\(\int \frac {(c+a^2 c x^2)^{3/2}}{\text {arcsinh}(a x)^{3/2}} \, dx\) [502]

Optimal result

Integrand size = 23, antiderivative size = 256 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\text {arcsinh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{4 a \sqrt {1+a^2 x^2}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{4 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{a \sqrt {1+a^2 x^2}} \]

[Out]

-1/2*c*erf(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+1/2*c*erfi(2^(
1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)-1/4*c*erf(2*arcsinh(a*x)^(1/
2))*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+1/4*c*erfi(2*arcsinh(a*x)^(1/2))*Pi^(1/2)*(a^2*c*x^2+c)^(
1/2)/a/(a^2*x^2+1)^(1/2)-2*(a^2*c*x^2+c)^(3/2)*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5790, 5819, 5556, 3389, 2211, 2235, 2236} \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\text {arcsinh}(a x)^{3/2}} \, dx=-\frac {\sqrt {\pi } c \sqrt {a^2 c x^2+c} \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{4 a \sqrt {a^2 x^2+1}}-\frac {\sqrt {\frac {\pi }{2}} c \sqrt {a^2 c x^2+c} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{a \sqrt {a^2 x^2+1}}+\frac {\sqrt {\pi } c \sqrt {a^2 c x^2+c} \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{4 a \sqrt {a^2 x^2+1}}+\frac {\sqrt {\frac {\pi }{2}} c \sqrt {a^2 c x^2+c} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{a \sqrt {a^2 x^2+1}}-\frac {2 \sqrt {a^2 x^2+1} \left (a^2 c x^2+c\right )^{3/2}}{a \sqrt {\text {arcsinh}(a x)}} \]

[In]

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

[Out]

(-2*Sqrt[1 + a^2*x^2]*(c + a^2*c*x^2)^(3/2))/(a*Sqrt[ArcSinh[a*x]]) - (c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erf[2*Sq
rt[ArcSinh[a*x]]])/(4*a*Sqrt[1 + a^2*x^2]) - (c*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]]
)/(a*Sqrt[1 + a^2*x^2]) + (c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(4*a*Sqrt[1 + a^2*x^2])
+ (c*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(a*Sqrt[1 + a^2*x^2])

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 2235

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 2236

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3389

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

Rule 5556

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 5790

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[Simp[Sqrt[1 + c^2*
x^2]*(d + e*x^2)^p]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[c*((2*p + 1)/(b*(n + 1)))*Simp[(d
+ e*x^2)^p/(1 + c^2*x^2)^p], Int[x*(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] && LtQ[n, -1]

Rule 5819

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*
c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1),
x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {\left (8 a c \sqrt {c+a^2 c x^2}\right ) \int \frac {x \left (1+a^2 x^2\right )}{\sqrt {\text {arcsinh}(a x)}} \, dx}{\sqrt {1+a^2 x^2}} \\ & = -\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {\left (8 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a \sqrt {1+a^2 x^2}} \\ & = -\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {\left (8 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 \sqrt {x}}+\frac {\sinh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a \sqrt {1+a^2 x^2}} \\ & = -\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a \sqrt {1+a^2 x^2}}+\frac {\left (2 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a \sqrt {1+a^2 x^2}} \\ & = -\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2 a \sqrt {1+a^2 x^2}}+\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2 a \sqrt {1+a^2 x^2}}-\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a \sqrt {1+a^2 x^2}}+\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a \sqrt {1+a^2 x^2}} \\ & = -\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{a \sqrt {1+a^2 x^2}}+\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{a \sqrt {1+a^2 x^2}}-\frac {\left (2 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{a \sqrt {1+a^2 x^2}}+\frac {\left (2 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{a \sqrt {1+a^2 x^2}} \\ & = -\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{4 a \sqrt {1+a^2 x^2}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{4 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{a \sqrt {1+a^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\text {arcsinh}(a x)^{3/2}} \, dx=-\frac {c e^{-4 \text {arcsinh}(a x)} \sqrt {c+a^2 c x^2} \left (1+14 e^{4 \text {arcsinh}(a x)}+e^{8 \text {arcsinh}(a x)}+16 a^2 e^{4 \text {arcsinh}(a x)} x^2+4 e^{4 \text {arcsinh}(a x)} \sqrt {2 \pi } \sqrt {\text {arcsinh}(a x)} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-4 e^{4 \text {arcsinh}(a x)} \sqrt {2 \pi } \sqrt {\text {arcsinh}(a x)} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-2 e^{4 \text {arcsinh}(a x)} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-4 \text {arcsinh}(a x)\right )-2 e^{4 \text {arcsinh}(a x)} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},4 \text {arcsinh}(a x)\right )\right )}{8 a \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \]

[In]

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

[Out]

-1/8*(c*Sqrt[c + a^2*c*x^2]*(1 + 14*E^(4*ArcSinh[a*x]) + E^(8*ArcSinh[a*x]) + 16*a^2*E^(4*ArcSinh[a*x])*x^2 +
4*E^(4*ArcSinh[a*x])*Sqrt[2*Pi]*Sqrt[ArcSinh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]] - 4*E^(4*ArcSinh[a*x])*Sqrt
[2*Pi]*Sqrt[ArcSinh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]] - 2*E^(4*ArcSinh[a*x])*Sqrt[-ArcSinh[a*x]]*Gamma[1/
2, -4*ArcSinh[a*x]] - 2*E^(4*ArcSinh[a*x])*Sqrt[ArcSinh[a*x]]*Gamma[1/2, 4*ArcSinh[a*x]]))/(a*E^(4*ArcSinh[a*x
])*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}{\operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

Integral((c*(a**2*x**2 + 1))**(3/2)/asinh(a*x)**(3/2), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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