3.35 \(\int \frac {\text {Shi}(d (a+b \log (c x^n)))}{x} \, dx\)

Optimal. Leaf size=55 \[ \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\cosh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b d n} \]

[Out]

-cosh(d*(a+b*ln(c*x^n)))/b/d/n+(a+b*ln(c*x^n))*Shi(d*(a+b*ln(c*x^n)))/b/n

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Rubi [A]  time = 0.03, antiderivative size = 55, 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 = {6528} \[ \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\cosh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b d n} \]

Antiderivative was successfully verified.

[In]

Int[SinhIntegral[d*(a + b*Log[c*x^n])]/x,x]

[Out]

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

Rule 6528

Int[SinhIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*SinhIntegral[a + b*x])/b, x] - Simp[Cosh[a
+ b*x]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 96, normalized size = 1.75 \[ \frac {\log \left (c x^n\right ) \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{n}+\frac {a \text {Shi}\left (a d+b \log \left (c x^n\right ) d\right )}{b n}-\frac {\sinh (a d) \sinh \left (b d \log \left (c x^n\right )\right )}{b d n}-\frac {\cosh (a d) \cosh \left (b d \log \left (c x^n\right )\right )}{b d n} \]

Antiderivative was successfully verified.

[In]

Integrate[SinhIntegral[d*(a + b*Log[c*x^n])]/x,x]

[Out]

-((Cosh[a*d]*Cosh[b*d*Log[c*x^n]])/(b*d*n)) - (Sinh[a*d]*Sinh[b*d*Log[c*x^n]])/(b*d*n) + (Log[c*x^n]*SinhInteg
ral[d*(a + b*Log[c*x^n])])/n + (a*SinhIntegral[a*d + b*d*Log[c*x^n]])/(b*n)

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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Shi}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sinh_integral(b*d*log(c*x^n) + a*d)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Shi((b*log(c*x^n) + a)*d)/x, x)

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maple [A]  time = 0.04, size = 73, normalized size = 1.33 \[ \frac {\ln \left (c \,x^{n}\right ) \Shi \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n}+\frac {\Shi \left (a d +b d \ln \left (c \,x^{n}\right )\right ) a}{n b}-\frac {\cosh \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Shi(d*(a+b*ln(c*x^n)))/x,x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Shi((b*log(c*x^n) + a)*d)/x, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \frac {\mathrm {sinhint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,\ln \left (c\,x^n\right )}{n}+\frac {a\,\mathrm {sinhint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b\,n}-\frac {{\mathrm {e}}^{a\,d}\,{\left (c\,x^n\right )}^{b\,d}}{2\,b\,d\,n}-\frac {{\mathrm {e}}^{-a\,d}}{2\,b\,d\,n\,{\left (c\,x^n\right )}^{b\,d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinhint(d*(a + b*log(c*x^n)))/x,x)

[Out]

(sinhint(d*(a + b*log(c*x^n)))*log(c*x^n))/n + (a*sinhint(d*(a + b*log(c*x^n))))/(b*n) - (exp(a*d)*(c*x^n)^(b*
d))/(2*b*d*n) - exp(-a*d)/(2*b*d*n*(c*x^n)^(b*d))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Shi(d*(a+b*ln(c*x**n)))/x,x)

[Out]

Integral(Shi(a*d + b*d*log(c*x**n))/x, x)

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