\(\int (c+d x) \text {csch}^3(a+b x) \, dx\) [35]

Optimal result

Integrand size = 14, antiderivative size = 92 \[ \int (c+d x) \text {csch}^3(a+b x) \, dx=\frac {(c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {d \text {csch}(a+b x)}{2 b^2}-\frac {(c+d x) \coth (a+b x) \text {csch}(a+b x)}{2 b}+\frac {d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}-\frac {d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{2 b^2} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(271\) vs. \(2(92)=184\).

Time = 0.20 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.95 \[ \int (c+d x) \text {csch}^3(a+b x) \, dx=-\frac {d x \text {csch}^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b}-\frac {c \text {csch}^2\left (\frac {1}{2} (a+b x)\right )}{8 b}-\frac {d x \log \left (1-e^{a+b x}\right )}{2 b}+\frac {d x \log \left (1+e^{a+b x}\right )}{2 b}+\frac {c \log \left (\cosh \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}-\frac {c \log \left (\sinh \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}+\frac {d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}-\frac {d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{2 b^2}-\frac {d x \text {sech}^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b}-\frac {c \text {sech}^2\left (\frac {1}{2} (a+b x)\right )}{8 b}+\frac {d \text {csch}\left (\frac {a}{2}\right ) \text {csch}\left (\frac {a}{2}+\frac {b x}{2}\right ) \sinh \left (\frac {b x}{2}\right )}{4 b^2}+\frac {d \text {sech}\left (\frac {a}{2}\right ) \text {sech}\left (\frac {a}{2}+\frac {b x}{2}\right ) \sinh \left (\frac {b x}{2}\right )}{4 b^2} \] Input:

Integrate[(c + d*x)*Csch[a + b*x]^3,x]
 

Output:

-1/8*(d*x*Csch[a/2 + (b*x)/2]^2)/b - (c*Csch[(a + b*x)/2]^2)/(8*b) - (d*x* 
Log[1 - E^(a + b*x)])/(2*b) + (d*x*Log[1 + E^(a + b*x)])/(2*b) + (c*Log[Co 
sh[(a + b*x)/2]])/(2*b) - (c*Log[Sinh[(a + b*x)/2]])/(2*b) + (d*PolyLog[2, 
 -E^(a + b*x)])/(2*b^2) - (d*PolyLog[2, E^(a + b*x)])/(2*b^2) - (d*x*Sech[ 
a/2 + (b*x)/2]^2)/(8*b) - (c*Sech[(a + b*x)/2]^2)/(8*b) + (d*Csch[a/2]*Csc 
h[a/2 + (b*x)/2]*Sinh[(b*x)/2])/(4*b^2) + (d*Sech[a/2]*Sech[a/2 + (b*x)/2] 
*Sinh[(b*x)/2])/(4*b^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.44 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.17, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {3042, 26, 4673, 26, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int -i (c+d x) \csc (i a+i b x)^3dx\)

\(\Big \downarrow \) 26

\(\displaystyle -i \int (c+d x) \csc (i a+i b x)^3dx\)

\(\Big \downarrow \) 4673

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

\(\displaystyle -i \left (-\frac {1}{2} i \int i (c+d x) \csc (i a+i b x)dx-\frac {i d \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x) \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -i \left (\frac {1}{2} \int (c+d x) \csc (i a+i b x)dx-\frac {i d \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x) \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\)

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

\(\displaystyle -i \left (\frac {1}{2} \left (\frac {2 i (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )-\frac {i d \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x) \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\)

Input:

Int[(c + d*x)*Csch[a + b*x]^3,x]
 

Output:

(-I)*(((-1/2*I)*d*Csch[a + b*x])/b^2 - ((I/2)*(c + d*x)*Coth[a + b*x]*Csch 
[a + b*x])/b + (((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)/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]
 

rule 4673

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> 
 Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), 
 x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S 
imp[b^2*((n - 2)/(n - 1))   Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) 
 /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(196\) vs. \(2(81)=162\).

