\(\int \frac {(e+f x) \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [267]

Optimal result

Integrand size = 29, antiderivative size = 98 \[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {i f x}{4 a d}-\frac {f \cosh (c+d x)}{a d^2}+\frac {(e+f x) \sinh (c+d x)}{a d}+\frac {i f \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac {i (e+f x) \sinh ^2(c+d x)}{2 a d} \] Output:

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

Mathematica [A] (verified)

Time = 2.95 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.61 \[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {i f \cosh (c+d x) (4 i+\sinh (c+d x))+d (e+f x) (-i \cosh (2 (c+d x))+4 \sinh (c+d x))}{4 a d^2} \] Input:

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

Output:

(I*f*Cosh[c + d*x]*(4*I + Sinh[c + d*x]) + d*(e + f*x)*((-I)*Cosh[2*(c + d 
*x)] + 4*Sinh[c + d*x]))/(4*a*d^2)
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.96, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {6097, 3042, 3777, 26, 3042, 26, 3118, 5969, 3042, 25, 3115, 24}

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.

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

\(\Big \downarrow \) 6097

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 5969

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 24

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 5969

Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)* 
(x_)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sinh[a + b*x]^(n + 1)/(b*(n + 1 
))), x] - Simp[d*(m/(b*(n + 1)))   Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^(n + 
 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

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 = 5.76 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.15

method	result	size
risch	\(-\frac {i \left (2 d x f +2 d e -f \right ) {\mathrm e}^{2 d x +2 c}}{16 a \,d^{2}}+\frac {\left (d x f +d e -f \right ) {\mathrm e}^{d x +c}}{2 a \,d^{2}}-\frac {\left (d x f +d e +f \right ) {\mathrm e}^{-d x -c}}{2 a \,d^{2}}-\frac {i \left (2 d x f +2 d e +f \right ) {\mathrm e}^{-2 d x -2 c}}{16 a \,d^{2}}\)	\(113\)
derivativedivides	\(\frac {\frac {i c f \cosh \left (d x +c \right )^{2}}{2}-\frac {i d e \cosh \left (d x +c \right )^{2}}{2}-i f \left (\frac {\left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{2}-\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )-\sinh \left (d x +c \right ) c f +\sinh \left (d x +c \right ) d e +f \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2} a}\)	\(118\)
default	\(\frac {\frac {i c f \cosh \left (d x +c \right )^{2}}{2}-\frac {i d e \cosh \left (d x +c \right )^{2}}{2}-i f \left (\frac {\left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{2}-\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )-\sinh \left (d x +c \right ) c f +\sinh \left (d x +c \right ) d e +f \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2} a}\)	\(118\)

Input:

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

Output:

-1/16*I*(2*d*f*x+2*d*e-f)/a/d^2*exp(2*d*x+2*c)+1/2*(d*f*x+d*e-f)/a/d^2*exp 
(d*x+c)-1/2*(d*f*x+d*e+f)/a/d^2*exp(-d*x-c)-1/16*I*(2*d*f*x+2*d*e+f)/a/d^2 
*exp(-2*d*x-2*c)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.94 \[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {{\left (-2 i \, d f x - 2 i \, d e + {\left (-2 i \, d f x - 2 i \, d e + i \, f\right )} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, {\left (d f x + d e - f\right )} e^{\left (3 \, d x + 3 \, c\right )} - 8 \, {\left (d f x + d e + f\right )} e^{\left (d x + c\right )} - i \, f\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{16 \, a d^{2}} \] Input:

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

Output:

1/16*(-2*I*d*f*x - 2*I*d*e + (-2*I*d*f*x - 2*I*d*e + I*f)*e^(4*d*x + 4*c) 
+ 8*(d*f*x + d*e - f)*e^(3*d*x + 3*c) - 8*(d*f*x + d*e + f)*e^(d*x + c) - 
I*f)*e^(-2*d*x - 2*c)/(a*d^2)
 

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (82) = 164\).

