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

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.83 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=-\frac {i \left (d (e+f x) \cosh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (d \left (\frac {e}{f}+x\right )\right )-f (i+\sinh (c+d x))+d (e+f x) \sinh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (d \left (\frac {e}{f}+x\right )\right )\right )}{a f^2 (e+f x)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {6097, 17, 3042, 26, 3778, 3042, 3784, 26, 3042, 26, 3779, 3782}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6097

\(\displaystyle \frac {\int \frac {1}{(e+f x)^2}dx}{a}-\frac {i \int \frac {\sinh (c+d x)}{(e+f x)^2}dx}{a}\)

\(\Big \downarrow \) 17

\(\displaystyle -\frac {1}{a f (e+f x)}-\frac {i \int \frac {\sinh (c+d x)}{(e+f x)^2}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{a f (e+f x)}-\frac {i \int -\frac {i \sin (i c+i d x)}{(e+f x)^2}dx}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {1}{a f (e+f x)}-\frac {\int \frac {\sin (i c+i d x)}{(e+f x)^2}dx}{a}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{a f (e+f x)}-\frac {\frac {i d \int \frac {\cosh (c+d x)}{e+f x}dx}{f}-\frac {i \sinh (c+d x)}{f (e+f x)}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{a f (e+f x)}-\frac {\frac {i d \int \frac {\sin \left (i c+i d x+\frac {\pi }{2}\right )}{e+f x}dx}{f}-\frac {i \sinh (c+d x)}{f (e+f x)}}{a}\)

\(\Big \downarrow \) 3784

\(\displaystyle -\frac {1}{a f (e+f x)}-\frac {\frac {i d \left (\cosh \left (c-\frac {d e}{f}\right ) \int \frac {\cosh \left (\frac {d e}{f}+d x\right )}{e+f x}dx-i \sinh \left (c-\frac {d e}{f}\right ) \int \frac {i \sinh \left (\frac {d e}{f}+d x\right )}{e+f x}dx\right )}{f}-\frac {i \sinh (c+d x)}{f (e+f x)}}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {1}{a f (e+f x)}-\frac {\frac {i d \left (\sinh \left (c-\frac {d e}{f}\right ) \int \frac {\sinh \left (\frac {d e}{f}+d x\right )}{e+f x}dx+\cosh \left (c-\frac {d e}{f}\right ) \int \frac {\cosh \left (\frac {d e}{f}+d x\right )}{e+f x}dx\right )}{f}-\frac {i \sinh (c+d x)}{f (e+f x)}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{a f (e+f x)}-\frac {\frac {i d \left (\sinh \left (c-\frac {d e}{f}\right ) \int -\frac {i \sin \left (\frac {i d e}{f}+i d x\right )}{e+f x}dx+\cosh \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {i d e}{f}+i d x+\frac {\pi }{2}\right )}{e+f x}dx\right )}{f}-\frac {i \sinh (c+d x)}{f (e+f x)}}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {1}{a f (e+f x)}-\frac {\frac {i d \left (\cosh \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {i d e}{f}+i d x+\frac {\pi }{2}\right )}{e+f x}dx-i \sinh \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {i d e}{f}+i d x\right )}{e+f x}dx\right )}{f}-\frac {i \sinh (c+d x)}{f (e+f x)}}{a}\)

\(\Big \downarrow \) 3779

\(\displaystyle -\frac {1}{a f (e+f x)}-\frac {\frac {i d \left (\frac {\sinh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{f}+\cosh \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {i d e}{f}+i d x+\frac {\pi }{2}\right )}{e+f x}dx\right )}{f}-\frac {i \sinh (c+d x)}{f (e+f x)}}{a}\)

\(\Big \downarrow \) 3782

\(\displaystyle -\frac {1}{a f (e+f x)}-\frac {\frac {i d \left (\frac {\cosh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (\frac {d e}{f}+d x\right )}{f}+\frac {\sinh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{f}\right )}{f}-\frac {i \sinh (c+d x)}{f (e+f x)}}{a}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 17

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

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3778

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] - Simp[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 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> 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 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> 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 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[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 6097

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_. 
)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a   Int[(e + f*x)^m*Cosh[c 
 + d*x]^(n - 2), x], x] + Simp[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] && E 
qQ[a^2 + b^2, 0]
 

Maple [A] (verified)

Time = 25.81 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.59

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

Input:

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

Output:

-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)

Time = 0.09 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.25 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\frac {{\left (i \, f e^{\left (2 \, d x + 2 \, c\right )} + {\left ({\left (-i \, d f x - i \, d e\right )} {\rm Ei}\left (-\frac {d f x + d e}{f}\right ) e^{\left (\frac {d e - c f}{f}\right )} + {\left (-i \, d f x - i \, d e\right )} {\rm Ei}\left (\frac {d f x + d e}{f}\right ) e^{\left (-\frac {d e - c f}{f}\right )} - 2 \, f\right )} e^{\left (d x + c\right )} - i \, f\right )} e^{\left (-d x - c\right )}}{2 \, {\left (a f^{3} x + a e f^{2}\right )}} \] Input:

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

Output:

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.89 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=-\frac {1}{a f^{2} x + a e f} - \frac {i \, e^{\left (-c + \frac {d e}{f}\right )} E_{2}\left (\frac {{\left (f x + e\right )} d}{f}\right )}{2 \, {\left (f x + e\right )} a f} + \frac {i \, e^{\left (c - \frac {d e}{f}\right )} E_{2}\left (-\frac {{\left (f x + e\right )} d}{f}\right )}{2 \, {\left (f x + e\right )} a f} \] Input:

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

Output:

-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 = 0.18 (sec) , antiderivative size = 572, normalized size of antiderivative = 5.55 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=-\frac {{\left (i \, {\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} d^{2} {\rm Ei}\left (-\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (\frac {d e - c f}{f}\right )} + i \, d^{3} e {\rm Ei}\left (-\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (\frac {d e - c f}{f}\right )} - i \, c d^{2} f {\rm Ei}\left (-\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (\frac {d e - c f}{f}\right )} + i \, {\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} d^{2} {\rm Ei}\left (\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (-\frac {d e - c f}{f}\right )} + i \, d^{3} e {\rm Ei}\left (\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (-\frac {d e - c f}{f}\right )} - i \, c d^{2} f {\rm Ei}\left (\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (-\frac {d e - c f}{f}\right )} - i \, d^{2} f e^{\left (\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )}}{f}\right )} + i \, d^{2} f e^{\left (-\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )}}{f}\right )} + 2 \, d^{2} f\right )} f^{2}}{2 \, {\left ({\left (f x + e\right )} a {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} f^{4} + a d e f^{4} - a c f^{5}\right )} d} \] Input:

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

Output:

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

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

Output:

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

Reduce [F]

\[ \int \frac {\cosh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\frac {\int \frac {\cosh \left (d x +c \right )^{2}}{\sinh \left (d x +c \right ) e^{2} i +2 \sinh \left (d x +c \right ) e f i x +\sinh \left (d x +c \right ) f^{2} i \,x^{2}+e^{2}+2 e f x +f^{2} x^{2}}d x}{a} \] Input:

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

Output:

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