3.1.5 \(\int \frac {\sinh (a+b x)}{c+d x} \, dx\) [5]

Optimal. Leaf size=51 \[ \frac {\text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{d}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3384, 3379, 3382} \begin {gather*} \frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x]/(c + d*x),x]

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 49, normalized size = 0.96 \begin {gather*} \frac {\text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )+\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x]/(c + d*x),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.36, size = 82, normalized size = 1.61

method	result	size
risch	\(\frac {{\mathrm e}^{-\frac {a d -b c}{d}} \expIntegral \left (1, b x +a -\frac {a d -b c}{d}\right )}{2 d}-\frac {{\mathrm e}^{\frac {a d -b c}{d}} \expIntegral \left (1, -b x -a -\frac {-a d +b c}{d}\right )}{2 d}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(b*x+a)/(d*x+c),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 57, normalized size = 1.12 \begin {gather*} \frac {e^{\left (-a + \frac {b c}{d}\right )} E_{1}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{2 \, d} - \frac {e^{\left (a - \frac {b c}{d}\right )} E_{1}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

1/2*e^(-a + b*c/d)*exp_integral_e(1, (d*x + c)*b/d)/d - 1/2*e^(a - b*c/d)*exp_integral_e(1, -(d*x + c)*b/d)/d

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 94, normalized size = 1.84 \begin {gather*} \frac {{\left ({\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) + {\left ({\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

1/2*((Ei((b*d*x + b*c)/d) - Ei(-(b*d*x + b*c)/d))*cosh(-(b*c - a*d)/d) + (Ei((b*d*x + b*c)/d) + Ei(-(b*d*x + b
*c)/d))*sinh(-(b*c - a*d)/d))/d

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh {\left (a + b x \right )}}{c + d x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)/(d*x+c),x)

[Out]

Integral(sinh(a + b*x)/(c + d*x), x)

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 57, normalized size = 1.12 \begin {gather*} \frac {{\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} - {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {sinh}\left (a+b\,x\right )}{c+d\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x)/(c + d*x),x)

[Out]

int(sinh(a + b*x)/(c + d*x), x)

________________________________________________________________________________________