Time = 0.28 (sec) , antiderivative size = 321, normalized size of antiderivative = 3.28 \[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\begin {cases} \frac {\left (\left (- 512 a^{3} d^{7} e e^{2 c} - 512 a^{3} d^{7} f x e^{2 c} - 512 a^{3} d^{6} f e^{2 c}\right ) e^{- d x} + \left (512 a^{3} d^{7} e e^{4 c} + 512 a^{3} d^{7} f x e^{4 c} - 512 a^{3} d^{6} f e^{4 c}\right ) e^{d x} + \left (- 128 i a^{3} d^{7} e e^{c} - 128 i a^{3} d^{7} f x e^{c} - 64 i a^{3} d^{6} f e^{c}\right ) e^{- 2 d x} + \left (- 128 i a^{3} d^{7} e e^{5 c} - 128 i a^{3} d^{7} f x e^{5 c} + 64 i a^{3} d^{6} f e^{5 c}\right ) e^{2 d x}\right ) e^{- 3 c}}{1024 a^{4} d^{8}} & \text {for}\: a^{4} d^{8} e^{3 c} \neq 0 \\\frac {x^{2} \left (- i f e^{4 c} + 2 f e^{3 c} + 2 f e^{c} + i f\right ) e^{- 2 c}}{8 a} + \frac {x \left (- i e e^{4 c} + 2 e e^{3 c} + 2 e e^{c} + i e\right ) e^{- 2 c}}{4 a} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((((-512*a**3*d**7*e*exp(2*c) - 512*a**3*d**7*f*x*exp(2*c) - 512* 
a**3*d**6*f*exp(2*c))*exp(-d*x) + (512*a**3*d**7*e*exp(4*c) + 512*a**3*d** 
7*f*x*exp(4*c) - 512*a**3*d**6*f*exp(4*c))*exp(d*x) + (-128*I*a**3*d**7*e* 
exp(c) - 128*I*a**3*d**7*f*x*exp(c) - 64*I*a**3*d**6*f*exp(c))*exp(-2*d*x) 
 + (-128*I*a**3*d**7*e*exp(5*c) - 128*I*a**3*d**7*f*x*exp(5*c) + 64*I*a**3 
*d**6*f*exp(5*c))*exp(2*d*x))*exp(-3*c)/(1024*a**4*d**8), Ne(a**4*d**8*exp 
(3*c), 0)), (x**2*(-I*f*exp(4*c) + 2*f*exp(3*c) + 2*f*exp(c) + I*f)*exp(-2 
*c)/(8*a) + x*(-I*e*exp(4*c) + 2*e*exp(3*c) + 2*e*exp(c) + I*e)*exp(-2*c)/ 
(4*a), True))
 

Maxima [F(-2)]

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

integrate((f*x+e)*cosh(d*x+c)^3/(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 = 138, normalized size of antiderivative = 1.41 \[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {{\left (2 i \, d f x e^{\left (4 \, d x + 4 \, c\right )} - 8 \, d f x e^{\left (3 \, d x + 3 \, c\right )} + 8 \, d f x e^{\left (d x + c\right )} + 2 i \, d f x + 2 i \, d e e^{\left (4 \, d x + 4 \, c\right )} - 8 \, d e e^{\left (3 \, d x + 3 \, c\right )} + 8 \, d e e^{\left (d x + c\right )} + 2 i \, d e - i \, f e^{\left (4 \, d x + 4 \, c\right )} + 8 \, f e^{\left (3 \, d x + 3 \, c\right )} + 8 \, f e^{\left (d x + c\right )} + i \, f\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{16 \, a d^{2}} \] Input:

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

Output:

-1/16*(2*I*d*f*x*e^(4*d*x + 4*c) - 8*d*f*x*e^(3*d*x + 3*c) + 8*d*f*x*e^(d* 
x + c) + 2*I*d*f*x + 2*I*d*e*e^(4*d*x + 4*c) - 8*d*e*e^(3*d*x + 3*c) + 8*d 
*e*e^(d*x + c) + 2*I*d*e - I*f*e^(4*d*x + 4*c) + 8*f*e^(3*d*x + 3*c) + 8*f 
*e^(d*x + c) + I*f)*e^(-2*d*x - 2*c)/(a*d^2)
 

Mupad [B] (verification not implemented)

Time = 1.38 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.47 \[ \int \frac {(e+f x) \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-{\mathrm {e}}^{-c-d\,x}\,\left (\frac {f+d\,e}{2\,a\,d^2}+\frac {f\,x}{2\,a\,d}\right )-{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (\frac {\left (f+2\,d\,e\right )\,1{}\mathrm {i}}{16\,a\,d^2}+\frac {f\,x\,1{}\mathrm {i}}{8\,a\,d}\right )+{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (\frac {\left (f-2\,d\,e\right )\,1{}\mathrm {i}}{16\,a\,d^2}-\frac {f\,x\,1{}\mathrm {i}}{8\,a\,d}\right )-{\mathrm {e}}^{c+d\,x}\,\left (\frac {f-d\,e}{2\,a\,d^2}-\frac {f\,x}{2\,a\,d}\right ) \] Input:

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

Output:

exp(2*c + 2*d*x)*(((f - 2*d*e)*1i)/(16*a*d^2) - (f*x*1i)/(8*a*d)) - exp(- 
2*c - 2*d*x)*(((f + 2*d*e)*1i)/(16*a*d^2) + (f*x*1i)/(8*a*d)) - exp(- c - 
d*x)*((f + d*e)/(2*a*d^2) + (f*x)/(2*a*d)) - exp(c + d*x)*((f - d*e)/(2*a* 
d^2) - (f*x)/(2*a*d))
 

Reduce [F]

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

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

Output:

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