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

Optimal. Leaf size=64 \[ \frac{a \log (c+d x)}{d}+\frac{b \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{d}+\frac{b \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.124337, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3317, 3303, 3298, 3301} \[ \frac{a \log (c+d x)}{d}+\frac{b \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{d}+\frac{b \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[e + f*x])/(c + d*x),x]

[Out]

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

Rule 3317

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 0])

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

Mathematica [A]  time = 0.144335, size = 57, normalized size = 0.89 \[ \frac{a \log (c+d x)+b \text{Chi}\left (f \left (\frac{c}{d}+x\right )\right ) \sinh \left (e-\frac{c f}{d}\right )+b \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sinh[e + f*x])/(c + d*x),x]

[Out]

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

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Maple [A]  time = 0.023, size = 94, normalized size = 1.5 \begin{align*}{\frac{a\ln \left ( dx+c \right ) }{d}}+{\frac{b}{2\,d}{{\rm e}^{{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,fx+e+{\frac{cf-de}{d}} \right ) }-{\frac{b}{2\,d}{{\rm e}^{-{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,-fx-e-{\frac{cf-de}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.42502, size = 96, normalized size = 1.5 \begin{align*} \frac{1}{2} \, b{\left (\frac{e^{\left (-e + \frac{c f}{d}\right )} E_{1}\left (\frac{{\left (d x + c\right )} f}{d}\right )}{d} - \frac{e^{\left (e - \frac{c f}{d}\right )} E_{1}\left (-\frac{{\left (d x + c\right )} f}{d}\right )}{d}\right )} + \frac{a \log \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.4936, size = 230, normalized size = 3.59 \begin{align*} \frac{{\left (b{\rm Ei}\left (\frac{d f x + c f}{d}\right ) - b{\rm Ei}\left (-\frac{d f x + c f}{d}\right )\right )} \cosh \left (-\frac{d e - c f}{d}\right ) + 2 \, a \log \left (d x + c\right ) -{\left (b{\rm Ei}\left (\frac{d f x + c f}{d}\right ) + b{\rm Ei}\left (-\frac{d f x + c f}{d}\right )\right )} \sinh \left (-\frac{d e - c f}{d}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*sinh(e + f*x))/(c + d*x), x)

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Giac [A]  time = 1.24907, size = 95, normalized size = 1.48 \begin{align*} -\frac{b{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} - b{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} - 2 \, a \log \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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