3.2.41 \(\int (a+b \text {arcsinh}(c x))^{3/2} \, dx\) [141]

3.2.41.1 Optimal result

Integrand size = 12, antiderivative size = 135 \[ \int (a+b \text {arcsinh}(c x))^{3/2} \, dx=-\frac {3 b \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}+x (a+b \text {arcsinh}(c x))^{3/2}+\frac {3 b^{3/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 b^{3/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c} \]

output

x*(a+b*arcsinh(c*x))^(3/2)+3/8*b^(3/2)*exp(a/b)*erf((a+b*arcsinh(c*x))^(1/ 
2)/b^(1/2))*Pi^(1/2)/c+3/8*b^(3/2)*erfi((a+b*arcsinh(c*x))^(1/2)/b^(1/2))* 
Pi^(1/2)/c/exp(a/b)-3/2*b*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^(1/2)/c
 

3.2.41.2 Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.86 \[ \int (a+b \text {arcsinh}(c x))^{3/2} \, dx=\frac {a e^{-\frac {a}{b}} \sqrt {a+b \text {arcsinh}(c x)} \left (-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )}{\sqrt {\frac {a}{b}+\text {arcsinh}(c x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}}}\right )}{2 c}+\frac {\sqrt {b} \left (4 \sqrt {b} \sqrt {a+b \text {arcsinh}(c x)} \left (-3 \sqrt {1+c^2 x^2}+2 c x \text {arcsinh}(c x)\right )+(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+(-2 a+3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )}{8 c} \]

input

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

output

(a*Sqrt[a + b*ArcSinh[c*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c*x]] 
)/Sqrt[a/b + ArcSinh[c*x]]) + Gamma[3/2, -((a + b*ArcSinh[c*x])/b)]/Sqrt[- 
((a + b*ArcSinh[c*x])/b)]))/(2*c*E^(a/b)) + (Sqrt[b]*(4*Sqrt[b]*Sqrt[a + b 
*ArcSinh[c*x]]*(-3*Sqrt[1 + c^2*x^2] + 2*c*x*ArcSinh[c*x]) + (2*a + 3*b)*S 
qrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + ( 
-2*a + 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Si 
nh[a/b])))/(8*c)
 

3.2.41.3 Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6187, 6213, 6189, 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.

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

\(\Big \downarrow \) 6187

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 6189

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

input

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

output

x*(a + b*ArcSinh[c*x])^(3/2) - (3*b*c*((Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSi 
nh[c*x]])/c^2 - ((Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sq 
rt[b]])/2 + (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(2*E 
^(a/b)))/(2*c^2)))/2
 

3.2.41.3.1 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 6187

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A 
rcSinh[c*x])^n, x] - Simp[b*c*n   Int[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[ 
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
 

rule 6189

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

rule 6213

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p 
+ 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] 
 Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ 
{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
 

3.2.41.4 Maple [F]

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

input

int((a+b*arcsinh(c*x))^(3/2),x)
 

output

int((a+b*arcsinh(c*x))^(3/2),x)
 

3.2.41.5 Fricas [F(-2)]

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

input

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

output

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

3.2.41.6 Sympy [F]

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

input

integrate((a+b*asinh(c*x))**(3/2),x)
 

output

Integral((a + b*asinh(c*x))**(3/2), x)
 

3.2.41.7 Maxima [F]

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

input

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

output

integrate((b*arcsinh(c*x) + a)^(3/2), x)
 

3.2.41.8 Giac [F]

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

input

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

output

integrate((b*arcsinh(c*x) + a)^(3/2), x)
 

3.2.41.9 Mupad [F(-1)]

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

input

int((a + b*asinh(c*x))^(3/2),x)
 

output

int((a + b*asinh(c*x))^(3/2), x)