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

Optimal result

Integrand size = 18, antiderivative size = 274 \[ \int \frac {c+d x}{(a+b \cosh (e+f x))^2} \, dx=\frac {a (c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac {a (c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {d \log (a+b \cosh (e+f x))}{\left (a^2-b^2\right ) f^2}+\frac {a d \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}-\frac {a d \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}-\frac {b (c+d x) \sinh (e+f x)}{\left (a^2-b^2\right ) f (a+b \cosh (e+f x))} \] 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*cosh( 
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)*sinh(f*x+e)/(a^2-b^2)/f/(a+b*cosh(f*x+e))
 

Mathematica [A] (warning: unable to verify)

Time = 2.78 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.86 \[ \int \frac {c+d x}{(a+b \cosh (e+f x))^2} \, dx=-\frac {\frac {\left (a^2-b^2\right ) \left (-\sqrt {-\left (a^2-b^2\right )^2} d (e+f x)+2 a \sqrt {a^2-b^2} d \arctan \left (\frac {a+b e^{e+f x}}{\sqrt {-a^2+b^2}}\right )+2 a \sqrt {-a^2+b^2} d \text {arctanh}\left (\frac {a+b e^{e+f x}}{\sqrt {a^2-b^2}}\right )+2 a \sqrt {-a^2+b^2} d e \text {arctanh}\left (\frac {a+b e^{e+f x}}{\sqrt {a^2-b^2}}\right )-2 a \sqrt {-a^2+b^2} c f \text {arctanh}\left (\frac {a+b e^{e+f x}}{\sqrt {a^2-b^2}}\right )+a \sqrt {-a^2+b^2} d (e+f x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )-a \sqrt {-a^2+b^2} d (e+f x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )+\sqrt {-\left (a^2-b^2\right )^2} d \log \left (b+2 a e^{e+f x}+b e^{2 (e+f x)}\right )+a \sqrt {-a^2+b^2} d \operatorname {PolyLog}\left (2,\frac {b e^{e+f x}}{-a+\sqrt {a^2-b^2}}\right )-a \sqrt {-a^2+b^2} d \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )\right )}{\left (-\left (a^2-b^2\right )^2\right )^{3/2}}+\frac {b f (c+d x) \sinh (e+f x)}{(a-b) (a+b) (a+b \cosh (e+f x))}}{f^2} \] Input:

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

Output:

-((((a^2 - b^2)*(-(Sqrt[-(a^2 - b^2)^2]*d*(e + f*x)) + 2*a*Sqrt[a^2 - b^2] 
*d*ArcTan[(a + b*E^(e + f*x))/Sqrt[-a^2 + b^2]] + 2*a*Sqrt[-a^2 + b^2]*d*A 
rcTanh[(a + b*E^(e + f*x))/Sqrt[a^2 - b^2]] + 2*a*Sqrt[-a^2 + b^2]*d*e*Arc 
Tanh[(a + b*E^(e + f*x))/Sqrt[a^2 - b^2]] - 2*a*Sqrt[-a^2 + b^2]*c*f*ArcTa 
nh[(a + b*E^(e + f*x))/Sqrt[a^2 - b^2]] + a*Sqrt[-a^2 + b^2]*d*(e + f*x)*L 
og[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 - b^2])] - a*Sqrt[-a^2 + b^2]*d*(e + 
f*x)*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 - b^2])] + Sqrt[-(a^2 - b^2)^2] 
*d*Log[b + 2*a*E^(e + f*x) + b*E^(2*(e + f*x))] + a*Sqrt[-a^2 + b^2]*d*Pol 
yLog[2, (b*E^(e + f*x))/(-a + Sqrt[a^2 - b^2])] - a*Sqrt[-a^2 + b^2]*d*Pol 
yLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))]))/(-(a^2 - b^2)^2)^(3/2) 
 + (b*f*(c + d*x)*Sinh[e + f*x])/((a - b)*(a + b)*(a + b*Cosh[e + f*x])))/ 
f^2)
 

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {3042, 3805, 26, 3042, 26, 3147, 16, 3801, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3805

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3147

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 3801

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

(d*Log[a + b*Cosh[e + f*x]])/((a^2 - b^2)*f^2) + (2*a*((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)))/(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*PolyLog 
[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)*Sinh[e + f*x])/((a^2 - b^2)*f*(a + b*Cosh[ 
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 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 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 3801

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Comple 
x[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Simp[2   Int[((c + d*x)^m*(E^((-I)*e 
+ f*fz*x)/(b + (2*a*E^((-I)*e + f*fz*x))/E^(I*Pi*(k - 1/2)) - (b*E^(2*((-I) 
*e + f*fz*x)))/E^(2*I*k*Pi))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c 
, d, e, f, fz}, x] && IntegerQ[2*k] && 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. \(584\) vs. \(2(254)=508\).

