\(\int \frac {(e+f x) \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx\) [330]

Optimal result

Integrand size = 24, antiderivative size = 112 \[ \int \frac {(e+f x) \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=-\frac {a f \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac {e+f x}{2 b d (a+b \sinh (c+d x))^2}-\frac {f \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))} \] Output:

-a*f*arctanh((b-a*tanh(1/2*d*x+1/2*c))/(a^2+b^2)^(1/2))/b/(a^2+b^2)^(3/2)/ 
d^2-1/2*(f*x+e)/b/d/(a+b*sinh(d*x+c))^2-1/2*f*cosh(d*x+c)/(a^2+b^2)/d^2/(a 
+b*sinh(d*x+c))
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int \frac {(e+f x) \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=-\frac {\frac {f \cosh (c+d x)}{\left (a^2+b^2\right ) (a+b \sinh (c+d x))}+\frac {\frac {2 a f \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+\frac {d (e+f x)}{(a+b \sinh (c+d x))^2}}{b}}{2 d^2} \] Input:

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

Output:

-1/2*((f*Cosh[c + d*x])/((a^2 + b^2)*(a + b*Sinh[c + d*x])) + ((2*a*f*ArcT 
an[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/(-a^2 - b^2)^(3/2) + (d*(e 
 + f*x))/(a + b*Sinh[c + d*x])^2)/b)/d^2
 

Rubi [A] (warning: unable to verify)

Time = 0.46 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5987, 3042, 3143, 25, 27, 3042, 3139, 1083, 217}

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.

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

\(\Big \downarrow \) 5987

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3143

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3139

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

\(\Big \downarrow \) 1083

\(\displaystyle -\frac {e+f x}{2 b d (a+b \sinh (c+d x))^2}+\frac {f \left (-\frac {b \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}+\frac {4 i a \int \frac {1}{\tanh ^2\left (\frac {1}{2} (c+d x)\right )-4 \left (a^2+b^2\right )}d\left (2 i a \tanh \left (\frac {1}{2} (c+d x)\right )-2 i b\right )}{d \left (a^2+b^2\right )}\right )}{2 b d}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {f \left (\frac {2 a \text {arctanh}\left (\frac {\tanh \left (\frac {1}{2} (c+d x)\right )}{2 \sqrt {a^2+b^2}}\right )}{d \left (a^2+b^2\right )^{3/2}}-\frac {b \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{2 b d}-\frac {e+f x}{2 b d (a+b \sinh (c+d x))^2}\)

Input:

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

Output:

-1/2*(e + f*x)/(b*d*(a + b*Sinh[c + d*x])^2) + (f*((2*a*ArcTanh[Tanh[(c + 
d*x)/2]/(2*Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (b*Cosh[c + d*x])/(( 
a^2 + b^2)*d*(a + b*Sinh[c + d*x]))))/(2*b*d)
 

Defintions of rubi rules used

rule 25

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

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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 3042

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

rule 3139

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre 
eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + 2*b*e*x + a 
*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ 
[a^2 - b^2, 0]
 

rule 3143

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos 
[c + d*x]*((a + b*Sin[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 - b^2))), x] + Simp 
[1/((n + 1)*(a^2 - b^2))   Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1) 
- b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
 

rule 5987

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(307\) vs. \(2(103)=206\).

Time = 18.87 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.75

method	result	size
risch	\(-\frac {2 \,{\mathrm e}^{2 d x +2 c} a^{2} d f x +2 b^{2} d f x \,{\mathrm e}^{2 d x +2 c}+2 \,{\mathrm e}^{2 d x +2 c} a^{2} d e -a b f \,{\mathrm e}^{3 d x +3 c}+2 b^{2} d e \,{\mathrm e}^{2 d x +2 c}-2 a^{2} f \,{\mathrm e}^{2 d x +2 c}+b^{2} f \,{\mathrm e}^{2 d x +2 c}+3 a f \,{\mathrm e}^{d x +c} b -b^{2} f}{b \,d^{2} \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )^{2} \left (a^{2}+b^{2}\right )}+\frac {f a \ln \left ({\mathrm e}^{d x +c}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}} d^{2} b}-\frac {f a \ln \left ({\mathrm e}^{d x +c}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}} d^{2} b}\)	\(308\)

