3.3.64 \(\int \frac {c-a^2 c x^2}{\cosh ^{-1}(a x)} \, dx\) [264]

Optimal. Leaf size=29 \[ \frac {3 c \text {Shi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac {c \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{4 a} \]

[Out]

3/4*c*Shi(arccosh(a*x))/a-1/4*c*Shi(3*arccosh(a*x))/a

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Rubi [A]
time = 0.05, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5906, 3393, 3379} \begin {gather*} \frac {3 c \text {Shi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac {c \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)/ArcCosh[a*x],x]

[Out]

(3*c*SinhIntegral[ArcCosh[a*x]])/(4*a) - (c*SinhIntegral[3*ArcCosh[a*x]])/(4*a)

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[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])
)

Rule 5906

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(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]

Rubi steps

\begin {align*} \int \frac {c-a^2 c x^2}{\cosh ^{-1}(a x)} \, dx &=-\frac {c \text {Subst}\left (\int \frac {\sinh ^3(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {(i c) \text {Subst}\left (\int \left (\frac {3 i \sinh (x)}{4 x}-\frac {i \sinh (3 x)}{4 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {c \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}+\frac {(3 c) \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}\\ &=\frac {3 c \text {Shi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac {c \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{4 a}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 25, normalized size = 0.86 \begin {gather*} \frac {c \left (3 \text {Shi}\left (\cosh ^{-1}(a x)\right )-\text {Shi}\left (3 \cosh ^{-1}(a x)\right )\right )}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(3*SinhIntegral[ArcCosh[a*x]] - SinhIntegral[3*ArcCosh[a*x]]))/(4*a)

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Maple [A]
time = 1.74, size = 24, normalized size = 0.83

method	result	size
derivativedivides	\(\frac {c \left (3 \hyperbolicSineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )-\hyperbolicSineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right )\right )}{4 a}\)	\(24\)
default	\(\frac {c \left (3 \hyperbolicSineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )-\hyperbolicSineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right )\right )}{4 a}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)/arccosh(a*x),x,method=_RETURNVERBOSE)

[Out]

1/4/a*c*(3*Shi(arccosh(a*x))-Shi(3*arccosh(a*x)))

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Maxima [F]
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((-a^2*c*x^2+c)/arccosh(a*x),x, algorithm="maxima")

[Out]

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

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Fricas [F]
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((-a^2*c*x^2+c)/arccosh(a*x),x, algorithm="fricas")

[Out]

integral(-(a^2*c*x^2 - c)/arccosh(a*x), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - c \left (\int \frac {a^{2} x^{2}}{\operatorname {acosh}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {acosh}{\left (a x \right )}}\right )\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
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((-a^2*c*x^2+c)/arccosh(a*x),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {c-a^2\,c\,x^2}{\mathrm {acosh}\left (a\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - a^2*c*x^2)/acosh(a*x),x)

[Out]

int((c - a^2*c*x^2)/acosh(a*x), x)

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