\(\int \frac {\cosh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\) [264]

Optimal result

Integrand size = 31, antiderivative size = 103 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=-\frac {1}{a f (e+f x)}-\frac {i d \cosh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (\frac {d e}{f}+d x\right )}{a f^2}+\frac {i \sinh (c+d x)}{a f (e+f x)}-\frac {i d \sinh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f^2} \]

[Out]

-1/a/f/(f*x+e)-I*d*Chi(d*e/f+d*x)*cosh(c-d*e/f)/a/f^2-I*d*Shi(d*e/f+d*x)*sinh(c-d*e/f)/a/f^2+I*sinh(d*x+c)/a/f
/(f*x+e)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {5682, 32, 3378, 3384, 3379, 3382} \[ \int \frac {\cosh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=-\frac {i d \cosh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (\frac {d e}{f}+d x\right )}{a f^2}-\frac {i d \sinh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f^2}+\frac {i \sinh (c+d x)}{a f (e+f x)}-\frac {1}{a f (e+f x)} \]

[In]

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

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 3378

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.83 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=-\frac {i \left (d (e+f x) \cosh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (d \left (\frac {e}{f}+x\right )\right )-f (i+\sinh (c+d x))+d (e+f x) \sinh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (d \left (\frac {e}{f}+x\right )\right )\right )}{a f^2 (e+f x)} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 29.96 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.59

method	result	size
risch	\(-\frac {1}{a f \left (f x +e \right )}+\frac {i d \,{\mathrm e}^{d x +c}}{2 a \,f^{2} \left (\frac {d e}{f}+d x \right )}+\frac {i d \,{\mathrm e}^{\frac {c f -d e}{f}} \operatorname {Ei}_{1}\left (-d x -c -\frac {-c f +d e}{f}\right )}{2 a \,f^{2}}-\frac {i d \,{\mathrm e}^{-d x -c}}{2 a f \left (d f x +d e \right )}+\frac {i d \,{\mathrm e}^{-\frac {c f -d e}{f}} \operatorname {Ei}_{1}\left (d x +c -\frac {c f -d e}{f}\right )}{2 a \,f^{2}}\)	\(164\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.25 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\frac {{\left (i \, f e^{\left (2 \, d x + 2 \, c\right )} + {\left ({\left (-i \, d f x - i \, d e\right )} {\rm Ei}\left (-\frac {d f x + d e}{f}\right ) e^{\left (\frac {d e - c f}{f}\right )} + {\left (-i \, d f x - i \, d e\right )} {\rm Ei}\left (\frac {d f x + d e}{f}\right ) e^{\left (-\frac {d e - c f}{f}\right )} - 2 \, f\right )} e^{\left (d x + c\right )} - i \, f\right )} e^{\left (-d x - c\right )}}{2 \, {\left (a f^{3} x + a e f^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\cosh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.89 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=-\frac {1}{a f^{2} x + a e f} - \frac {i \, e^{\left (-c + \frac {d e}{f}\right )} E_{2}\left (\frac {{\left (f x + e\right )} d}{f}\right )}{2 \, {\left (f x + e\right )} a f} + \frac {i \, e^{\left (c - \frac {d e}{f}\right )} E_{2}\left (-\frac {{\left (f x + e\right )} d}{f}\right )}{2 \, {\left (f x + e\right )} a f} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 572 vs. \(2 (97) = 194\).

Time = 0.32 (sec) , antiderivative size = 572, normalized size of antiderivative = 5.55 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=-\frac {{\left (i \, {\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} d^{2} {\rm Ei}\left (-\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (\frac {d e - c f}{f}\right )} + i \, d^{3} e {\rm Ei}\left (-\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (\frac {d e - c f}{f}\right )} - i \, c d^{2} f {\rm Ei}\left (-\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (\frac {d e - c f}{f}\right )} + i \, {\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} d^{2} {\rm Ei}\left (\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (-\frac {d e - c f}{f}\right )} + i \, d^{3} e {\rm Ei}\left (\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (-\frac {d e - c f}{f}\right )} - i \, c d^{2} f {\rm Ei}\left (\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (-\frac {d e - c f}{f}\right )} - i \, d^{2} f e^{\left (\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )}}{f}\right )} + i \, d^{2} f e^{\left (-\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )}}{f}\right )} + 2 \, d^{2} f\right )} f^{2}}{2 \, {\left ({\left (f x + e\right )} a {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} f^{4} + a d e f^{4} - a c f^{5}\right )} d} \]

[In]

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

[Out]

-1/2*(I*(f*x + e)*(d - d*e/(f*x + e) + c*f/(f*x + e))*d^2*Ei(-((f*x + e)*(d - d*e/(f*x + e) + c*f/(f*x + e)) +
d*e - c*f)/f)*e^((d*e - c*f)/f) + I*d^3*e*Ei(-((f*x + e)*(d - d*e/(f*x + e) + c*f/(f*x + e)) + d*e - c*f)/f)*
e^((d*e - c*f)/f) - I*c*d^2*f*Ei(-((f*x + e)*(d - d*e/(f*x + e) + c*f/(f*x + e)) + d*e - c*f)/f)*e^((d*e - c*f
)/f) + I*(f*x + e)*(d - d*e/(f*x + e) + c*f/(f*x + e))*d^2*Ei(((f*x + e)*(d - d*e/(f*x + e) + c*f/(f*x + e)) +
d*e - c*f)/f)*e^(-(d*e - c*f)/f) + I*d^3*e*Ei(((f*x + e)*(d - d*e/(f*x + e) + c*f/(f*x + e)) + d*e - c*f)/f)*
e^(-(d*e - c*f)/f) - I*c*d^2*f*Ei(((f*x + e)*(d - d*e/(f*x + e) + c*f/(f*x + e)) + d*e - c*f)/f)*e^(-(d*e - c*
f)/f) - I*d^2*f*e^((f*x + e)*(d - d*e/(f*x + e) + c*f/(f*x + e))/f) + I*d^2*f*e^(-(f*x + e)*(d - d*e/(f*x + e)
+ c*f/(f*x + e))/f) + 2*d^2*f)*f^2/(((f*x + e)*a*(d - d*e/(f*x + e) + c*f/(f*x + e))*f^4 + a*d*e*f^4 - a*c*f^
5)*d)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2}{{\left (e+f\,x\right )}^2\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

[In]

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

[Out]

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