\(\int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [311]

Optimal result

Integrand size = 26, antiderivative size = 295 \[ \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b f \arctan (\sinh (c+d x))}{\left (a^2+b^2\right ) d^2}+\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a f \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {b (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}+\frac {a (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.28 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.13 \[ \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {2 f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{i a-b}+\frac {2 i f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a-i b}-\frac {f \log (\cosh (c+d x))}{a-i b}-\frac {f \log (\cosh (c+d x))}{a+i b}+\frac {2 b^2 \left (-2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {2 d (e+f x) \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^2+b^2}}{2 d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.58 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.93, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {6107, 3042, 3803, 25, 2694, 27, 2620, 2715, 2838, 7293, 2009}

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.

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3803

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7293

\(\displaystyle \frac {\int \left (a (e+f x) \text {sech}^2(c+d x)-b (e+f x) \text {sech}(c+d x) \tanh (c+d x)\right )dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 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 6107

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ 
.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b^2/(a^2 + b^2)   Int[(e + 
f*x)^m*(Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x] + Simp[1/(a^2 
+ b^2)   Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; F 
reeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0 
]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1927\) vs. \(2(275)=550\).

Time = 12.43 (sec) , antiderivative size = 1928, normalized size of antiderivative = 6.54

method	result	size
risch	\(\text {Expression too large to display}\)	\(1928\)

Input:

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

Output:

-1/(a^2+b^2)^2/d^2*a^3*f*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+2/(a^2+b^2) 
/d^2*a*f*ln(exp(d*x+c))+2/(a^2+b^2)^(5/2)/d^2*a^4*f*arctanh(1/2*(2*b*exp(d 
*x+c)+2*a)/(a^2+b^2)^(1/2))-4/(a^2+b^2)/d^2*b^3*f/(2*a^2+2*b^2)*arctan(exp 
(d*x+c))-2*(f*x+e)*(-b*exp(d*x+c)+a)/d/(a^2+b^2)/(1+exp(2*d*x+2*c))-2/(a^2 
+b^2)/d^2*a^3*f/(2*a^2+2*b^2)*ln(1+exp(2*d*x+2*c))+2/(a^2+b^2)/d^2*a^3*f/( 
2*a^2+2*b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-1/2/(a^2+b^2)^2/d^2*a*b 
^2*f*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+2/(a^2+b^2)^(3/2)/d^2*b^2*f/(2* 
a^2+2*b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*a^2* 
c-2/(a^2+b^2)^(3/2)/d^2*b^2*f/(2*a^2+2*b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/ 
2)+a)/(a+(a^2+b^2)^(1/2)))*a^2*c+2/(a^2+b^2)^(3/2)/d*b^2*f/(2*a^2+2*b^2)*l 
n((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*a^2*x-2/(a^2+b^2 
)^(3/2)/d*b^2*f/(2*a^2+2*b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+ 
b^2)^(1/2)))*a^2*x+2/(a^2+b^2)^(3/2)/d^2*c*b^2*f/(2*a^2+2*b^2)*arctanh(1/2 
*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^2+2/(a^2+b^2)^(3/2)/d^2*c*b^4*f/( 
2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+2/(a^2+b^2) 
^(3/2)/d^2*b^4*f/(2*a^2+2*b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a 
^2+b^2)^(1/2)))*c-2/(a^2+b^2)^(3/2)/d^2*b^4*f/(2*a^2+2*b^2)*ln((b*exp(d*x+ 
c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+2/(a^2+b^2)^(3/2)/d^2*b^2*f/( 
2*a^2+2*b^2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2))) 
*a^2-2/(a^2+b^2)^(3/2)/d^2*b^2*f/(2*a^2+2*b^2)*dilog((b*exp(d*x+c)+(a^2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1296 vs. \(2 (273) = 546\).

