3.5 \(\int \frac{\sinh (a+b x)}{c+d x} \, dx\)

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

[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

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Rubi [A]  time = 0.109284, antiderivative size = 51, normalized size of antiderivative = 1., 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 = {3303, 3298, 3301} \[ \frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{d}+\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{d} \]

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 3303

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]

Rule 3298

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 3301

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]

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 = 0.0848559, size = 49, normalized size = 0.96 \[ \frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )+\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{d} \]

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

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Maple [A]  time = 0.059, size = 82, normalized size = 1.6 \begin{align*}{\frac{1}{2\,d}{{\rm e}^{-{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{da-cb}{d}} \right ) }-{\frac{1}{2\,d}{{\rm e}^{{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,-bx-a-{\frac{-da+cb}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [A]  time = 1.2888, size = 77, normalized size = 1.51 \begin{align*} \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{align*}

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

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Fricas [A]  time = 2.58023, size = 193, normalized size = 3.78 \begin{align*} \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{align*}

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

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

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)

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Giac [A]  time = 1.15718, size = 77, normalized size = 1.51 \begin{align*} \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{align*}

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