\(\int \frac {(e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [364]

Optimal result

Integrand size = 32, antiderivative size = 278 \[ \int \frac {(e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {f x}{4 b d}-\frac {a^2 (e+f x)^2}{2 b^3 f}+\frac {a f \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {a (e+f x) \sinh (c+d x)}{b^2 d}-\frac {f \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {(e+f x) \sinh ^2(c+d x)}{2 b d} \] Output:

1/4*f*x/b/d-1/2*a^2*(f*x+e)^2/b^3/f+a*f*cosh(d*x+c)/b^2/d^2+a^2*(f*x+e)*ln 
(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d+a^2*(f*x+e)*ln(1+b*exp(d*x+c)/( 
a+(a^2+b^2)^(1/2)))/b^3/d+a^2*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2) 
))/b^3/d^2+a^2*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d^2-a*(f 
*x+e)*sinh(d*x+c)/b^2/d-1/4*f*cosh(d*x+c)*sinh(d*x+c)/b/d^2+1/2*(f*x+e)*si 
nh(d*x+c)^2/b/d
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.74 \[ \int \frac {(e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {e \log (a+b \sinh (c+d x))}{4 b d}-\frac {1}{4} f \left (-\frac {x^2}{2 b}+\frac {x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d}+\frac {x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d}+\frac {\operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}\right )+\frac {e \left (\left (4 a^2+b^2\right ) \log (a+b \sinh (c+d x))-4 a b \sinh (c+d x)+2 b^2 \sinh ^2(c+d x)\right )}{4 b^3 d}+\frac {f \left (8 a b \cosh (c+d x)+2 b^2 d x \cosh (2 (c+d x))+\left (4 a^2+b^2\right ) \left (2 c (c+d x)-(c+d x)^2+2 (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )-8 a b d x \sinh (c+d x)-b^2 \sinh (2 (c+d x))\right )}{8 b^3 d^2} \] Input:

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

Output:

-1/4*(e*Log[a + b*Sinh[c + d*x]])/(b*d) - (f*(-1/2*x^2/b + (x*Log[1 + (b*E 
^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + (x*Log[1 + (b*E^(c + d*x))/(a 
+ Sqrt[a^2 + b^2])])/(b*d) + PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b 
^2])]/(b*d^2) + PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]/(b*d^ 
2)))/4 + (e*((4*a^2 + b^2)*Log[a + b*Sinh[c + d*x]] - 4*a*b*Sinh[c + d*x] 
+ 2*b^2*Sinh[c + d*x]^2))/(4*b^3*d) + (f*(8*a*b*Cosh[c + d*x] + 2*b^2*d*x* 
Cosh[2*(c + d*x)] + (4*a^2 + b^2)*(2*c*(c + d*x) - (c + d*x)^2 + 2*(c + d* 
x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*(c + d*x)*Log[1 + (b 
*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*c*Log[b - 2*a*E^(c + d*x) - b*E^( 
2*(c + d*x))] + 2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*P 
olyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]) - 8*a*b*d*x*Sinh[c + 
d*x] - b^2*Sinh[2*(c + d*x)]))/(8*b^3*d^2)
 

Rubi [A] (verified)

Time = 1.69 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.99, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.531, Rules used = {6113, 5969, 3042, 25, 3115, 24, 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.

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

\(\Big \downarrow \) 6113

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

\(\Big \downarrow \) 5969

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 6113

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-((a*(-((a*(-1/2*(e + f*x)^2/(b*f) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a 
 - Sqrt[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)) 
/b) + (((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))/b
 

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 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 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 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. \(564\) vs. \(2(258)=516\).

Time = 7.20 (sec) , antiderivative size = 565, normalized size of antiderivative = 2.03

method	result	size
risch	\(-\frac {a^{2} f \,x^{2}}{2 b^{3}}+\frac {a^{2} e x}{b^{3}}+\frac {\left (2 d x f +2 d e -f \right ) {\mathrm e}^{2 d x +2 c}}{16 b \,d^{2}}-\frac {a \left (d x f +d e -f \right ) {\mathrm e}^{d x +c}}{2 b^{2} d^{2}}+\frac {a \left (d x f +d e +f \right ) {\mathrm e}^{-d x -c}}{2 b^{2} d^{2}}+\frac {\left (2 d x f +2 d e +f \right ) {\mathrm e}^{-2 d x -2 c}}{16 b \,d^{2}}-\frac {2 a^{2} f c x}{d \,b^{3}}-\frac {2 a^{2} e \ln \left ({\mathrm e}^{d x +c}\right )}{d \,b^{3}}+\frac {2 c \,a^{2} f \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b^{3}}-\frac {c \,a^{2} f \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} b^{3}}-\frac {a^{2} f \,c^{2}}{d^{2} b^{3}}+\frac {a^{2} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{3}}+\frac {a^{2} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{3}}+\frac {a^{2} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{3}}+\frac {a^{2} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{3}}+\frac {a^{2} f \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{3}}+\frac {a^{2} f \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{3}}+\frac {a^{2} e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d \,b^{3}}\)	\(565\)

Input:

int((f*x+e)*cosh(d*x+c)*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURNVE 
RBOSE)
 

Output:

-1/2/b^3*a^2*f*x^2+1/b^3*a^2*e*x+1/16*(2*d*f*x+2*d*e-f)/b/d^2*exp(2*d*x+2* 
c)-1/2*a*(d*f*x+d*e-f)/b^2/d^2*exp(d*x+c)+1/2*a*(d*f*x+d*e+f)/b^2/d^2*exp( 
-d*x-c)+1/16*(2*d*f*x+2*d*e+f)/b/d^2*exp(-2*d*x-2*c)-2/d/b^3*a^2*f*c*x-2/d 
/b^3*a^2*e*ln(exp(d*x+c))+2/d^2/b^3*c*a^2*f*ln(exp(d*x+c))-1/d^2/b^3*c*a^2 
*f*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-1/d^2/b^3*a^2*f*c^2+1/d^2/b^3*a^2 
*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+1/d^2/b^3* 
a^2*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+1/d/b^3*a 
^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/b^3* 
a^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/b^3 
*a^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 
/b^3*a^2*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+1/d 
/b^3*a^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. 1248 vs. \(2 (256) = 512\).

