\(\int (a+b \text {arccosh}(c+d x))^{5/2} \, dx\) [86]

Optimal result

Integrand size = 14, antiderivative size = 186 \[ \int (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\frac {15 b^2 (c+d x) \sqrt {a+b \text {arccosh}(c+d x)}}{4 d}-\frac {5 b \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{2 d}+\frac {(c+d x) (a+b \text {arccosh}(c+d x))^{5/2}}{d}-\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16 d} \] Output:

15/4*b^2*(d*x+c)*(a+b*arccosh(d*x+c))^(1/2)/d-5/2*b*(d*x+c-1)^(1/2)*(d*x+c 
+1)^(1/2)*(a+b*arccosh(d*x+c))^(3/2)/d+(d*x+c)*(a+b*arccosh(d*x+c))^(5/2)/ 
d-15/16*b^(5/2)*exp(a/b)*Pi^(1/2)*erf((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/ 
d-15/16*b^(5/2)*Pi^(1/2)*erfi((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/d/exp(a/ 
b)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(494\) vs. \(2(186)=372\).

Time = 2.29 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.66 \[ \int (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\frac {4 b \sqrt {a+b \text {arccosh}(c+d x)} \left (2 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) (a-5 b \text {arccosh}(c+d x))+b (c+d x) \left (15+4 \text {arccosh}(c+d x)^2\right )\right )+8 a^2 e^{-\frac {a}{b}} \sqrt {a+b \text {arccosh}(c+d x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )}{\sqrt {\frac {a}{b}+\text {arccosh}(c+d x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}}}\right )-\sqrt {b} \left (4 a^2+12 a b+15 b^2\right ) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )-\sqrt {b} \left (4 a^2-12 a b+15 b^2\right ) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )+4 a b \left (-12 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \sqrt {a+b \text {arccosh}(c+d x)}+8 (c+d x) \text {arccosh}(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}+\frac {(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}+\frac {(2 a-3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}\right )}{16 d} \] Input:

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

Output:

(4*b*Sqrt[a + b*ArcCosh[c + d*x]]*(2*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 
 + c + d*x)*(a - 5*b*ArcCosh[c + d*x]) + b*(c + d*x)*(15 + 4*ArcCosh[c + d 
*x]^2)) + (8*a^2*Sqrt[a + b*ArcCosh[c + d*x]]*((E^((2*a)/b)*Gamma[3/2, a/b 
 + ArcCosh[c + d*x]])/Sqrt[a/b + ArcCosh[c + d*x]] + Gamma[3/2, -((a + b*A 
rcCosh[c + d*x])/b)]/Sqrt[-((a + b*ArcCosh[c + d*x])/b)]))/E^(a/b) - Sqrt[ 
b]*(4*a^2 + 12*a*b + 15*b^2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sq 
rt[b]]*(Cosh[a/b] - Sinh[a/b]) - Sqrt[b]*(4*a^2 - 12*a*b + 15*b^2)*Sqrt[Pi 
]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]) + 4*a* 
b*(-12*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*Sqrt[a + b*ArcCosh 
[c + d*x]] + 8*(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]))/Sqrt[b] + ((2*a - 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + 
d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]))/Sqrt[b]))/(16*d)
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6410, 6294, 6330, 6294, 6368, 3042, 3788, 26, 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.

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

\(\Big \downarrow \) 6410

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

\(\Big \downarrow \) 6294

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

\(\Big \downarrow \) 6330

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

\(\Big \downarrow \) 6294

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

\(\Big \downarrow \) 6368

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

Input:

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

Output:

((c + d*x)*(a + b*ArcCosh[c + d*x])^(5/2) - (5*b*(Sqrt[-1 + c + d*x]*Sqrt[ 
1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2) - (3*b*((c + d*x)*Sqrt[a + b*A 
rcCosh[c + d*x]] + (-1/2*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[ 
c + d*x]]/Sqrt[b]]) - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/ 
Sqrt[b]])/(2*E^(a/b)))/2))/2))/2)/d
 

Defintions of rubi rules used

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 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 3788

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

rule 6294

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A 
rcCosh[c*x])^n, x] - Simp[b*c*n   Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt 
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 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 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 6410

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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