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

Optimal. Leaf size=25 \[ \frac {\sinh (a) \text {Chi}\left (b x^n\right )}{n}+\frac {\cosh (a) \text {Shi}\left (b x^n\right )}{n} \]

[Out]

cosh(a)*Shi(b*x^n)/n+Chi(b*x^n)*sinh(a)/n

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Rubi [A]  time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5318, 5317, 5316} \[ \frac {\sinh (a) \text {Chi}\left (b x^n\right )}{n}+\frac {\cosh (a) \text {Shi}\left (b x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x^n]/x,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 23, normalized size = 0.92 \[ \frac {\sinh (a) \text {Chi}\left (b x^n\right )+\cosh (a) \text {Shi}\left (b x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x^n]/x,x]

[Out]

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

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fricas [B]  time = 0.58, size = 55, normalized size = 2.20 \[ \frac {{\left (\cosh \relax (a) + \sinh \relax (a)\right )} {\rm Ei}\left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right )\right ) - {\left (\cosh \relax (a) - \sinh \relax (a)\right )} {\rm Ei}\left (-b \cosh \left (n \log \relax (x)\right ) - b \sinh \left (n \log \relax (x)\right )\right )}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((cosh(a) + sinh(a))*Ei(b*cosh(n*log(x)) + b*sinh(n*log(x))) - (cosh(a) - sinh(a))*Ei(-b*cosh(n*log(x)) -
b*sinh(n*log(x))))/n

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sinh(b*x^n + a)/x, x)

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maple [A]  time = 0.02, size = 33, normalized size = 1.32 \[ \frac {{\mathrm e}^{-a} \Ei \left (1, b \,x^{n}\right )}{2 n}-\frac {{\mathrm e}^{a} \Ei \left (1, -b \,x^{n}\right )}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/n*exp(-a)*Ei(1,b*x^n)-1/2/n*exp(a)*Ei(1,-b*x^n)

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maxima [A]  time = 0.61, size = 30, normalized size = 1.20 \[ -\frac {{\rm Ei}\left (-b x^{n}\right ) e^{\left (-a\right )}}{2 \, n} + \frac {{\rm Ei}\left (b x^{n}\right ) e^{a}}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*Ei(-b*x^n)*e^(-a)/n + 1/2*Ei(b*x^n)*e^a/n

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {sinh}\left (a+b\,x^n\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x^n)/x,x)

[Out]

int(sinh(a + b*x^n)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(a + b*x**n)/x, x)

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