)
 

3.2.62.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.12 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.02, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {6411, 27, 6299, 6354, 6302, 25, 5971, 27, 3042, 26, 3789, 2611, 2633, 2634, 6308}

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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6299

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 6302

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3789

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

\(\Big \downarrow \) 6308

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

input

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

output

(e*(((c + d*x)^2*(a + b*ArcCosh[c + d*x])^(3/2))/2 - (3*b*((Sqrt[-1 + c + 
d*x]*(c + d*x)*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/2 + (a + b* 
ArcCosh[c + d*x])^(3/2)/(3*b) - (I/8)*((I/2)*Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2 
]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqr 
t[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/E^((2*a)/b)) 
))/4))/d
 

3.2.62.3.1 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 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 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 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 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]
 

3.2.62.4 Maple [F]

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

input

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

output

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

3.2.62.5 Fricas [F(-2)]

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

input

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

3.2.62.6 Sympy [F]

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

input

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

output

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

3.2.62.7 Maxima [F]

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

input

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

output

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

3.2.62.8 Giac [F]

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

input

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

output

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

3.2.62.9 Mupad [F(-1)]

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

input

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

output

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