\(\int (c+d x) \text {csch}(a+b x) \, dx\) [3]

Optimal result

Integrand size = 12, antiderivative size = 50 \[ \int (c+d x) \text {csch}(a+b x) \, dx=-\frac {2 (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.80 \[ \int (c+d x) \text {csch}(a+b x) \, dx=-\frac {c \text {arctanh}(\cosh (a+b x))}{b}+2 d \left (\frac {x \log \left (1-e^{a+b x}\right )}{2 b}-\frac {x \log \left (1+e^{a+b x}\right )}{2 b}-\frac {\operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}+\frac {\operatorname {PolyLog}\left (2,e^{a+b x}\right )}{2 b^2}\right ) \] Input:

Integrate[(c + d*x)*Csch[a + b*x],x]
 

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3042, 26, 4670, 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.

\(\displaystyle \int (c+d x) \text {csch}(a+b x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int i (c+d x) \csc (i a+i b x)dx\)

\(\Big \downarrow \) 26

\(\displaystyle i \int (c+d x) \csc (i a+i b x)dx\)

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

\(\displaystyle i \left (\frac {2 i (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )\)

Input:

Int[(c + d*x)*Csch[a + b*x],x]
 

Output:

I*(((2*I)*(c + d*x)*ArcTanh[E^(a + b*x)])/b + (I*d*PolyLog[2, -E^(a + b*x) 
])/b^2 - (I*d*PolyLog[2, E^(a + b*x)])/b^2)
 

Defintions of rubi rules used

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 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 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.78

method	result	size
derivativedivides	\(\frac {\frac {d \left (\left (b x +a \right ) \ln \left (1-{\mathrm e}^{b x +a}\right )+\operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )-\left (b x +a \right ) \ln \left (1+{\mathrm e}^{b x +a}\right )-\operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )\right )}{b}+\frac {2 d a \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}-2 c \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}\)	\(89\)
default	\(\frac {\frac {d \left (\left (b x +a \right ) \ln \left (1-{\mathrm e}^{b x +a}\right )+\operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )-\left (b x +a \right ) \ln \left (1+{\mathrm e}^{b x +a}\right )-\operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )\right )}{b}+\frac {2 d a \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}-2 c \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}\)	\(89\)
parts	\(\frac {\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d x}{b}+\frac {\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) c}{b}+\frac {2 d \left (-\frac {\operatorname {dilog}\left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2}-\frac {\operatorname {dilog}\left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{2}-\frac {\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{2}\right )}{b^{2}}\)	\(90\)
risch	\(-\frac {2 c \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}-\frac {d \ln \left (1+{\mathrm e}^{b x +a}\right ) x}{b}-\frac {d \ln \left (1+{\mathrm e}^{b x +a}\right ) a}{b^{2}}-\frac {d \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {d \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {d \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {d \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 d a \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{2}}\)	\(124\)

Input:

int((d*x+c)*csch(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (45) = 90\).

Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.38 \[ \int (c+d x) \text {csch}(a+b x) \, dx=\frac {d {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - d {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - {\left (b d x + b c\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + {\left (b c - a d\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + {\left (b d x + a d\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right )}{b^{2}} \] Input:

integrate((d*x+c)*csch(b*x+a),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int (c+d x) \text {csch}(a+b x) \, dx=\int \left (c + d x\right ) \operatorname {csch}{\left (a + b x \right )}\, dx \] Input:

integrate((d*x+c)*csch(b*x+a),x)
 

Output:

Integral((c + d*x)*csch(a + b*x), x)
 

Maxima [F]

\[ \int (c+d x) \text {csch}(a+b x) \, dx=\int { {\left (d x + c\right )} \operatorname {csch}\left (b x + a\right ) \,d x } \] Input:

integrate((d*x+c)*csch(b*x+a),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int (c+d x) \text {csch}(a+b x) \, dx=\int { {\left (d x + c\right )} \operatorname {csch}\left (b x + a\right ) \,d x } \] Input:

integrate((d*x+c)*csch(b*x+a),x, algorithm="giac")
 

Output:

integrate((d*x + c)*csch(b*x + a), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c+d x) \text {csch}(a+b x) \, dx=\frac {\left (\int \mathrm {csch}\left (b x +a \right ) x d x \right ) b d +\mathrm {log}\left (e^{b x +a}-1\right ) c -\mathrm {log}\left (e^{b x +a}+1\right ) c}{b} \] Input:

int((d*x+c)*csch(b*x+a),x)
 

Output:

(int(csch(a + b*x)*x,x)*b*d + log(e**(a + b*x) - 1)*c - log(e**(a + b*x) + 
 1)*c)/b