Integrand size = 24, antiderivative size = 175 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\sqrt {\text {arccosh}(a x)}} \, dx=-\frac {\sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{4 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{4 a \sqrt {-1+a x} \sqrt {1+a x}} \] Output:
-(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+1/8 *2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*erf(2^(1/2)*arccosh(a*x)^(1/2))/a/( a*x-1)^(1/2)/(a*x+1)^(1/2)+1/8*2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*erfi( 2^(1/2)*arccosh(a*x)^(1/2))/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)
Time = 0.14 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\sqrt {\text {arccosh}(a x)}} \, dx=-\frac {\sqrt {-c (-1+a x) (1+a x)} \left (8 \text {arccosh}(a x)-\sqrt {2} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-2 \text {arccosh}(a x)\right )+\sqrt {2} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},2 \text {arccosh}(a x)\right )\right )}{8 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\text {arccosh}(a x)}} \] Input:
Integrate[Sqrt[c - a^2*c*x^2]/Sqrt[ArcCosh[a*x]],x]
Output:
-1/8*(Sqrt[-(c*(-1 + a*x)*(1 + a*x))]*(8*ArcCosh[a*x] - Sqrt[2]*Sqrt[-ArcC osh[a*x]]*Gamma[1/2, -2*ArcCosh[a*x]] + Sqrt[2]*Sqrt[ArcCosh[a*x]]*Gamma[1 /2, 2*ArcCosh[a*x]]))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh [a*x]])
Time = 0.39 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.59, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6321, 3042, 25, 3793, 2009}
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.
\(\displaystyle \int \frac {\sqrt {c-a^2 c x^2}}{\sqrt {\text {arccosh}(a x)}} \, dx\) |
\(\Big \downarrow \) 6321 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int \frac {(a x-1) (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a \sqrt {a x-1} \sqrt {a x+1}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int -\frac {\sin (i \text {arccosh}(a x))^2}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a \sqrt {a x-1} \sqrt {a x+1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {c-a^2 c x^2} \int \frac {\sin (i \text {arccosh}(a x))^2}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a \sqrt {a x-1} \sqrt {a x+1}}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle -\frac {\sqrt {c-a^2 c x^2} \int \left (\frac {1}{2 \sqrt {\text {arccosh}(a x)}}-\frac {\cosh (2 \text {arccosh}(a x))}{2 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a \sqrt {a x-1} \sqrt {a x+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \left (\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\sqrt {\text {arccosh}(a x)}\right )}{a \sqrt {a x-1} \sqrt {a x+1}}\) |
Input:
Int[Sqrt[c - a^2*c*x^2]/Sqrt[ArcCosh[a*x]],x]
Output:
(Sqrt[c - a^2*c*x^2]*(-Sqrt[ArcCosh[a*x]] + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[A rcCosh[a*x]]])/4 + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/4))/(a*Sq rt[-1 + a*x]*Sqrt[1 + a*x])
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Subst[Int[x^n*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0]
\[\int \frac {\sqrt {-a^{2} c \,x^{2}+c}}{\sqrt {\operatorname {arccosh}\left (a x \right )}}d x\]
Input:
int((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(1/2),x)
Output:
int((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(1/2),x)
Exception generated. \[ \int \frac {\sqrt {c-a^2 c x^2}}{\sqrt {\text {arccosh}(a x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {\sqrt {c-a^2 c x^2}}{\sqrt {\text {arccosh}(a x)}} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt {\operatorname {acosh}{\left (a x \right )}}}\, dx \] Input:
integrate((-a**2*c*x**2+c)**(1/2)/acosh(a*x)**(1/2),x)
Output:
Integral(sqrt(-c*(a*x - 1)*(a*x + 1))/sqrt(acosh(a*x)), x)
\[ \int \frac {\sqrt {c-a^2 c x^2}}{\sqrt {\text {arccosh}(a x)}} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{\sqrt {\operatorname {arcosh}\left (a x\right )}} \,d x } \] Input:
integrate((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-a^2*c*x^2 + c)/sqrt(arccosh(a*x)), x)
\[ \int \frac {\sqrt {c-a^2 c x^2}}{\sqrt {\text {arccosh}(a x)}} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{\sqrt {\operatorname {arcosh}\left (a x\right )}} \,d x } \] Input:
integrate((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-a^2*c*x^2 + c)/sqrt(arccosh(a*x)), x)
Timed out. \[ \int \frac {\sqrt {c-a^2 c x^2}}{\sqrt {\text {arccosh}(a x)}} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}}{\sqrt {\mathrm {acosh}\left (a\,x\right )}} \,d x \] Input:
int((c - a^2*c*x^2)^(1/2)/acosh(a*x)^(1/2),x)
Output:
int((c - a^2*c*x^2)^(1/2)/acosh(a*x)^(1/2), x)
\[ \int \frac {\sqrt {c-a^2 c x^2}}{\sqrt {\text {arccosh}(a x)}} \, dx=\frac {2 \sqrt {c}\, \left (\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acosh} \left (a x \right )}-2 \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x}{a^{2} x^{2}-1}d x \right ) a^{2}\right )}{a} \] Input:
int((-a^2*c*x^2+c)^(1/2)/acosh(a*x)^(1/2),x)
Output:
(2*sqrt(c)*(sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt( - a**2*x**2 + 1)*sqrt(acosh( a*x)) - 2*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt( - a**2*x**2 + 1)*sqrt(aco sh(a*x))*x)/(a**2*x**2 - 1),x)*a**2))/a