Input:

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

Output:

-1/b*(2*exp(2*d*x+2*c)*a^2*d*f*x+2*b^2*d*f*x*exp(2*d*x+2*c)+2*exp(2*d*x+2* 
c)*a^2*d*e-a*b*f*exp(3*d*x+3*c)+2*b^2*d*e*exp(2*d*x+2*c)-2*a^2*f*exp(2*d*x 
+2*c)+b^2*f*exp(2*d*x+2*c)+3*a*f*exp(d*x+c)*b-b^2*f)/d^2/(b*exp(2*d*x+2*c) 
+2*a*exp(d*x+c)-b)^2/(a^2+b^2)+1/2/(a^2+b^2)^(3/2)*f*a/d^2/b*ln(exp(d*x+c) 
+(a*(a^2+b^2)^(3/2)-a^4-2*a^2*b^2-b^4)/b/(a^2+b^2)^(3/2))-1/2/(a^2+b^2)^(3 
/2)*f*a/d^2/b*ln(exp(d*x+c)+(a*(a^2+b^2)^(3/2)+a^4+2*a^2*b^2+b^4)/b/(a^2+b 
^2)^(3/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1230 vs. \(2 (105) = 210\).

Time = 0.13 (sec) , antiderivative size = 1230, normalized size of antiderivative = 10.98 \[ \int \frac {(e+f x) \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(2*(a^3*b + a*b^3)*f*cosh(d*x + c)^3 + 2*(a^3*b + a*b^3)*f*sinh(d*x + 
c)^3 - 6*(a^3*b + a*b^3)*f*cosh(d*x + c) - 2*(2*(a^4 + 2*a^2*b^2 + b^4)*d* 
f*x + 2*(a^4 + 2*a^2*b^2 + b^4)*d*e - (2*a^4 + a^2*b^2 - b^4)*f)*cosh(d*x 
+ c)^2 - 2*(2*(a^4 + 2*a^2*b^2 + b^4)*d*f*x + 2*(a^4 + 2*a^2*b^2 + b^4)*d* 
e - 3*(a^3*b + a*b^3)*f*cosh(d*x + c) - (2*a^4 + a^2*b^2 - b^4)*f)*sinh(d* 
x + c)^2 + (a*b^2*f*cosh(d*x + c)^4 + a*b^2*f*sinh(d*x + c)^4 + 4*a^2*b*f* 
cosh(d*x + c)^3 - 4*a^2*b*f*cosh(d*x + c) + a*b^2*f + 2*(2*a^3 - a*b^2)*f* 
cosh(d*x + c)^2 + 4*(a*b^2*f*cosh(d*x + c) + a^2*b*f)*sinh(d*x + c)^3 + 2* 
(3*a*b^2*f*cosh(d*x + c)^2 + 6*a^2*b*f*cosh(d*x + c) + (2*a^3 - a*b^2)*f)* 
sinh(d*x + c)^2 + 4*(a*b^2*f*cosh(d*x + c)^3 + 3*a^2*b*f*cosh(d*x + c)^2 - 
 a^2*b*f + (2*a^3 - a*b^2)*f*cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 + b^2) 
*log((b^2*cosh(d*x + c)^2 + b^2*sinh(d*x + c)^2 + 2*a*b*cosh(d*x + c) + 2* 
a^2 + b^2 + 2*(b^2*cosh(d*x + c) + a*b)*sinh(d*x + c) - 2*sqrt(a^2 + b^2)* 
(b*cosh(d*x + c) + b*sinh(d*x + c) + a))/(b*cosh(d*x + c)^2 + b*sinh(d*x + 
 c)^2 + 2*a*cosh(d*x + c) + 2*(b*cosh(d*x + c) + a)*sinh(d*x + c) - b)) + 
2*(a^2*b^2 + b^4)*f + 2*(3*(a^3*b + a*b^3)*f*cosh(d*x + c)^2 - 3*(a^3*b + 
a*b^3)*f - 2*(2*(a^4 + 2*a^2*b^2 + b^4)*d*f*x + 2*(a^4 + 2*a^2*b^2 + b^4)* 
d*e - (2*a^4 + a^2*b^2 - b^4)*f)*cosh(d*x + c))*sinh(d*x + c))/((a^4*b^3 + 
 2*a^2*b^5 + b^7)*d^2*cosh(d*x + c)^4 + (a^4*b^3 + 2*a^2*b^5 + b^7)*d^2*si 
nh(d*x + c)^4 + 4*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*d^2*cosh(d*x + c)^3 + 2...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (105) = 210\).

