\(\int \frac {c+d x}{(a+b \sinh (e+f x))^2} \, dx\) [175]

Optimal result

Integrand size = 18, antiderivative size = 254 \[ \int \frac {c+d x}{(a+b \sinh (e+f x))^2} \, dx=\frac {a (c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f}-\frac {a (c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f}+\frac {d \log (a+b \sinh (e+f x))}{\left (a^2+b^2\right ) f^2}+\frac {a d \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f^2}-\frac {a d \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f^2}-\frac {b (c+d x) \cosh (e+f x)}{\left (a^2+b^2\right ) f (a+b \sinh (e+f x))} \] Output:

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

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.76 \[ \int \frac {c+d x}{(a+b \sinh (e+f x))^2} \, dx=\frac {d \log (a+b \sinh (e+f x))+\frac {a \left (f (c+d x) \left (\log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )-\log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )\right )+d \operatorname {PolyLog}\left (2,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )-d \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}-\frac {b f (c+d x) \cosh (e+f x)}{a+b \sinh (e+f x)}}{\left (a^2+b^2\right ) f^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3805, 3042, 3147, 16, 3803, 25, 2694, 27, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3805

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3147

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 3803

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

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 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 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && 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 3147

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_.), x_Symbol] :> Simp[1/(b^p*f)   Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) 
/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p 
 - 1)/2] && NeQ[a^2 - b^2, 0]
 

rule 3803

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

rule 3805

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(518\) vs. \(2(234)=468\).

Time = 0.32 (sec) , antiderivative size = 519, normalized size of antiderivative = 2.04

method	result	size
risch	\(\frac {2 \left (d x +c \right ) \left (a \,{\mathrm e}^{f x +e}-b \right )}{f \left (a^{2}+b^{2}\right ) \left ({\mathrm e}^{2 f x +2 e} b +2 a \,{\mathrm e}^{f x +e}-b \right )}-\frac {2 d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2} \left (a^{2}+b^{2}\right )}+\frac {d \ln \left ({\mathrm e}^{2 f x +2 e} b +2 a \,{\mathrm e}^{f x +e}-b \right )}{f^{2} \left (a^{2}+b^{2}\right )}-\frac {2 a c \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{f x +e}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{f \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {d a \ln \left (\frac {-b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{f \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {d a \ln \left (\frac {b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{f \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {d a \ln \left (\frac {-b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) e}{f^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {d a \ln \left (\frac {b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) e}{f^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {d a \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{f^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {d a \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{f^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {2 a d e \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{f x +e}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{f^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\)	\(519\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1717 vs. \(2 (232) = 464\).

Time = 0.12 (sec) , antiderivative size = 1717, normalized size of antiderivative = 6.76 \[ \int \frac {c+d x}{(a+b \sinh (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(2*(a^2*b + b^3)*d*e - 2*(a^2*b + b^3)*c*f - 2*((a^2*b + b^3)*d*f*x + (a^2 
*b + b^3)*d*e)*cosh(f*x + e)^2 - 2*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d* 
e)*sinh(f*x + e)^2 + (a*b^2*d*cosh(f*x + e)^2 + a*b^2*d*sinh(f*x + e)^2 + 
2*a^2*b*d*cosh(f*x + e) - a*b^2*d + 2*(a*b^2*d*cosh(f*x + e) + a^2*b*d)*si 
nh(f*x + e))*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(f*x + e) + a*sinh(f*x + e 
) + (b*cosh(f*x + e) + b*sinh(f*x + e))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) 
- (a*b^2*d*cosh(f*x + e)^2 + a*b^2*d*sinh(f*x + e)^2 + 2*a^2*b*d*cosh(f*x 
+ e) - a*b^2*d + 2*(a*b^2*d*cosh(f*x + e) + a^2*b*d)*sinh(f*x + e))*sqrt(( 
a^2 + b^2)/b^2)*dilog((a*cosh(f*x + e) + a*sinh(f*x + e) - (b*cosh(f*x + e 
) + b*sinh(f*x + e))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - (a*b^2*d*f*x + a* 
b^2*d*e - (a*b^2*d*f*x + a*b^2*d*e)*cosh(f*x + e)^2 - (a*b^2*d*f*x + a*b^2 
*d*e)*sinh(f*x + e)^2 - 2*(a^2*b*d*f*x + a^2*b*d*e)*cosh(f*x + e) - 2*(a^2 
*b*d*f*x + a^2*b*d*e + (a*b^2*d*f*x + a*b^2*d*e)*cosh(f*x + e))*sinh(f*x + 
 e))*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(f*x + e) + a*sinh(f*x + e) + (b*co 
sh(f*x + e) + b*sinh(f*x + e))*sqrt((a^2 + b^2)/b^2) - b)/b) + (a*b^2*d*f* 
x + a*b^2*d*e - (a*b^2*d*f*x + a*b^2*d*e)*cosh(f*x + e)^2 - (a*b^2*d*f*x + 
 a*b^2*d*e)*sinh(f*x + e)^2 - 2*(a^2*b*d*f*x + a^2*b*d*e)*cosh(f*x + e) - 
2*(a^2*b*d*f*x + a^2*b*d*e + (a*b^2*d*f*x + a*b^2*d*e)*cosh(f*x + e))*sinh 
(f*x + e))*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(f*x + e) + a*sinh(f*x + e) - 
 (b*cosh(f*x + e) + b*sinh(f*x + e))*sqrt((a^2 + b^2)/b^2) - b)/b) - 2*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

(2*a*f*integrate(x*e^(f*x + e)/(a^2*b*f*e^(2*f*x + 2*e) + b^3*f*e^(2*f*x + 
 2*e) + 2*a^3*f*e^(f*x + e) + 2*a*b^2*f*e^(f*x + e) - a^2*b*f - b^3*f), x) 
 + b*(a*log((b*e^(f*x + e) + a - sqrt(a^2 + b^2))/(b*e^(f*x + e) + a + sqr 
t(a^2 + b^2)))/((a^2*b + b^3)*sqrt(a^2 + b^2)*f^2) - 2*(f*x + e)/((a^2*b + 
 b^3)*f^2) + log(b*e^(2*f*x + 2*e) + 2*a*e^(f*x + e) - b)/((a^2*b + b^3)*f 
^2)) - 2*(a*x*e^(f*x + e) - b*x)/(a^2*b*f + b^3*f - (a^2*b*f*e^(2*e) + b^3 
*f*e^(2*e))*e^(2*f*x) - 2*(a^3*f*e^e + a*b^2*f*e^e)*e^(f*x)) - a*log((b*e^ 
(f*x + e) + a - sqrt(a^2 + b^2))/(b*e^(f*x + e) + a + sqrt(a^2 + b^2)))/(( 
a^2 + b^2)^(3/2)*f^2))*d + c*(a*log((b*e^(-f*x - e) - a - sqrt(a^2 + b^2)) 
/(b*e^(-f*x - e) - a + sqrt(a^2 + b^2)))/((a^2 + b^2)^(3/2)*f) - 2*(a*e^(- 
f*x - e) + b)/((a^2*b + b^3 + 2*(a^3 + a*b^2)*e^(-f*x - e) - (a^2*b + b^3) 
*e^(-2*f*x - 2*e))*f))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {c+d x}{(a+b \sinh (e+f x))^2} \, dx=\text {too large to display} \] Input:

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

Output:

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