Time = 0.13 (sec) , antiderivative size = 1248, normalized size of antiderivative = 4.49 \[ \int \frac {(e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

1/16*(2*b^2*d*f*x + (2*b^2*d*f*x + 2*b^2*d*e - b^2*f)*cosh(d*x + c)^4 + (2 
*b^2*d*f*x + 2*b^2*d*e - b^2*f)*sinh(d*x + c)^4 + 2*b^2*d*e - 8*(a*b*d*f*x 
 + a*b*d*e - a*b*f)*cosh(d*x + c)^3 - 4*(2*a*b*d*f*x + 2*a*b*d*e - 2*a*b*f 
 - (2*b^2*d*f*x + 2*b^2*d*e - b^2*f)*cosh(d*x + c))*sinh(d*x + c)^3 + b^2* 
f - 8*(a^2*d^2*f*x^2 + 2*a^2*d^2*e*x + 4*a^2*c*d*e - 2*a^2*c^2*f)*cosh(d*x 
 + c)^2 - 2*(4*a^2*d^2*f*x^2 + 8*a^2*d^2*e*x + 16*a^2*c*d*e - 8*a^2*c^2*f 
- 3*(2*b^2*d*f*x + 2*b^2*d*e - b^2*f)*cosh(d*x + c)^2 + 12*(a*b*d*f*x + a* 
b*d*e - a*b*f)*cosh(d*x + c))*sinh(d*x + c)^2 + 8*(a*b*d*f*x + a*b*d*e + a 
*b*f)*cosh(d*x + c) + 16*(a^2*f*cosh(d*x + c)^2 + 2*a^2*f*cosh(d*x + c)*si 
nh(d*x + c) + a^2*f*sinh(d*x + c)^2)*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 
) + 16*(a^2*f*cosh(d*x + c)^2 + 2*a^2*f*cosh(d*x + c)*sinh(d*x + c) + a^2* 
f*sinh(d*x + c)^2)*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) + 16*((a^2*d*e - 
 a^2*c*f)*cosh(d*x + c)^2 + 2*(a^2*d*e - a^2*c*f)*cosh(d*x + c)*sinh(d*x + 
 c) + (a^2*d*e - a^2*c*f)*sinh(d*x + c)^2)*log(2*b*cosh(d*x + c) + 2*b*sin 
h(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 16*((a^2*d*e - a^2*c*f)*co 
sh(d*x + c)^2 + 2*(a^2*d*e - a^2*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a^2*d 
*e - a^2*c*f)*sinh(d*x + c)^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 
 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 16*((a^2*d*f*x + a^2*c*f)*cosh(d*x ...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/8*e*(8*(d*x + c)*a^2/(b^3*d) - (4*a*e^(-d*x - c) - b)*e^(2*d*x + 2*c)/(b 
^2*d) + 8*a^2*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^3*d) + (4 
*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c))/(b^2*d)) + 1/16*f*((8*a^2*d^2*x^2*e^ 
(2*c) + (2*b^2*d*x*e^(4*c) - b^2*e^(4*c))*e^(2*d*x) - 8*(a*b*d*x*e^(3*c) - 
 a*b*e^(3*c))*e^(d*x) + 8*(a*b*d*x*e^c + a*b*e^c)*e^(-d*x) + (2*b^2*d*x + 
b^2)*e^(-2*d*x))*e^(-2*c)/(b^3*d^2) - 2*integrate(16*(a^3*x*e^(d*x + c) - 
a^2*b*x)/(b^4*e^(2*d*x + 2*c) + 2*a*b^3*e^(d*x + c) - b^4), x))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 \cosh \left (d x +c \right )^{2} b^{2} d e +\cosh \left (d x +c \right )^{2} b^{2} d f x -\cosh \left (d x +c \right ) \sinh \left (d x +c \right ) b^{2} f +4 \cosh \left (d x +c \right ) a b f +4 e^{2 c} \left (\int \frac {e^{2 d x} x}{e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a^{2} b \,d^{2} f +4 \left (\int \frac {x}{e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a^{2} b \,d^{2} f +4 \,\mathrm {log}\left (a +b \sinh \left (d x +c \right )\right ) a^{2} d e +\sinh \left (d x +c \right )^{2} b^{2} d f x -4 \sinh \left (d x +c \right ) a b d e -4 \sinh \left (d x +c \right ) a b d f x}{4 b^{3} d^{2}} \] Input:

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

Output:

(2*cosh(c + d*x)**2*b**2*d*e + cosh(c + d*x)**2*b**2*d*f*x - cosh(c + d*x) 
*sinh(c + d*x)*b**2*f + 4*cosh(c + d*x)*a*b*f + 4*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**2*b*d**2*f + 4*int(x 
/(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**2*b*d**2*f + 4*log(sinh 
(c + d*x)*b + a)*a**2*d*e + sinh(c + d*x)**2*b**2*d*f*x - 4*sinh(c + d*x)* 
a*b*d*e - 4*sinh(c + d*x)*a*b*d*f*x)/(4*b**3*d**2)