\(\int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [296]

Optimal result

Integrand size = 26, antiderivative size = 250 \[ \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a (e+f x)^2}{2 b^2 f}+\frac {(e+f x) \cosh (c+d x)}{b d}+\frac {\sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {\sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {\sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {f \sinh (c+d x)}{b d^2} \] Output:

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

Mathematica [A] (verified)

Time = 1.44 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.03 \[ \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a (c+d x) (c f-d (2 e+f x))+2 b d (e+f x) \cosh (c+d x)+2 \sqrt {a^2+b^2} \left (-2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )-2 b f \sinh (c+d x)}{2 b^2 d^2} \] Input:

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

Output:

(a*(c + d*x)*(c*f - d*(2*e + f*x)) + 2*b*d*(e + f*x)*Cosh[c + d*x] + 2*Sqr 
t[a^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]))]) - 2*b*f*Sinh[c + 
 d*x])/(2*b^2*d^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.15 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {6099, 17, 3042, 26, 3777, 3042, 3117, 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.

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

\(\Big \downarrow \) 6099

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 3803

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

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 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 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 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 6099

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_. 
)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-a/b^2   Int[(e + f*x)^m*Cos 
h[c + d*x]^(n - 2), x], x] + (Simp[1/b   Int[(e + f*x)^m*Cosh[c + d*x]^(n - 
 2)*Sinh[c + d*x], x], x] + Simp[(a^2 + b^2)/b^2   Int[(e + f*x)^m*(Cosh[c 
+ d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, 
 x] && IGtQ[n, 1] && 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. \(900\) vs. \(2(228)=456\).

Time = 2.13 (sec) , antiderivative size = 901, normalized size of antiderivative = 3.60

method	result	size
risch	\(-\frac {a f \,x^{2}}{2 b^{2}}-\frac {a e x}{b^{2}}+\frac {\left (d x f +d e -f \right ) {\mathrm e}^{d x +c}}{2 b \,d^{2}}+\frac {\left (d x f +d e +f \right ) {\mathrm e}^{-d x -c}}{2 b \,d^{2}}-\frac {2 a^{2} e \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,b^{2} \sqrt {a^{2}+b^{2}}}-\frac {2 e \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \sqrt {a^{2}+b^{2}}}+\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^{2} \sqrt {a^{2}+b^{2}}}-\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^{2} \sqrt {a^{2}+b^{2}}}+\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^{2} \sqrt {a^{2}+b^{2}}}-\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^{2} \sqrt {a^{2}+b^{2}}}+\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^{2} \sqrt {a^{2}+b^{2}}}-\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^{2} \sqrt {a^{2}+b^{2}}}+\frac {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 \sqrt {a^{2}+b^{2}}}-\frac {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 \sqrt {a^{2}+b^{2}}}+\frac {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} \sqrt {a^{2}+b^{2}}}-\frac {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} \sqrt {a^{2}+b^{2}}}+\frac {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} \sqrt {a^{2}+b^{2}}}-\frac {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} \sqrt {a^{2}+b^{2}}}+\frac {2 a^{2} f c \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{2} \sqrt {a^{2}+b^{2}}}+\frac {2 f c \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} \sqrt {a^{2}+b^{2}}}\)	\(901\)

Input:

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

Output:

-1/2*a/b^2*f*x^2-a/b^2*e*x+1/2*(d*f*x+d*e-f)/b/d^2*exp(d*x+c)+1/2*(d*f*x+d 
*e+f)/b/d^2*exp(-d*x-c)-2/d/b^2*a^2*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp 
(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/d*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d 
*x+c)+2*a)/(a^2+b^2)^(1/2))+1/d*a^2/b^2*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c 
)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/d/b^2*a^2*f/(a^2+b^2)^(1/2) 
*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^2*a^2* 
f/(a^2+b^2)^(1/2)*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^2*a^2*f/(a^2+b^2)^(1/2)*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^2*a^2*f/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c) 
+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/d^2/b^2*a^2*f/(a^2+b^2)^(1/2)* 
dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+1/d*f/(a^2+b^2 
)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/d*f 
/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))* 
x+1/d^2*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^ 
2)^(1/2)))*c-1/d^2*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/( 
a+(a^2+b^2)^(1/2)))*c+1/d^2*f/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c)+(a^2+b^ 
2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/d^2*f/(a^2+b^2)^(1/2)*dilog((b*exp(d*x 
+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+2/d^2/b^2*a^2*f*c/(a^2+b^2)^(1 
/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+2/d^2*f*c/(a^2+b^2)^ 
(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+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. 742 vs. \(2 (226) = 452\).

Time = 0.14 (sec) , antiderivative size = 742, normalized size of antiderivative = 2.97 \[ \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*(b*d*f*x + b*d*e + (b*d*f*x + b*d*e - b*f)*cosh(d*x + c)^2 + (b*d*f*x 
+ b*d*e - b*f)*sinh(d*x + c)^2 + 2*(b*f*cosh(d*x + c) + b*f*sinh(d*x + c)) 
*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) - 2*(b*f*cos 
h(d*x + c) + b*f*sinh(d*x + c))*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) - 2*((b*d*e - b*c*f)*cosh(d*x + c) + (b*d*e - b*c*f)*si 
nh(d*x + c))*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) + 2*((b*d*e - b*c*f)*cosh(d*x + c) + 
 (b*d*e - b*c*f)*sinh(d*x + c))*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) + 2*((b*d*f*x + b 
*c*f)*cosh(d*x + c) + (b*d*f*x + b*c*f)*sinh(d*x + c))*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) - 2*((b*d*f*x + b*c*f)*cosh(d*x + c) 
+ (b*d*f*x + b*c*f)*sinh(d*x + c))*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*f - (a*d^2*f*x^2 + 2*a*d^2*e*x)*cosh(d*x + c) - (a*d^ 
2*f*x^2 + 2*a*d^2*e*x - 2*(b*d*f*x + b*d*e - b*f)*cosh(d*x + c))*sinh(d*x 
+ c))/(b^2*d^2*cosh(d*x + c) + b^2*d^2*sinh(d*x + c))
                                                                                    
                                                                                    
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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