\(\int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\) [269]

Optimal result

Integrand size = 31, antiderivative size = 131 \[ \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\frac {\cosh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {i \text {Chi}\left (\frac {2 d e}{f}+2 d x\right ) \sinh \left (2 c-\frac {2 d e}{f}\right )}{2 a f}+\frac {\sinh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {i \cosh \left (2 c-\frac {2 d e}{f}\right ) \text {Shi}\left (\frac {2 d e}{f}+2 d x\right )}{2 a f} \] Output:

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

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\frac {2 \cosh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (d \left (\frac {e}{f}+x\right )\right )-i \left (\text {Chi}\left (\frac {2 d (e+f x)}{f}\right ) \sinh \left (2 c-\frac {2 d e}{f}\right )+2 i \sinh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (d \left (\frac {e}{f}+x\right )\right )+\cosh \left (2 c-\frac {2 d e}{f}\right ) \text {Shi}\left (\frac {2 d (e+f x)}{f}\right )\right )}{2 a f} \] Input:

Integrate[Cosh[c + d*x]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])),x]
 

Output:

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

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, Rules used = {6097, 3042, 3784, 26, 3042, 26, 3779, 3782, 5971, 27, 3042, 26, 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.

\(\displaystyle \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\)

\(\Big \downarrow \) 6097

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

\(\displaystyle \frac {\frac {\cosh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (\frac {d e}{f}+d x\right )}{f}+\frac {\sinh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{f}}{a}-\frac {\frac {i \sinh \left (2 c-\frac {2 d e}{f}\right ) \text {Chi}\left (\frac {2 d e}{f}+2 d x\right )}{f}+\frac {i \cosh \left (2 c-\frac {2 d e}{f}\right ) \text {Shi}\left (\frac {2 d e}{f}+2 d x\right )}{f}}{2 a}\)

Input:

Int[Cosh[c + d*x]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])),x]
 

Output:

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

Defintions of rubi rules used

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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

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

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 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 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 = 27.98 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.37

method	result	size
risch	\(-\frac {{\mathrm e}^{-\frac {c f -d e}{f}} \operatorname {expIntegral}_{1}\left (d x +c -\frac {c f -d e}{f}\right )}{2 a f}+\frac {i {\mathrm e}^{\frac {2 c f -2 d e}{f}} \operatorname {expIntegral}_{1}\left (-2 d x -2 c -\frac {2 \left (-c f +d e \right )}{f}\right )}{4 a f}-\frac {{\mathrm e}^{\frac {c f -d e}{f}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {-c f +d e}{f}\right )}{2 a f}-\frac {i {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{f}} \operatorname {expIntegral}_{1}\left (2 d x +2 c -\frac {2 \left (c f -d e \right )}{f}\right )}{4 a f}\)	\(180\)

Input:

int(cosh(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

-1/2/a/f*exp(-(c*f-d*e)/f)*Ei(1,d*x+c-(c*f-d*e)/f)+1/4*I/a/f*exp(2*(c*f-d* 
e)/f)*Ei(1,-2*d*x-2*c-2*(-c*f+d*e)/f)-1/2/a/f*exp((c*f-d*e)/f)*Ei(1,-d*x-c 
-(-c*f+d*e)/f)-1/4*I/a/f*exp(-2*(c*f-d*e)/f)*Ei(1,2*d*x+2*c-2*(c*f-d*e)/f)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.97 \[ \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\frac {i \, {\rm Ei}\left (-\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) e^{\left (\frac {2 \, {\left (d e - c f\right )}}{f}\right )} + 2 \, {\rm Ei}\left (-\frac {d f x + d e}{f}\right ) e^{\left (\frac {d e - c f}{f}\right )} + 2 \, {\rm Ei}\left (\frac {d f x + d e}{f}\right ) e^{\left (-\frac {d e - c f}{f}\right )} - i \, {\rm Ei}\left (\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right )}}{4 \, a f} \] Input:

integrate(cosh(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=- \frac {i \int \frac {\cosh ^{3}{\left (c + d x \right )}}{e \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a} \] Input:

integrate(cosh(d*x+c)**3/(f*x+e)/(a+I*a*sinh(d*x+c)),x)
 

Output:

-I*Integral(cosh(c + d*x)**3/(e*sinh(c + d*x) - I*e + f*x*sinh(c + d*x) - 
I*f*x), x)/a
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(cosh(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.10 \[ \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=-\frac {{\left (i \, {\rm Ei}\left (\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) e^{\left (4 \, c - \frac {2 \, d e}{f}\right )} - 2 \, {\rm Ei}\left (\frac {d f x + d e}{f}\right ) e^{\left (3 \, c - \frac {d e}{f}\right )} - 2 \, {\rm Ei}\left (-\frac {d f x + d e}{f}\right ) e^{\left (c + \frac {d e}{f}\right )} - i \, {\rm Ei}\left (-\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) e^{\left (\frac {2 \, d e}{f}\right )} + 3 i \, e^{\left (2 \, c\right )} \log \left (f x + e\right ) - 3 i \, e^{\left (2 \, c\right )} \log \left (i \, f x + i \, e\right )\right )} e^{\left (-2 \, c\right )}}{4 \, a f} \] Input:

integrate(cosh(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3}{\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \] Input:

int(cosh(c + d*x)^3/((e + f*x)*(a + a*sinh(c + d*x)*1i)),x)
 

Output:

int(cosh(c + d*x)^3/((e + f*x)*(a + a*sinh(c + d*x)*1i)), x)
 

Reduce [F]

\[ \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\frac {\int \frac {\cosh \left (d x +c \right )^{3}}{\sinh \left (d x +c \right ) e i +\sinh \left (d x +c \right ) f i x +e +f x}d x}{a} \] Input:

int(cosh(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x)
 

Output:

int(cosh(c + d*x)**3/(sinh(c + d*x)*e*i + sinh(c + d*x)*f*i*x + e + f*x),x 
)/a