Time = 0.13 (sec) , antiderivative size = 1296, normalized size of antiderivative = 4.39 \[ \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

(2*(a^3 + a*b^2)*d*f*x*cosh(d*x + c)^2 + 2*(a^3 + a*b^2)*d*f*x*sinh(d*x + 
c)^2 - 2*(a^3 + a*b^2)*d*e + (b^3*f*cosh(d*x + c)^2 + 2*b^3*f*cosh(d*x + c 
)*sinh(d*x + c) + b^3*f*sinh(d*x + c)^2 + b^3*f)*sqrt((a^2 + b^2)/b^2)*dil 
og((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) - (b^3*f*cosh(d*x + c)^2 + 2*b^3*f*cos 
h(d*x + c)*sinh(d*x + c) + b^3*f*sinh(d*x + c)^2 + b^3*f)*sqrt((a^2 + b^2) 
/b^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) - (b^3*d*e - b^3*c*f + (b^3*d 
*e - b^3*c*f)*cosh(d*x + c)^2 + 2*(b^3*d*e - b^3*c*f)*cosh(d*x + c)*sinh(d 
*x + c) + (b^3*d*e - b^3*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^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) + 
(b^3*d*e - b^3*c*f + (b^3*d*e - b^3*c*f)*cosh(d*x + c)^2 + 2*(b^3*d*e - b^ 
3*c*f)*cosh(d*x + c)*sinh(d*x + c) + (b^3*d*e - b^3*c*f)*sinh(d*x + c)^2)* 
sqrt((a^2 + b^2)/b^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) + (b^3*d*f*x + b^3*c*f + (b^3*d*f*x + b^3*c*f)*co 
sh(d*x + c)^2 + 2*(b^3*d*f*x + b^3*c*f)*cosh(d*x + c)*sinh(d*x + c) + (b^3 
*d*f*x + b^3*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*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) - (b^3*d*f*x + b^3*c*f + (b^3*d*f*x + b^3*c*f)*cosh(d*x + 
 c)^2 + 2*(b^3*d*f*x + b^3*c*f)*cosh(d*x + c)*sinh(d*x + c) + (b^3*d*f*...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 e^{2 d x +2 c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b^{2} e i +2 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b^{2} e i +8 e^{2 d x +5 c} \left (\int \frac {e^{3 d x} x}{e^{6 d x +6 c} b +2 e^{5 d x +5 c} a +e^{4 d x +4 c} b +4 e^{3 d x +3 c} a -e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a^{4} d f +16 e^{2 d x +5 c} \left (\int \frac {e^{3 d x} x}{e^{6 d x +6 c} b +2 e^{5 d x +5 c} a +e^{4 d x +4 c} b +4 e^{3 d x +3 c} a -e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a^{2} b^{2} d f +8 e^{2 d x +5 c} \left (\int \frac {e^{3 d x} x}{e^{6 d x +6 c} b +2 e^{5 d x +5 c} a +e^{4 d x +4 c} b +4 e^{3 d x +3 c} a -e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) b^{4} d f +2 e^{2 d x +2 c} a^{3} e +2 e^{2 d x +2 c} a \,b^{2} e +2 e^{d x +c} a^{2} b e +2 e^{d x +c} b^{3} e +8 e^{3 c} \left (\int \frac {e^{3 d x} x}{e^{6 d x +6 c} b +2 e^{5 d x +5 c} a +e^{4 d x +4 c} b +4 e^{3 d x +3 c} a -e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a^{4} d f +16 e^{3 c} \left (\int \frac {e^{3 d x} x}{e^{6 d x +6 c} b +2 e^{5 d x +5 c} a +e^{4 d x +4 c} b +4 e^{3 d x +3 c} a -e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a^{2} b^{2} d f +8 e^{3 c} \left (\int \frac {e^{3 d x} x}{e^{6 d x +6 c} b +2 e^{5 d x +5 c} a +e^{4 d x +4 c} b +4 e^{3 d x +3 c} a -e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) b^{4} d f}{d \left (e^{2 d x +2 c} a^{4}+2 e^{2 d x +2 c} a^{2} b^{2}+e^{2 d x +2 c} b^{4}+a^{4}+2 a^{2} b^{2}+b^{4}\right )} \] Input:

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

Output:

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