3.100 \(\int \frac{a+i a \sinh (e+f x)}{(c+d x)^2} \, dx\)

Optimal. Leaf size=95 \[ \frac{i a f \text{Chi}\left (x f+\frac{c f}{d}\right ) \cosh \left (e-\frac{c f}{d}\right )}{d^2}+\frac{i a f \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d^2}-\frac{i a \sinh (e+f x)}{d (c+d x)}-\frac{a}{d (c+d x)} \]

[Out]

-(a/(d*(c + d*x))) + (I*a*f*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/d^2 - (I*a*Sinh[e + f*x])/(d*(c + d
*x)) + (I*a*f*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^2

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Rubi [A]  time = 0.189107, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3317, 3297, 3303, 3298, 3301} \[ \frac{i a f \text{Chi}\left (x f+\frac{c f}{d}\right ) \cosh \left (e-\frac{c f}{d}\right )}{d^2}+\frac{i a f \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d^2}-\frac{i a \sinh (e+f x)}{d (c+d x)}-\frac{a}{d (c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Sinh[e + f*x])/(c + d*x)^2,x]

[Out]

-(a/(d*(c + d*x))) + (I*a*f*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/d^2 - (I*a*Sinh[e + f*x])/(d*(c + d
*x)) + (I*a*f*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^2

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 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

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+i a \sinh (e+f x)}{(c+d x)^2} \, dx &=\int \left (\frac{a}{(c+d x)^2}+\frac{i a \sinh (e+f x)}{(c+d x)^2}\right ) \, dx\\ &=-\frac{a}{d (c+d x)}+(i a) \int \frac{\sinh (e+f x)}{(c+d x)^2} \, dx\\ &=-\frac{a}{d (c+d x)}-\frac{i a \sinh (e+f x)}{d (c+d x)}+\frac{(i a f) \int \frac{\cosh (e+f x)}{c+d x} \, dx}{d}\\ &=-\frac{a}{d (c+d x)}-\frac{i a \sinh (e+f x)}{d (c+d x)}+\frac{\left (i a f \cosh \left (e-\frac{c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx}{d}+\frac{\left (i a f \sinh \left (e-\frac{c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx}{d}\\ &=-\frac{a}{d (c+d x)}+\frac{i a f \cosh \left (e-\frac{c f}{d}\right ) \text{Chi}\left (\frac{c f}{d}+f x\right )}{d^2}-\frac{i a \sinh (e+f x)}{d (c+d x)}+\frac{i a f \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (\frac{c f}{d}+f x\right )}{d^2}\\ \end{align*}

Mathematica [A]  time = 0.4877, size = 83, normalized size = 0.87 \[ \frac{i a \left (f (c+d x) \text{Chi}\left (f \left (\frac{c}{d}+x\right )\right ) \cosh \left (e-\frac{c f}{d}\right )+f (c+d x) \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )-d (\sinh (e+f x)-i)\right )}{d^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Sinh[e + f*x])/(c + d*x)^2,x]

[Out]

(I*a*(f*(c + d*x)*Cosh[e - (c*f)/d]*CoshIntegral[f*(c/d + x)] - d*(-I + Sinh[e + f*x]) + f*(c + d*x)*Sinh[e -
(c*f)/d]*SinhIntegral[f*(c/d + x)]))/(d^2*(c + d*x))

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Maple [A]  time = 0.053, size = 153, normalized size = 1.6 \begin{align*} -{\frac{a}{d \left ( dx+c \right ) }}+{\frac{{\frac{i}{2}}fa{{\rm e}^{-fx-e}}}{d \left ( dfx+cf \right ) }}-{\frac{{\frac{i}{2}}fa}{{d}^{2}}{{\rm e}^{{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,fx+e+{\frac{cf-de}{d}} \right ) }-{\frac{{\frac{i}{2}}fa{{\rm e}^{fx+e}}}{{d}^{2}} \left ({\frac{cf}{d}}+fx \right ) ^{-1}}-{\frac{{\frac{i}{2}}fa}{{d}^{2}}{{\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+I*a*sinh(f*x+e))/(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.22058, size = 119, normalized size = 1.25 \begin{align*} \frac{1}{2} i \, a{\left (\frac{e^{\left (-e + \frac{c f}{d}\right )} E_{2}\left (\frac{{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} - \frac{e^{\left (e - \frac{c f}{d}\right )} E_{2}\left (-\frac{{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d}\right )} - \frac{a}{d^{2} x + c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*sinh(f*x+e))/(d*x+c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.7878, size = 290, normalized size = 3.05 \begin{align*} \frac{{\left (-i \, a d e^{\left (2 \, f x + 2 \, e\right )} + i \, a d +{\left ({\left (i \, a d f x + i \, a c f\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (\frac{d e - c f}{d}\right )} +{\left (i \, a d f x + i \, a c f\right )}{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (-\frac{d e - c f}{d}\right )} - 2 \, a d\right )} e^{\left (f x + e\right )}\right )} e^{\left (-f x - e\right )}}{2 \,{\left (d^{3} x + c d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*sinh(f*x+e))/(d*x+c)^2,x, algorithm="fricas")

[Out]

1/2*(-I*a*d*e^(2*f*x + 2*e) + I*a*d + ((I*a*d*f*x + I*a*c*f)*Ei((d*f*x + c*f)/d)*e^((d*e - c*f)/d) + (I*a*d*f*
x + I*a*c*f)*Ei(-(d*f*x + c*f)/d)*e^(-(d*e - c*f)/d) - 2*a*d)*e^(f*x + e))*e^(-f*x - e)/(d^3*x + c*d^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*sinh(f*x+e))/(d*x+c)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.28447, size = 225, normalized size = 2.37 \begin{align*} \frac{i \,{\left (d f x{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} + d f x{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} + c f{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} + c f{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} - d e^{\left (f x + e\right )} + d e^{\left (-f x - e\right )}\right )} a}{2 \,{\left (d^{3} x + c d^{2}\right )}} - \frac{a}{{\left (d x + c\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*sinh(f*x+e))/(d*x+c)^2,x, algorithm="giac")

[Out]

1/2*I*(d*f*x*Ei(-(d*f*x + c*f)/d)*e^(c*f/d - e) + d*f*x*Ei((d*f*x + c*f)/d)*e^(-c*f/d + e) + c*f*Ei(-(d*f*x +
c*f)/d)*e^(c*f/d - e) + c*f*Ei((d*f*x + c*f)/d)*e^(-c*f/d + e) - d*e^(f*x + e) + d*e^(-f*x - e))*a/(d^3*x + c*
d^2) - a/((d*x + c)*d)