\(\int \frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [335]

Optimal result

Integrand size = 30, antiderivative size = 212 \[ \int \frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a (e+f x)^2}{2 b^2 f}-\frac {f \cosh (c+d x)}{b d^2}-\frac {a (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {a (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {a f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {a f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}+\frac {(e+f x) \sinh (c+d x)}{b d} \] Output:

1/2*a*(f*x+e)^2/b^2/f-f*cosh(d*x+c)/b/d^2-a*(f*x+e)*ln(1+b*exp(d*x+c)/(a-( 
a^2+b^2)^(1/2)))/b^2/d-a*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^ 
2/d-a*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/d^2-a*f*polylog(2 
,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/d^2+(f*x+e)*sinh(d*x+c)/b/d
 

Mathematica [A] (verified)

Time = 2.40 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.78 \[ \int \frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-2 b f \cosh (c+d x)-a \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )+2 b d (e+f x) \sinh (c+d x)}{2 b^2 d^2} \] Input:

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

Output:

(-2*b*f*Cosh[c + d*x] - a*(-2*d*e*(c + d*x) + 2*c*f*(c + d*x) - f*(c + d*x 
)^2 + (4*a*Sqrt[a^2 + b^2]*d*e*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2] 
])/Sqrt[-(a^2 + b^2)^2] - (4*a*Sqrt[-(a^2 + b^2)^2]*d*e*ArcTanh[(a + b*E^( 
c + d*x))/Sqrt[a^2 + b^2]])/(-a^2 - b^2)^(3/2) + 2*f*(c + d*x)*Log[1 + (b* 
E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*f*(c + d*x)*Log[1 + (b*E^(c + d*x) 
)/(a + Sqrt[a^2 + b^2])] - 2*c*f*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x 
))] + 2*d*e*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))] + 2*f*PolyLog[ 
2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*f*PolyLog[2, -((b*E^(c + d* 
x))/(a + Sqrt[a^2 + b^2]))]) + 2*b*d*(e + f*x)*Sinh[c + d*x])/(2*b^2*d^2)
 

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {6113, 3042, 3777, 26, 3042, 26, 3118, 6095, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 6113

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}}{b}-\frac {a \left (-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{b}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}}{b}-\frac {a \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{b}\)

Input:

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

Output:

-((a*(-1/2*(e + f*x)^2/(b*f) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqr 
t[a^2 + b^2])])/(b*d) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + 
 b^2])])/(b*d) + (f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/ 
(b*d^2) + (f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^2) 
))/b) + (-((f*Cosh[c + d*x])/d^2) + ((e + f*x)*Sinh[c + d*x])/d)/b
 

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 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

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 6095

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin 
h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), 
 x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
 

rule 6113

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + 
 (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S 
imp[1/b   Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x] - S 
imp[a/b   Int[(e + f*x)^m*Cosh[c + d*x]^p*(Sinh[c + d*x]^(n - 1)/(a + b*Sin 
h[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[ 
n, 0] && IGtQ[p, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(482\) vs. \(2(198)=396\).

Time = 1.76 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.28

method	result	size
risch	\(\frac {a f \,x^{2}}{2 b^{2}}-\frac {a e x}{b^{2}}+\frac {\left (d x f +d e -f \right ) {\mathrm e}^{d x +c}}{2 b \,d^{2}}-\frac {\left (d x f +d e +f \right ) {\mathrm e}^{-d x -c}}{2 b \,d^{2}}-\frac {a f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{2}}-\frac {a f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{2}}-\frac {a f \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{2}}-\frac {a f \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{2}}+\frac {a f \,c^{2}}{d^{2} b^{2}}-\frac {a f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{2}}-\frac {2 a c f \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b^{2}}+\frac {a c f \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} b^{2}}-\frac {a f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{2}}+\frac {2 a e \ln \left ({\mathrm e}^{d x +c}\right )}{d \,b^{2}}+\frac {2 a f c x}{d \,b^{2}}-\frac {a e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d \,b^{2}}\)	\(483\)

Input:

int((f*x+e)*cosh(d*x+c)*sinh(d*x+c)/(a+b*sinh(d*x+c)),x,method=_RETURNVERB 
OSE)
 

