3.4.83 \(\int \frac {\sqrt {\cosh ^{-1}(a x)}}{(c-a^2 c x^2)^{5/2}} \, dx\) [383]

Optimal. Leaf size=194 \[ \frac {x \sqrt {\cosh ^{-1}(a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sqrt {\cosh ^{-1}(a x)}}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right ) \sqrt {\cosh ^{-1}(a x)}},x\right )}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {Int}\left (\frac {x}{\left (-1+a^2 x^2\right )^2 \sqrt {\cosh ^{-1}(a x)}},x\right )}{6 c^2 \sqrt {c-a^2 c x^2}} \]

[Out]

1/3*x*arccosh(a*x)^(1/2)/c/(-a^2*c*x^2+c)^(3/2)+2/3*x*arccosh(a*x)^(1/2)/c^2/(-a^2*c*x^2+c)^(1/2)+1/3*a*(a*x-1
)^(1/2)*(a*x+1)^(1/2)*Unintegrable(x/(-a^2*x^2+1)/arccosh(a*x)^(1/2),x)/c^2/(-a^2*c*x^2+c)^(1/2)+1/6*a*(a*x-1)
^(1/2)*(a*x+1)^(1/2)*Unintegrable(x/(a^2*x^2-1)^2/arccosh(a*x)^(1/2),x)/c^2/(-a^2*c*x^2+c)^(1/2)

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Rubi [A]
time = 0.22, 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 {\sqrt {\cosh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[ArcCosh[a*x]]/(c - a^2*c*x^2)^(5/2),x]

[Out]

(x*Sqrt[ArcCosh[a*x]])/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*x*Sqrt[ArcCosh[a*x]])/(3*c^2*Sqrt[c - a^2*c*x^2]) + (a
*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Defer[Int][x/((1 - a^2*x^2)*Sqrt[ArcCosh[a*x]]), x])/(3*c^2*Sqrt[c - a^2*c*x^2])
+ (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Defer[Int][x/((-1 + a^2*x^2)^2*Sqrt[ArcCosh[a*x]]), x])/(6*c^2*Sqrt[c - a^2
*c*x^2])

Rubi steps

\begin {align*} \int \frac {\sqrt {\cosh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\sqrt {\cosh ^{-1}(a x)}}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sqrt {\cosh ^{-1}(a x)}}{3 c^2 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}-\frac {\left (2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\sqrt {\cosh ^{-1}(a x)}}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{\left (-1+a^2 x^2\right )^2 \sqrt {\cosh ^{-1}(a x)}} \, dx}{6 c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {2 x \sqrt {\cosh ^{-1}(a x)}}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {x \sqrt {\cosh ^{-1}(a x)}}{3 c^2 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{\left (-1+a^2 x^2\right )^2 \sqrt {\cosh ^{-1}(a x)}} \, dx}{6 c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{\left (1-a^2 x^2\right ) \sqrt {\cosh ^{-1}(a x)}} \, dx}{3 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}

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Mathematica [A]
time = 1.86, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\cosh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sqrt[ArcCosh[a*x]]/(c - a^2*c*x^2)^(5/2),x]

[Out]

Integrate[Sqrt[ArcCosh[a*x]]/(c - a^2*c*x^2)^(5/2), x]

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Maple [A]
time = 4.57, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\mathrm {arccosh}\left (a x \right )}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^(1/2)/(-a^2*c*x^2+c)^(5/2),x)

[Out]

int(arccosh(a*x)^(1/2)/(-a^2*c*x^2+c)^(5/2),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(arccosh(a*x)^(1/2)/(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(arccosh(a*x))/(-a^2*c*x^2 + c)^(5/2), 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(arccosh(a*x)^(1/2)/(-a^2*c*x^2+c)^(5/2),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*} \int \frac {\sqrt {\operatorname {acosh}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**(1/2)/(-a**2*c*x**2+c)**(5/2),x)

[Out]

Integral(sqrt(acosh(a*x))/(-c*(a*x - 1)*(a*x + 1))**(5/2), x)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^(1/2)/(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(arccosh(a*x))/(-a^2*c*x^2 + c)^(5/2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {\mathrm {acosh}\left (a\,x\right )}}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)^(1/2)/(c - a^2*c*x^2)^(5/2),x)

[Out]

int(acosh(a*x)^(1/2)/(c - a^2*c*x^2)^(5/2), x)

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