\(\int \frac {d-c^2 d x^2}{x (a+b \text {arccosh}(c x))^{3/2}} \, dx\) [373]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {d-c^2 d x^2}{x (a+b \text {arccosh}(c x))^{3/2}} \, dx=\frac {2 d (-1+c x)^{3/2} (1+c x)^{3/2}}{b c x \sqrt {a+b \text {arccosh}(c x)}}-\frac {d e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {d e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2}}+\frac {2 d \text {Int}\left (\frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \text {arccosh}(c x)}},x\right )}{b c} \]

[Out]

-1/2*d*exp(2*a/b)*erf(2^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)-1/2*d*erfi(2^(1/2)*(a
+b*arccosh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)/exp(2*a/b)+2*d*(c*x-1)^(3/2)*(c*x+1)^(3/2)/b/c/x/(a+b
*arccosh(c*x))^(1/2)+2*d*Unintegrable(1/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(a+b*arccosh(c*x))^(1/2),x)/b/c

Rubi [N/A]

Not integrable

Time = 1.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {d-c^2 d x^2}{x (a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {d-c^2 d x^2}{x (a+b \text {arccosh}(c x))^{3/2}} \, dx \]

[In]

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

[Out]

(2*d*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(b*c*x*Sqrt[a + b*ArcCosh[c*x]]) - (d*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[
2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/b^(3/2) - (d*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]
])/(b^(3/2)*E^((2*a)/b)) + (2*d*Defer[Int][1/(x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]]), x])/
(b*c)

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

Mathematica [N/A]

Not integrable

Time = 1.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {d-c^2 d x^2}{x (a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {d-c^2 d x^2}{x (a+b \text {arccosh}(c x))^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 8.80 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.22 \[ \int \frac {d-c^2 d x^2}{x (a+b \text {arccosh}(c x))^{3/2}} \, dx=- d \left (\int \frac {c^{2} x^{2}}{a x \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx + \int \left (- \frac {1}{a x \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\right )\, dx\right ) \]

[In]

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

[Out]

-d*(Integral(c**2*x**2/(a*x*sqrt(a + b*acosh(c*x)) + b*x*sqrt(a + b*acosh(c*x))*acosh(c*x)), x) + Integral(-1/
(a*x*sqrt(a + b*acosh(c*x)) + b*x*sqrt(a + b*acosh(c*x))*acosh(c*x)), x))

Maxima [N/A]

Not integrable

Time = 1.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {d-c^2 d x^2}{x (a+b \text {arccosh}(c x))^{3/2}} \, dx=\int { -\frac {c^{2} d x^{2} - d}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}} x} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 3.92 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {d-c^2 d x^2}{x (a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {d-c^2\,d\,x^2}{x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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