Output:

1/2*a/b^2*f*x^2-a/b^2*e*x+1/2*(d*f*x+d*e-f)/b/d^2*exp(d*x+c)-1/2*(d*f*x+d* 
e+f)/b/d^2*exp(-d*x-c)-1/d*a/b^2*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(- 
a+(a^2+b^2)^(1/2)))*x-1/d*a/b^2*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+( 
a^2+b^2)^(1/2)))*x-1/d^2*a/b^2*f*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/( 
-a+(a^2+b^2)^(1/2)))-1/d^2*a/b^2*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/ 
(a+(a^2+b^2)^(1/2)))+1/d^2*a/b^2*f*c^2-1/d^2*a/b^2*f*ln((-b*exp(d*x+c)+(a^ 
2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-2/d^2*a/b^2*c*f*ln(exp(d*x+c))+1/d 
^2*a/b^2*c*f*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-1/d^2*a/b^2*f*ln((b*exp 
(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+2/d*a/b^2*e*ln(exp(d*x+c 
))+2/d*a/b^2*f*c*x-1/d*a/b^2*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (196) = 392\).

Time = 0.10 (sec) , antiderivative size = 692, normalized size of antiderivative = 3.26 \[ \int \frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*(b*d*f*x + b*d*e - (b*d*f*x + b*d*e - b*f)*cosh(d*x + c)^2 - (b*d*f*x 
 + b*d*e - b*f)*sinh(d*x + c)^2 + b*f - (a*d^2*f*x^2 + 2*a*d^2*e*x + 4*a*c 
*d*e - 2*a*c^2*f)*cosh(d*x + c) + 2*(a*f*cosh(d*x + c) + a*f*sinh(d*x + c) 
)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x 
 + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 2*(a*f*cosh(d*x + c) + a*f*sinh 
(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b 
*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 2*((a*d*e - a*c*f)*cos 
h(d*x + c) + (a*d*e - a*c*f)*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*si 
nh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 2*((a*d*e - a*c*f)*cosh(d 
*x + c) + (a*d*e - a*c*f)*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh( 
d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 2*((a*d*f*x + a*c*f)*cosh(d* 
x + c) + (a*d*f*x + a*c*f)*sinh(d*x + c))*log(-(a*cosh(d*x + c) + a*sinh(d 
*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b 
) + 2*((a*d*f*x + a*c*f)*cosh(d*x + c) + (a*d*f*x + a*c*f)*sinh(d*x + c))* 
log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + 
c))*sqrt((a^2 + b^2)/b^2) - b)/b) - (a*d^2*f*x^2 + 2*a*d^2*e*x + 4*a*c*d*e 
 - 2*a*c^2*f + 2*(b*d*f*x + b*d*e - b*f)*cosh(d*x + c))*sinh(d*x + c))/(b^ 
2*d^2*cosh(d*x + c) + b^2*d^2*sinh(d*x + c))
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

-1/2*e*(2*(d*x + c)*a/(b^2*d) - e^(d*x + c)/(b*d) + e^(-d*x - c)/(b*d) + 2 
*a*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^2*d)) - 1/4*f*(2*(a* 
d^2*x^2*e^c - (b*d*x*e^(2*c) - b*e^(2*c))*e^(d*x) + (b*d*x + b)*e^(-d*x))* 
e^(-c)/(b^2*d^2) - integrate(8*(a^2*x*e^(d*x + c) - a*b*x)/(b^3*e^(2*d*x + 
 2*c) + 2*a*b^2*e^(d*x + c) - b^3), x))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - cosh(c + d*x)*b*f - e**(2*c)*int((e**(2*d*x)*x)/(e**(2*c + 2*d*x)*b + 
2*e**(c + d*x)*a - b),x)*a*b*d**2*f - int(x/(e**(2*c + 2*d*x)*b + 2*e**(c 
+ d*x)*a - b),x)*a*b*d**2*f - log(sinh(c + d*x)*b + a)*a*d*e + sinh(c + d* 
x)*b*d*e + sinh(c + d*x)*b*d*f*x)/(b**2*d**2)