Time = 0.25 (sec) , antiderivative size = 413, normalized size of antiderivative = 3.69 \[ \int \frac {(e+f x) \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\frac {1}{2} \, f {\left (\frac {2 \, {\left (a b e^{\left (3 \, d x + 3 \, c\right )} - 3 \, a b e^{\left (d x + c\right )} + b^{2} + {\left (2 \, a^{2} e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )} - 2 \, {\left (a^{2} d e^{\left (2 \, c\right )} + b^{2} d e^{\left (2 \, c\right )}\right )} x\right )} e^{\left (2 \, d x\right )}\right )}}{a^{2} b^{3} d^{2} + b^{5} d^{2} + {\left (a^{2} b^{3} d^{2} e^{\left (4 \, c\right )} + b^{5} d^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 4 \, {\left (a^{3} b^{2} d^{2} e^{\left (3 \, c\right )} + a b^{4} d^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + 2 \, {\left (2 \, a^{4} b d^{2} e^{\left (2 \, c\right )} + a^{2} b^{3} d^{2} e^{\left (2 \, c\right )} - b^{5} d^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 4 \, {\left (a^{3} b^{2} d^{2} e^{c} + a b^{4} d^{2} e^{c}\right )} e^{\left (d x\right )}} + \frac {a \log \left (\frac {b e^{\left (d x + 2 \, c\right )} + a e^{c} - \sqrt {a^{2} + b^{2}} e^{c}}{b e^{\left (d x + 2 \, c\right )} + a e^{c} + \sqrt {a^{2} + b^{2}} e^{c}}\right )}{{\left (a^{2} b + b^{3}\right )} \sqrt {a^{2} + b^{2}} d^{2}}\right )} - \frac {2 \, e e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (4 \, a b^{2} e^{\left (-d x - c\right )} - 4 \, a b^{2} e^{\left (-3 \, d x - 3 \, c\right )} + b^{3} e^{\left (-4 \, d x - 4 \, c\right )} + b^{3} + 2 \, {\left (2 \, a^{2} b - b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} \] Input:

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

Output:

