3.3.78 \(\int \frac {\sinh ^5(a+b \log (c x^n))}{x} \, dx\) [278]

Optimal. Leaf size=65 \[ \frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {2 \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\cosh ^5\left (a+b \log \left (c x^n\right )\right )}{5 b n} \]

[Out]

cosh(a+b*ln(c*x^n))/b/n-2/3*cosh(a+b*ln(c*x^n))^3/b/n+1/5*cosh(a+b*ln(c*x^n))^5/b/n

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Rubi [A]
time = 0.04, antiderivative size = 65, 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 ^5\left (a+b \log \left (c x^n\right )\right )}{5 b n}-\frac {2 \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]]^5/x,x]

[Out]

Cosh[a + b*Log[c*x^n]]/(b*n) - (2*Cosh[a + b*Log[c*x^n]]^3)/(3*b*n) + Cosh[a + b*Log[c*x^n]]^5/(5*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 ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \sinh ^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (1-2 x^2+x^4\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 {2 \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\cosh ^5\left (a+b \log \left (c x^n\right )\right )}{5 b n}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 68, normalized size = 1.05 \begin {gather*} \frac {5 \cosh \left (a+b \log \left (c x^n\right )\right )}{8 b n}-\frac {5 \cosh \left (3 \left (a+b \log \left (c x^n\right )\right )\right )}{48 b n}+\frac {\cosh \left (5 \left (a+b \log \left (c x^n\right )\right )\right )}{80 b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*Log[c*x^n]]^5/x,x]

[Out]

(5*Cosh[a + b*Log[c*x^n]])/(8*b*n) - (5*Cosh[3*(a + b*Log[c*x^n])])/(48*b*n) + Cosh[5*(a + b*Log[c*x^n])]/(80*
b*n)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sinh ^{5}\left (a +b \ln \left (c \,x^{n}\right )\right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a+b*ln(c*x^n))^5/x,x)

[Out]

int(sinh(a+b*ln(c*x^n))^5/x,x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (61) = 122\).
time = 0.26, size = 130, normalized size = 2.00 \begin {gather*} \frac {e^{\left (5 \, b \log \left (c x^{n}\right ) + 5 \, a\right )}}{160 \, b n} - \frac {5 \, e^{\left (3 \, b \log \left (c x^{n}\right ) + 3 \, a\right )}}{96 \, b n} + \frac {5 \, e^{\left (b \log \left (c x^{n}\right ) + a\right )}}{16 \, b n} + \frac {5 \, e^{\left (-b \log \left (c x^{n}\right ) - a\right )}}{16 \, b n} - \frac {5 \, e^{\left (-3 \, b \log \left (c x^{n}\right ) - 3 \, a\right )}}{96 \, b n} + \frac {e^{\left (-5 \, b \log \left (c x^{n}\right ) - 5 \, a\right )}}{160 \, b n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b*log(c*x^n))^5/x,x, algorithm="maxima")

[Out]

1/160*e^(5*b*log(c*x^n) + 5*a)/(b*n) - 5/96*e^(3*b*log(c*x^n) + 3*a)/(b*n) + 5/16*e^(b*log(c*x^n) + a)/(b*n) +
5/16*e^(-b*log(c*x^n) - a)/(b*n) - 5/96*e^(-3*b*log(c*x^n) - 3*a)/(b*n) + 1/160*e^(-5*b*log(c*x^n) - 5*a)/(b*
n)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (61) = 122\).
time = 0.39, size = 130, normalized size = 2.00 \begin {gather*} \frac {3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{5} + 15 \, \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 )^{4} - 25 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 15 \, {\left (2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 5 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 150 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{240 \, b n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b*log(c*x^n))^5/x,x, algorithm="fricas")

[Out]

1/240*(3*cosh(b*n*log(x) + b*log(c) + a)^5 + 15*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a
)^4 - 25*cosh(b*n*log(x) + b*log(c) + a)^3 + 15*(2*cosh(b*n*log(x) + b*log(c) + a)^3 - 5*cosh(b*n*log(x) + b*l
og(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^2 + 150*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. 110 vs. \(2 (53) = 106\).
time = 10.68, size = 110, normalized size = 1.69 \begin {gather*} \begin {cases} \log {\left (x \right )} \sinh ^{5}{\left (a \right )} & \text {for}\: b = 0 \wedge n = 0 \\\log {\left (x \right )} \sinh ^{5}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\log {\left (x \right )} \sinh ^{5}{\left (a \right )} & \text {for}\: b = 0 \\\frac {\sinh ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b n} - \frac {4 \sinh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{3 b n} + \frac {8 \cosh ^{5}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{15 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))**5/x,x)

[Out]

Piecewise((log(x)*sinh(a)**5, Eq(b, 0) & Eq(n, 0)), (log(x)*sinh(a + b*log(c))**5, Eq(n, 0)), (log(x)*sinh(a)*
*5, Eq(b, 0)), (sinh(a + b*log(c*x**n))**4*cosh(a + b*log(c*x**n))/(b*n) - 4*sinh(a + b*log(c*x**n))**2*cosh(a
+ b*log(c*x**n))**3/(3*b*n) + 8*cosh(a + b*log(c*x**n))**5/(15*b*n), True))

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Giac [A]
time = 0.43, size = 115, normalized size = 1.77 \begin {gather*} \frac {{\left (3 \, c^{10 \, b} x^{5 \, b n} e^{\left (10 \, a\right )} - 25 \, c^{8 \, b} x^{3 \, b n} e^{\left (8 \, a\right )} + 150 \, c^{6 \, b} x^{b n} e^{\left (6 \, a\right )} + \frac {150 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 25 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 3}{x^{5 \, b n}}\right )} e^{\left (-5 \, a\right )}}{480 \, b c^{5 \, b} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b*log(c*x^n))^5/x,x, algorithm="giac")

[Out]

1/480*(3*c^(10*b)*x^(5*b*n)*e^(10*a) - 25*c^(8*b)*x^(3*b*n)*e^(8*a) + 150*c^(6*b)*x^(b*n)*e^(6*a) + (150*c^(4*
b)*x^(4*b*n)*e^(4*a) - 25*c^(2*b)*x^(2*b*n)*e^(2*a) + 3)/x^(5*b*n))*e^(-5*a)/(b*c^(5*b)*n)

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Mupad [B]
time = 0.84, size = 49, normalized size = 0.75 \begin {gather*} \frac {\frac {{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^5}{5}-\frac {2\,{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{3}+\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}{b\,n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*log(c*x^n))^5/x,x)

[Out]

(cosh(a + b*log(c*x^n)) - (2*cosh(a + b*log(c*x^n))^3)/3 + cosh(a + b*log(c*x^n))^5/5)/(b*n)

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