Time = 0.78 (sec) , antiderivative size = 585, normalized size of antiderivative = 2.14

method	result	size
risch	\(\frac {2 \left (d x +c \right ) \left (a \,{\mathrm e}^{f x +e}+b \right )}{f \left (a^{2}-b^{2}\right ) \left ({\mathrm e}^{2 f x +2 e} b +2 a \,{\mathrm e}^{f x +e}+b \right )}+\frac {2 a c \arctan \left (\frac {2 b \,{\mathrm e}^{f x +e}+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{f \left (a^{2}-b^{2}\right ) \sqrt {-a^{2}+b^{2}}}+\frac {a d \ln \left (\frac {-b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) x}{f \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}-\frac {a d \ln \left (\frac {b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) x}{f \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}+\frac {a d \ln \left (\frac {-b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) e}{f^{2} \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}-\frac {a d \ln \left (\frac {b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) e}{f^{2} \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}+\frac {a d \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right )}{f^{2} \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}-\frac {a d \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right )}{f^{2} \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}+\frac {d \ln \left ({\mathrm e}^{2 f x +2 e} b +2 a \,{\mathrm e}^{f x +e}+b \right )}{f^{2} \left (a^{2}-b^{2}\right )}-\frac {2 d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2} \left (a^{2}-b^{2}\right )}-\frac {2 a d e \arctan \left (\frac {2 b \,{\mathrm e}^{f x +e}+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{f^{2} \left (a^{2}-b^{2}\right ) \sqrt {-a^{2}+b^{2}}}\)	\(585\)

Input:

int((d*x+c)/(a+b*cosh(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/(a^2-b^2)*a*c/(-a^2+b^2)^(1/2)*arctan(1/2*(2*b*exp(f*x+e)+2*a)/(-a^2+ 
b^2)^(1/2))+1/f/(a^2-b^2)^(3/2)*a*d*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)*a*d*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)*a*d*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)*a*d*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)*a*d*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)*a*d*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)*d*ln(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))-2/f^2/(a^2-b^2)*a*d*e/(-a^2+b^2)^(1/2)*arctan(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. 1765 vs. \(2 (252) = 504\).

Time = 0.21 (sec) , antiderivative size = 1765, normalized size of antiderivative = 6.44 \[ \int \frac {c+d x}{(a+b \cosh (e+f x))^2} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)/(a+b*cosh(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)*s 
inh(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* 
cosh(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))*si 
nh(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. \[ \int \frac {c+d x}{(a+b \cosh (e+f x))^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {c+d x}{(a+b \cosh (e+f x))^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f 
or more de
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*e**(2*e + 2*f*x)*sqrt( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - 
 a**2 + b**2))*a**3*b*d - 2*e**(2*e + 2*f*x)*sqrt( - a**2 + b**2)*atan((e* 
*(e + f*x)*b + a)/sqrt( - a**2 + b**2))*a*b**3*c*f - 2*e**(2*e + 2*f*x)*sq 
rt( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - a**2 + b**2))*a*b**3* 
d + 4*e**(e + f*x)*sqrt( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - 
a**2 + b**2))*a**4*d - 4*e**(e + f*x)*sqrt( - a**2 + b**2)*atan((e**(e + f 
*x)*b + a)/sqrt( - a**2 + b**2))*a**2*b**2*c*f - 4*e**(e + f*x)*sqrt( - a* 
*2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - a**2 + b**2))*a**2*b**2*d + 2 
*sqrt( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - a**2 + b**2))*a**3 
*b*d - 2*sqrt( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - a**2 + b** 
2))*a*b**3*c*f - 2*sqrt( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - 
a**2 + b**2))*a*b**3*d - 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**...