3.72 \(\int \frac {\sinh ^3(a+b x^n)}{x} \, dx\)

Optimal. Leaf size=67 \[ -\frac {3 \sinh (a) \text {Chi}\left (b x^n\right )}{4 n}+\frac {\sinh (3 a) \text {Chi}\left (3 b x^n\right )}{4 n}-\frac {3 \cosh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {\cosh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n} \]

[Out]

-3/4*cosh(a)*Shi(b*x^n)/n+1/4*cosh(3*a)*Shi(3*b*x^n)/n-3/4*Chi(b*x^n)*sinh(a)/n+1/4*Chi(3*b*x^n)*sinh(3*a)/n

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Rubi [A]  time = 0.10, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5362, 5318, 5317, 5316} \[ -\frac {3 \sinh (a) \text {Chi}\left (b x^n\right )}{4 n}+\frac {\sinh (3 a) \text {Chi}\left (3 b x^n\right )}{4 n}-\frac {3 \cosh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {\cosh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x^n]^3/x,x]

[Out]

(-3*CoshIntegral[b*x^n]*Sinh[a])/(4*n) + (CoshIntegral[3*b*x^n]*Sinh[3*a])/(4*n) - (3*Cosh[a]*SinhIntegral[b*x
^n])/(4*n) + (Cosh[3*a]*SinhIntegral[3*b*x^n])/(4*n)

Rule 5316

Int[Sinh[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinhIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 5317

Int[Cosh[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[CoshIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 5318

Int[Sinh[(c_) + (d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Dist[Sinh[c], Int[Cosh[d*x^n]/x, x], x] + Dist[Cosh[c], In
t[Sinh[d*x^n]/x, x], x] /; FreeQ[{c, d, n}, x]

Rule 5362

Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(
e*x)^m, (a + b*Sinh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 52, normalized size = 0.78 \[ \frac {-3 \sinh (a) \text {Chi}\left (b x^n\right )+\sinh (3 a) \text {Chi}\left (3 b x^n\right )-3 \cosh (a) \text {Shi}\left (b x^n\right )+\cosh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x^n]^3/x,x]

[Out]

(-3*CoshIntegral[b*x^n]*Sinh[a] + CoshIntegral[3*b*x^n]*Sinh[3*a] - 3*Cosh[a]*SinhIntegral[b*x^n] + Cosh[3*a]*
SinhIntegral[3*b*x^n])/(4*n)

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fricas [A]  time = 0.60, size = 115, normalized size = 1.72 \[ \frac {{\left (\cosh \left (3 \, a\right ) + \sinh \left (3 \, a\right )\right )} {\rm Ei}\left (3 \, b \cosh \left (n \log \relax (x)\right ) + 3 \, b \sinh \left (n \log \relax (x)\right )\right ) - 3 \, {\left (\cosh \relax (a) + \sinh \relax (a)\right )} {\rm Ei}\left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right )\right ) + 3 \, {\left (\cosh \relax (a) - \sinh \relax (a)\right )} {\rm Ei}\left (-b \cosh \left (n \log \relax (x)\right ) - b \sinh \left (n \log \relax (x)\right )\right ) - {\left (\cosh \left (3 \, a\right ) - \sinh \left (3 \, a\right )\right )} {\rm Ei}\left (-3 \, b \cosh \left (n \log \relax (x)\right ) - 3 \, b \sinh \left (n \log \relax (x)\right )\right )}{8 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((cosh(3*a) + sinh(3*a))*Ei(3*b*cosh(n*log(x)) + 3*b*sinh(n*log(x))) - 3*(cosh(a) + sinh(a))*Ei(b*cosh(n*l
og(x)) + b*sinh(n*log(x))) + 3*(cosh(a) - sinh(a))*Ei(-b*cosh(n*log(x)) - b*sinh(n*log(x))) - (cosh(3*a) - sin
h(3*a))*Ei(-3*b*cosh(n*log(x)) - 3*b*sinh(n*log(x))))/n

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (b x^{n} + a\right )^{3}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sinh(b*x^n + a)^3/x, x)

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maple [A]  time = 0.19, size = 67, normalized size = 1.00 \[ \frac {{\mathrm e}^{-3 a} \Ei \left (1, 3 b \,x^{n}\right )}{8 n}-\frac {3 \,{\mathrm e}^{-a} \Ei \left (1, b \,x^{n}\right )}{8 n}-\frac {{\mathrm e}^{3 a} \Ei \left (1, -3 b \,x^{n}\right )}{8 n}+\frac {3 \,{\mathrm e}^{a} \Ei \left (1, -b \,x^{n}\right )}{8 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/n*exp(-3*a)*Ei(1,3*b*x^n)-3/8/n*exp(-a)*Ei(1,b*x^n)-1/8/n*exp(3*a)*Ei(1,-3*b*x^n)+3/8/n*exp(a)*Ei(1,-b*x^n
)

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maxima [A]  time = 0.43, size = 62, normalized size = 0.93 \[ \frac {{\rm Ei}\left (3 \, b x^{n}\right ) e^{\left (3 \, a\right )}}{8 \, n} + \frac {3 \, {\rm Ei}\left (-b x^{n}\right ) e^{\left (-a\right )}}{8 \, n} - \frac {{\rm Ei}\left (-3 \, b x^{n}\right ) e^{\left (-3 \, a\right )}}{8 \, n} - \frac {3 \, {\rm Ei}\left (b x^{n}\right ) e^{a}}{8 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*Ei(3*b*x^n)*e^(3*a)/n + 3/8*Ei(-b*x^n)*e^(-a)/n - 1/8*Ei(-3*b*x^n)*e^(-3*a)/n - 3/8*Ei(b*x^n)*e^a/n

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {sinh}\left (a+b\,x^n\right )}^3}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x^n)^3/x,x)

[Out]

int(sinh(a + b*x^n)^3/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(a + b*x**n)**3/x, x)

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