3.3.76 \(\int \frac {\sinh ^3(a+b \log (c x^n))}{x} \, dx\) [276]

Optimal. Leaf size=43 \[ -\frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{b n}+\frac {\cosh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]

[Out]

-cosh(a+b*ln(c*x^n))/b/n+1/3*cosh(a+b*ln(c*x^n))^3/b/n

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Rubi [A]
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2713} \begin {gather*} \frac {\cosh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*Log[c*x^n]]^3/x,x]

[Out]

-(Cosh[a + b*Log[c*x^n]]/(b*n)) + Cosh[a + b*Log[c*x^n]]^3/(3*b*n)

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 45, normalized size = 1.05 \begin {gather*} -\frac {3 \cosh \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {\cosh \left (3 \left (a+b \log \left (c x^n\right )\right )\right )}{12 b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*Log[c*x^n]]^3/x,x]

[Out]

(-3*Cosh[a + b*Log[c*x^n]])/(4*b*n) + Cosh[3*(a + b*Log[c*x^n])]/(12*b*n)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sinh ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a+b*ln(c*x^n))^3/x,x)

[Out]

int(sinh(a+b*ln(c*x^n))^3/x,x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (41) = 82\).
time = 0.26, size = 86, normalized size = 2.00 \begin {gather*} \frac {e^{\left (3 \, b \log \left (c x^{n}\right ) + 3 \, a\right )}}{24 \, b n} - \frac {3 \, e^{\left (b \log \left (c x^{n}\right ) + a\right )}}{8 \, b n} - \frac {3 \, e^{\left (-b \log \left (c x^{n}\right ) - a\right )}}{8 \, b n} + \frac {e^{\left (-3 \, b \log \left (c x^{n}\right ) - 3 \, a\right )}}{24 \, b n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b*log(c*x^n))^3/x,x, algorithm="maxima")

[Out]

1/24*e^(3*b*log(c*x^n) + 3*a)/(b*n) - 3/8*e^(b*log(c*x^n) + a)/(b*n) - 3/8*e^(-b*log(c*x^n) - a)/(b*n) + 1/24*
e^(-3*b*log(c*x^n) - 3*a)/(b*n)

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Fricas [A]
time = 0.39, size = 65, normalized size = 1.51 \begin {gather*} \frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 9 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{12 \, b n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b*log(c*x^n))^3/x,x, algorithm="fricas")

[Out]

1/12*(cosh(b*n*log(x) + b*log(c) + a)^3 + 3*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^2
- 9*cosh(b*n*log(x) + b*log(c) + a))/(b*n)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (32) = 64\).
time = 1.78, size = 76, normalized size = 1.77 \begin {gather*} \begin {cases} \log {\left (x \right )} \sinh ^{3}{\left (a \right )} & \text {for}\: b = 0 \wedge n = 0 \\\log {\left (x \right )} \sinh ^{3}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\log {\left (x \right )} \sinh ^{3}{\left (a \right )} & \text {for}\: b = 0 \\\frac {\sinh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b n} - \frac {2 \cosh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{3 b n} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b*ln(c*x**n))**3/x,x)

[Out]

Piecewise((log(x)*sinh(a)**3, Eq(b, 0) & Eq(n, 0)), (log(x)*sinh(a + b*log(c))**3, Eq(n, 0)), (log(x)*sinh(a)*
*3, Eq(b, 0)), (sinh(a + b*log(c*x**n))**2*cosh(a + b*log(c*x**n))/(b*n) - 2*cosh(a + b*log(c*x**n))**3/(3*b*n
), True))

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Giac [A]
time = 0.42, size = 81, normalized size = 1.88 \begin {gather*} \frac {{\left (c^{6 \, b} x^{3 \, b n} e^{\left (6 \, a\right )} - 9 \, c^{4 \, b} x^{b n} e^{\left (4 \, a\right )} - \frac {9 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1}{x^{3 \, b n}}\right )} e^{\left (-3 \, a\right )}}{24 \, b c^{3 \, b} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b*log(c*x^n))^3/x,x, algorithm="giac")

[Out]

1/24*(c^(6*b)*x^(3*b*n)*e^(6*a) - 9*c^(4*b)*x^(b*n)*e^(4*a) - (9*c^(2*b)*x^(2*b*n)*e^(2*a) - 1)/x^(3*b*n))*e^(
-3*a)/(b*c^(3*b)*n)

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Mupad [B]
time = 0.72, size = 37, normalized size = 0.86 \begin {gather*} -\frac {3\,\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )-{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{3\,b\,n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*log(c*x^n))^3/x,x)

[Out]

-(3*cosh(a + b*log(c*x^n)) - cosh(a + b*log(c*x^n))^3)/(3*b*n)

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