\(\int \frac {x \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx\) [224]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [B] (verified)
Fricas [B] (verification not implemented)
Sympy [F]
Maxima [F]
Giac [F]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 20, antiderivative size = 161 \[ \int \frac {x \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=-\frac {x^2}{2 b}+\frac {x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d}+\frac {x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d}+\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^2} \] Output:

-1/2*x^2/b+x*ln(1+b*exp(d*x+c)/(a-(a^2-b^2)^(1/2)))/b/d+x*ln(1+b*exp(d*x+c 
)/(a+(a^2-b^2)^(1/2)))/b/d+polylog(2,-b*exp(d*x+c)/(a-(a^2-b^2)^(1/2)))/b/ 
d^2+polylog(2,-b*exp(d*x+c)/(a+(a^2-b^2)^(1/2)))/b/d^2
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.99 \[ \int \frac {x \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=-\frac {x^2}{2 b}+\frac {x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d}+\frac {x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d}+\frac {\operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6096, 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.

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

\(\Big \downarrow \) 6096

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 6096
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_ 
.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), 
 x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 - b^2, 2] + b*E^(c + d*x))) 
, x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x))) 
, x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0]
 
Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(367\) vs. \(2(147)=294\).

Time = 0.74 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.29

method result size
risch \(-\frac {x^{2}}{2 b}+\frac {\ln \left (\frac {-{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) x}{d b}+\frac {\ln \left (\frac {-{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) c}{d^{2} b}+\frac {\ln \left (\frac {{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) x}{d b}+\frac {\ln \left (\frac {{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) c}{d^{2} b}+\frac {\operatorname {dilog}\left (\frac {-{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right )}{d^{2} b}+\frac {\operatorname {dilog}\left (\frac {{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right )}{d^{2} b}-\frac {2 c x}{d b}-\frac {c^{2}}{d^{2} b}-\frac {c \ln \left ({\mathrm e}^{2 d x +2 c} b +2 \,{\mathrm e}^{d x +c} a +b \right )}{d^{2} b}+\frac {2 c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b}\) \(368\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (145) = 290\).

Time = 0.09 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.20 \[ \int \frac {x \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=-\frac {d^{2} x^{2} + 2 \, c \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) + 2 \, b \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + 2 \, a\right ) + 2 \, c \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) - 2 \, b \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + 2 \, a\right ) - 2 \, {\left (d x + c\right )} \log \left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b}\right ) - 2 \, {\left (d x + c\right )} \log \left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b}\right ) - 2 \, {\rm Li}_2\left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b} + 1\right ) - 2 \, {\rm Li}_2\left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b} + 1\right )}{2 \, b d^{2}} \] Input:

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

Output:

-1/2*(d^2*x^2 + 2*c*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*c*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*(d*x + c)*log((a*cosh(d*x + c) + a*sin 
h(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2) + b 
)/b) - 2*(d*x + c)*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*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*dilog(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*c 
osh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b + 1))/(b*d^2)
 

Sympy [F]

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

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

Output:

Integral(x*sinh(c + d*x)/(a + b*cosh(c + d*x)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int((x*sinh(c + d*x))/(a + b*cosh(c + d*x)),x)
 

Output:

int((x*sinh(c + d*x))/(a + b*cosh(c + d*x)), x)
 

Reduce [F]

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

int(x*sinh(d*x+c)/(a+b*cosh(d*x+c)),x)
                                                                                    
                                                                                    
 

Output:

e**(2*c)*int((e**(2*d*x)*x)/(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a + b),x) 
 - int(x/(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a + b),x)