1/2*f*(2*(a*b*e^(3*d*x + 3*c) - 3*a*b*e^(d*x + c) + b^2 + (2*a^2*e^(2*c) - 
 b^2*e^(2*c) - 2*(a^2*d*e^(2*c) + b^2*d*e^(2*c))*x)*e^(2*d*x))/(a^2*b^3*d^ 
2 + b^5*d^2 + (a^2*b^3*d^2*e^(4*c) + b^5*d^2*e^(4*c))*e^(4*d*x) + 4*(a^3*b 
^2*d^2*e^(3*c) + a*b^4*d^2*e^(3*c))*e^(3*d*x) + 2*(2*a^4*b*d^2*e^(2*c) + a 
^2*b^3*d^2*e^(2*c) - b^5*d^2*e^(2*c))*e^(2*d*x) - 4*(a^3*b^2*d^2*e^c + a*b 
^4*d^2*e^c)*e^(d*x)) + a*log((b*e^(d*x + 2*c) + a*e^c - sqrt(a^2 + b^2)*e^ 
c)/(b*e^(d*x + 2*c) + a*e^c + sqrt(a^2 + b^2)*e^c))/((a^2*b + b^3)*sqrt(a^ 
2 + b^2)*d^2)) - 2*e*e^(-2*d*x - 2*c)/((4*a*b^2*e^(-d*x - c) - 4*a*b^2*e^( 
-3*d*x - 3*c) + b^3*e^(-4*d*x - 4*c) + b^3 + 2*(2*a^2*b - b^3)*e^(-2*d*x - 
 2*c))*d)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 756, normalized size of antiderivative = 6.75 \[ \int \frac {(e+f x) \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\frac {4 e^{4 d x +4 c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{2} f i +16 e^{3 d x +3 c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b f i +16 e^{2 d x +2 c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{3} f i -8 e^{2 d x +2 c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{2} f i -16 e^{d x +c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b f i +4 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{2} f i -e^{4 d x +4 c} a^{2} b^{2} f -e^{4 d x +4 c} b^{4} f -8 e^{2 d x +2 c} a^{4} d e -8 e^{2 d x +2 c} a^{4} d f x +4 e^{2 d x +2 c} a^{4} f -16 e^{2 d x +2 c} a^{2} b^{2} d e -16 e^{2 d x +2 c} a^{2} b^{2} d f x +2 e^{2 d x +2 c} a^{2} b^{2} f -8 e^{2 d x +2 c} b^{4} d e -8 e^{2 d x +2 c} b^{4} d f x -2 e^{2 d x +2 c} b^{4} f -8 e^{d x +c} a^{3} b f -8 e^{d x +c} a \,b^{3} f +3 a^{2} b^{2} f +3 b^{4} f}{4 b \,d^{2} \left (e^{4 d x +4 c} a^{4} b^{2}+2 e^{4 d x +4 c} a^{2} b^{4}+e^{4 d x +4 c} b^{6}+4 e^{3 d x +3 c} a^{5} b +8 e^{3 d x +3 c} a^{3} b^{3}+4 e^{3 d x +3 c} a \,b^{5}+4 e^{2 d x +2 c} a^{6}+6 e^{2 d x +2 c} a^{4} b^{2}-2 e^{2 d x +2 c} b^{6}-4 e^{d x +c} a^{5} b -8 e^{d x +c} a^{3} b^{3}-4 e^{d x +c} a \,b^{5}+a^{4} b^{2}+2 a^{2} b^{4}+b^{6}\right )} \] Input:

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

Output:

(4*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a 
**2 + b**2))*a*b**2*f*i + 16*e**(3*c + 3*d*x)*sqrt(a**2 + b**2)*atan((e**( 
c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b*f*i + 16*e**(2*c + 2*d*x)*sq 
rt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**3*f*i 
- 8*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt( 
a**2 + b**2))*a*b**2*f*i - 16*e**(c + d*x)*sqrt(a**2 + b**2)*atan((e**(c + 
 d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b*f*i + 4*sqrt(a**2 + b**2)*atan( 
(e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a*b**2*f*i - e**(4*c + 4*d*x)* 
a**2*b**2*f - e**(4*c + 4*d*x)*b**4*f - 8*e**(2*c + 2*d*x)*a**4*d*e - 8*e* 
*(2*c + 2*d*x)*a**4*d*f*x + 4*e**(2*c + 2*d*x)*a**4*f - 16*e**(2*c + 2*d*x 
)*a**2*b**2*d*e - 16*e**(2*c + 2*d*x)*a**2*b**2*d*f*x + 2*e**(2*c + 2*d*x) 
*a**2*b**2*f - 8*e**(2*c + 2*d*x)*b**4*d*e - 8*e**(2*c + 2*d*x)*b**4*d*f*x 
 - 2*e**(2*c + 2*d*x)*b**4*f - 8*e**(c + d*x)*a**3*b*f - 8*e**(c + d*x)*a* 
b**3*f + 3*a**2*b**2*f + 3*b**4*f)/(4*b*d**2*(e**(4*c + 4*d*x)*a**4*b**2 + 
 2*e**(4*c + 4*d*x)*a**2*b**4 + e**(4*c + 4*d*x)*b**6 + 4*e**(3*c + 3*d*x) 
*a**5*b + 8*e**(3*c + 3*d*x)*a**3*b**3 + 4*e**(3*c + 3*d*x)*a*b**5 + 4*e** 
(2*c + 2*d*x)*a**6 + 6*e**(2*c + 2*d*x)*a**4*b**2 - 2*e**(2*c + 2*d*x)*b** 
6 - 4*e**(c + d*x)*a**5*b - 8*e**(c + d*x)*a**3*b**3 - 4*e**(c + d*x)*a*b* 
*5 + a**4*b**2 + 2*a**2*b**4 + b**6))