3.4.78 \(\int \frac {(d-c^2 d x^2)^2}{x (a+b \cosh ^{-1}(c x))^{3/2}} \, dx\) [378]

Optimal. Leaf size=289 \[ -\frac {2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c x \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {d^2 e^{\frac {4 a}{b}} \sqrt {\pi } \text {Erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2}}-\frac {3 d^2 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {d^2 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2}}-\frac {3 d^2 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {2 d^2 \text {Int}\left (\frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}},x\right )}{b c} \]

[Out]

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

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Rubi [A]
time = 1.56, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (d-c^2 d x^2\right )^2}{x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-2*d^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(b*c*x*Sqrt[a + b*ArcCosh[c*x]]) + (d^2*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*
Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b^(3/2)) + (d^2*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[
c*x]])/Sqrt[b]])/(2*b^(3/2)) - (d^2*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/b^
(3/2) + (d^2*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b^(3/2)*E^((4*a)/b)) + (d^2*Sqrt[Pi/2]*Er
fi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(2*b^(3/2)*E^((2*a)/b)) - (d^2*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a
+ b*ArcCosh[c*x]])/Sqrt[b]])/(b^(3/2)*E^((2*a)/b)) + (2*d^2*Defer[Int][1/(x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sq
rt[a + b*ArcCosh[c*x]]), x])/(b*c)

Rubi steps

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

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Mathematica [A]
time = 2.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d-c^2 d x^2\right )^2}{x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{2}}{x \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^2/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)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} d^{2} \left (\int \left (- \frac {2 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 )}}\right )\, dx + \int \frac {c^{4} x^{4}}{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 \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 )}}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**2*(Integral(-2*c**2*x**2/(a*x*sqrt(a + b*acosh(c*x)) + b*x*sqrt(a + b*acosh(c*x))*acosh(c*x)), x) + Integra
l(c**4*x**4/(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))

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^2/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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d-c^2\,d\,x^2\right )}^2}{x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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