3.3.63 \(\int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\) [263]

Optimal. Leaf size=76 \[ \frac {\log (e+f x)}{a f}-\frac {i \text {Chi}\left (\frac {d e}{f}+d x\right ) \sinh \left (c-\frac {d e}{f}\right )}{a f}-\frac {i \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f} \]

[Out]

ln(f*x+e)/a/f-I*cosh(c-d*e/f)*Shi(d*e/f+d*x)/a/f-I*Chi(d*e/f+d*x)*sinh(c-d*e/f)/a/f

________________________________________________________________________________________

Rubi [A]
time = 0.15, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {5682, 31, 3384, 3379, 3382} \begin {gather*} -\frac {i \sinh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {i \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f}+\frac {\log (e+f x)}{a f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[e + f*x]/(a*f) - (I*CoshIntegral[(d*e)/f + d*x]*Sinh[c - (d*e)/f])/(a*f) - (I*Cosh[c - (d*e)/f]*SinhIntegr
al[(d*e)/f + d*x])/(a*f)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, 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]

Rule 5682

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> Dist[1/a, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(n -
2)*Sinh[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 + b^2, 0]

Rubi steps

\begin {align*} \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx &=-\frac {i \int \frac {\sinh (c+d x)}{e+f x} \, dx}{a}+\frac {\int \frac {1}{e+f x} \, dx}{a}\\ &=\frac {\log (e+f x)}{a f}-\frac {\left (i \cosh \left (c-\frac {d e}{f}\right )\right ) \int \frac {\sinh \left (\frac {d e}{f}+d x\right )}{e+f x} \, dx}{a}-\frac {\left (i \sinh \left (c-\frac {d e}{f}\right )\right ) \int \frac {\cosh \left (\frac {d e}{f}+d x\right )}{e+f x} \, dx}{a}\\ &=\frac {\log (e+f x)}{a f}-\frac {i \text {Chi}\left (\frac {d e}{f}+d x\right ) \sinh \left (c-\frac {d e}{f}\right )}{a f}-\frac {i \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.18, size = 62, normalized size = 0.82 \begin {gather*} \frac {\log (e+f x)-i \text {Chi}\left (d \left (\frac {e}{f}+x\right )\right ) \sinh \left (c-\frac {d e}{f}\right )-i \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (d \left (\frac {e}{f}+x\right )\right )}{a f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 1.62, size = 103, normalized size = 1.36

method	result	size
risch	\(\frac {\ln \left (f x +e \right )}{a f}+\frac {i {\mathrm e}^{\frac {c f -d e}{f}} \expIntegral \left (1, -d x -c -\frac {-c f +d e}{f}\right )}{2 a f}-\frac {i {\mathrm e}^{-\frac {c f -d e}{f}} \expIntegral \left (1, d x +c -\frac {c f -d e}{f}\right )}{2 a f}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^2/(f*x+e)/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.37, size = 81, normalized size = 1.07 \begin {gather*} -\frac {i \, e^{\left (-c + \frac {d e}{f}\right )} E_{1}\left (\frac {{\left (f x + e\right )} d}{f}\right )}{2 \, a f} + \frac {i \, e^{\left (c - \frac {d e}{f}\right )} E_{1}\left (-\frac {{\left (f x + e\right )} d}{f}\right )}{2 \, a f} + \frac {\log \left (f x + e\right )}{a f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.34, size = 84, normalized size = 1.11 \begin {gather*} \frac {-i \, {\rm Ei}\left (\frac {d f x + d e}{f}\right ) e^{\left (\frac {c f - d e}{f}\right )} + i \, {\rm Ei}\left (-\frac {d f x + d e}{f}\right ) e^{\left (-\frac {c f - d e}{f}\right )} + 2 \, \log \left (\frac {f x + e}{f}\right )}{2 \, a f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {i \int \frac {\cosh ^{2}{\left (c + d x \right )}}{e \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*Integral(cosh(c + d*x)**2/(e*sinh(c + d*x) - I*e + f*x*sinh(c + d*x) - I*f*x), x)/a

________________________________________________________________________________________

Giac [A]
time = 0.45, size = 76, normalized size = 1.00 \begin {gather*} -\frac {{\left (i \, {\rm Ei}\left (\frac {d f x + d e}{f}\right ) e^{\left (2 \, c - \frac {d e}{f}\right )} - i \, {\rm Ei}\left (-\frac {d f x + d e}{f}\right ) e^{\left (\frac {d e}{f}\right )} - 2 \, e^{c} \log \left (i \, f x + i \, e\right )\right )} e^{\left (-c\right )}}{2 \, a f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2}{\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)^2/((e + f*x)*(a + a*sinh(c + d*x)*1i)),x)

[Out]

int(cosh(c + d*x)^2/((e + f*x)*(a + a*sinh(c + d*x)*1i)), x)

________________________________________________________________________________________