\(\int \log (x) \sinh (a+b x) \, dx\) [195]

Optimal result

Integrand size = 9, antiderivative size = 35 \[ \int \log (x) \sinh (a+b x) \, dx=-\frac {\cosh (a) \text {Chi}(b x)}{b}+\frac {\cosh (a+b x) \log (x)}{b}-\frac {\sinh (a) \text {Shi}(b x)}{b} \]

[Out]

-Chi(b*x)*cosh(a)/b+cosh(b*x+a)*ln(x)/b-Shi(b*x)*sinh(a)/b

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2718, 2634, 12, 3384, 3379, 3382} \[ \int \log (x) \sinh (a+b x) \, dx=-\frac {\cosh (a) \text {Chi}(b x)}{b}-\frac {\sinh (a) \text {Shi}(b x)}{b}+\frac {\log (x) \cosh (a+b x)}{b} \]

[In]

Int[Log[x]*Sinh[a + b*x],x]

[Out]

-((Cosh[a]*CoshIntegral[b*x])/b) + (Cosh[a + b*x]*Log[x])/b - (Sinh[a]*SinhIntegral[b*x])/b

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2634

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*(D[u, x]
/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\cosh (a+b x) \log (x)}{b}-\int \frac {\cosh (a+b x)}{b x} \, dx \\ & = \frac {\cosh (a+b x) \log (x)}{b}-\frac {\int \frac {\cosh (a+b x)}{x} \, dx}{b} \\ & = \frac {\cosh (a+b x) \log (x)}{b}-\frac {\cosh (a) \int \frac {\cosh (b x)}{x} \, dx}{b}-\frac {\sinh (a) \int \frac {\sinh (b x)}{x} \, dx}{b} \\ & = -\frac {\cosh (a) \text {Chi}(b x)}{b}+\frac {\cosh (a+b x) \log (x)}{b}-\frac {\sinh (a) \text {Shi}(b x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \log (x) \sinh (a+b x) \, dx=-\frac {\cosh (a) \text {Chi}(b x)-\cosh (a+b x) \log (x)+\sinh (a) \text {Shi}(b x)}{b} \]

[In]

Integrate[Log[x]*Sinh[a + b*x],x]

[Out]

-((Cosh[a]*CoshIntegral[b*x] - Cosh[a + b*x]*Log[x] + Sinh[a]*SinhIntegral[b*x])/b)

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66

method	result	size
risch	\(\frac {\ln \left (x \right ) {\mathrm e}^{-b x -a}}{2 b}+\frac {{\mathrm e}^{b x +a} \ln \left (x \right )}{2 b}+\frac {{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )}{2 b}+\frac {{\mathrm e}^{a} \operatorname {Ei}_{1}\left (-b x \right )}{2 b}\)	\(58\)
meijerg	\(-\frac {\sinh \left (a \right ) \sinh \left (b x \right )}{b}+\frac {\sinh \left (a \right ) \ln \left (x \right ) \sinh \left (b x \right )}{b}+\frac {\sinh \left (a \right ) b^{2} \left (\frac {9 \sinh \left (b x \right )}{b^{3}}-\frac {9 \,\operatorname {Shi}\left (b x \right )}{b^{3}}\right )}{9}-\frac {\cosh \left (a \right ) b \left (-\frac {2}{b^{2}}+\frac {2 \cosh \left (b x \right )}{b^{2}}\right )}{4}+\frac {\cosh \left (a \right ) b \ln \left (x \right ) \left (-\frac {2}{b^{2}}+\frac {2 \cosh \left (b x \right )}{b^{2}}\right )}{2}+\frac {\cosh \left (a \right ) b^{3} \left (-\frac {48}{b^{4}}+\frac {48 \cosh \left (b x \right )}{b^{4}}-\frac {96 \left (\operatorname {Chi}\left (b x \right )-\ln \left (b x \right )-\gamma \right )}{b^{4}}\right )}{96}\)	\(134\)

[In]

int(ln(x)*sinh(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/2/b*ln(x)*exp(-b*x-a)+1/2*exp(b*x+a)*ln(x)/b+1/2/b*exp(-a)*Ei(1,b*x)+1/2/b*exp(a)*Ei(1,-b*x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (35) = 70\).

Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 3.83 \[ \int \log (x) \sinh (a+b x) \, dx=-\frac {{\left ({\rm Ei}\left (b x\right ) + {\rm Ei}\left (-b x\right )\right )} \cosh \left (b x + a\right ) \cosh \left (a\right ) - \log \left (x\right ) \sinh \left (b x + a\right )^{2} + {\left ({\rm Ei}\left (b x\right ) - {\rm Ei}\left (-b x\right )\right )} \cosh \left (b x + a\right ) \sinh \left (a\right ) - {\left (\cosh \left (b x + a\right )^{2} + 1\right )} \log \left (x\right ) + {\left ({\left ({\rm Ei}\left (b x\right ) + {\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right ) - 2 \, \cosh \left (b x + a\right ) \log \left (x\right ) + {\left ({\rm Ei}\left (b x\right ) - {\rm Ei}\left (-b x\right )\right )} \sinh \left (a\right )\right )} \sinh \left (b x + a\right )}{2 \, {\left (b \cosh \left (b x + a\right ) + b \sinh \left (b x + a\right )\right )}} \]

[In]

integrate(log(x)*sinh(b*x+a),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \log (x) \sinh (a+b x) \, dx=\int \log {\left (x \right )} \sinh {\left (a + b x \right )}\, dx \]

[In]

integrate(ln(x)*sinh(b*x+a),x)

[Out]

Integral(log(x)*sinh(a + b*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03 \[ \int \log (x) \sinh (a+b x) \, dx=\frac {\cosh \left (b x + a\right ) \log \left (x\right )}{b} - \frac {{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + {\rm Ei}\left (b x\right ) e^{a}}{2 \, b} \]

[In]

integrate(log(x)*sinh(b*x+a),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.49 \[ \int \log (x) \sinh (a+b x) \, dx=\frac {1}{2} \, {\left (\frac {e^{\left (b x + a\right )}}{b} + \frac {e^{\left (-b x - a\right )}}{b}\right )} \log \left (x\right ) - \frac {{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + {\rm Ei}\left (b x\right ) e^{a}}{2 \, b} \]

[In]

integrate(log(x)*sinh(b*x+a),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \log (x) \sinh (a+b x) \, dx=\int \mathrm {sinh}\left (a+b\,x\right )\,\ln \left (x\right ) \,d x \]

[In]

int(sinh(a + b*x)*log(x),x)

[Out]

int(sinh(a + b*x)*log(x), x)