Time = 0.12 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.14

method	result	size
risch	\(-\frac {{\mathrm e}^{b x +a} \left (b d x \,{\mathrm e}^{2 b x +2 a}+b c \,{\mathrm e}^{2 b x +2 a}+d x b +d \,{\mathrm e}^{2 b x +2 a}+b c -d \right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}+\frac {c \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}-\frac {d \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{2 b}-\frac {d \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{2 b^{2}}-\frac {d \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {d \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{2 b}+\frac {d \ln \left ({\mathrm e}^{b x +a}+1\right ) a}{2 b^{2}}+\frac {d \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {d a \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{2}}\)	\(197\)

Input:

int((d*x+c)*csch(b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1026 vs. \(2 (79) = 158\).

Time = 0.12 (sec) , antiderivative size = 1026, normalized size of antiderivative = 11.15 \[ \int (c+d x) \text {csch}^3(a+b x) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)*csch(b*x+a)^3,x, algorithm="fricas")
 

Output:

-1/2*(2*(b*d*x + b*c + d)*cosh(b*x + a)^3 + 6*(b*d*x + b*c + d)*cosh(b*x + 
 a)*sinh(b*x + a)^2 + 2*(b*d*x + b*c + d)*sinh(b*x + a)^3 + 2*(b*d*x + b*c 
 - d)*cosh(b*x + a) + (d*cosh(b*x + a)^4 + 4*d*cosh(b*x + a)*sinh(b*x + a) 
^3 + d*sinh(b*x + a)^4 - 2*d*cosh(b*x + a)^2 + 2*(3*d*cosh(b*x + a)^2 - d) 
*sinh(b*x + a)^2 + 4*(d*cosh(b*x + a)^3 - d*cosh(b*x + a))*sinh(b*x + a) + 
 d)*dilog(cosh(b*x + a) + sinh(b*x + a)) - (d*cosh(b*x + a)^4 + 4*d*cosh(b 
*x + a)*sinh(b*x + a)^3 + d*sinh(b*x + a)^4 - 2*d*cosh(b*x + a)^2 + 2*(3*d 
*cosh(b*x + a)^2 - d)*sinh(b*x + a)^2 + 4*(d*cosh(b*x + a)^3 - d*cosh(b*x 
+ a))*sinh(b*x + a) + d)*dilog(-cosh(b*x + a) - sinh(b*x + a)) - ((b*d*x + 
 b*c)*cosh(b*x + a)^4 + 4*(b*d*x + b*c)*cosh(b*x + a)*sinh(b*x + a)^3 + (b 
*d*x + b*c)*sinh(b*x + a)^4 + b*d*x - 2*(b*d*x + b*c)*cosh(b*x + a)^2 - 2* 
(b*d*x - 3*(b*d*x + b*c)*cosh(b*x + a)^2 + b*c)*sinh(b*x + a)^2 + b*c + 4* 
((b*d*x + b*c)*cosh(b*x + a)^3 - (b*d*x + b*c)*cosh(b*x + a))*sinh(b*x + a 
))*log(cosh(b*x + a) + sinh(b*x + a) + 1) + ((b*c - a*d)*cosh(b*x + a)^4 + 
 4*(b*c - a*d)*cosh(b*x + a)*sinh(b*x + a)^3 + (b*c - a*d)*sinh(b*x + a)^4 
 - 2*(b*c - a*d)*cosh(b*x + a)^2 + 2*(3*(b*c - a*d)*cosh(b*x + a)^2 - b*c 
+ a*d)*sinh(b*x + a)^2 + b*c - a*d + 4*((b*c - a*d)*cosh(b*x + a)^3 - (b*c 
 - a*d)*cosh(b*x + a))*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) - 
1) + ((b*d*x + a*d)*cosh(b*x + a)^4 + 4*(b*d*x + a*d)*cosh(b*x + a)*sinh(b 
*x + a)^3 + (b*d*x + a*d)*sinh(b*x + a)^4 + b*d*x - 2*(b*d*x + a*d)*cos...
 

Sympy [F]

\[ \int (c+d x) \text {csch}^3(a+b x) \, dx=\int \left (c + d x\right ) \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \] Input:

integrate((d*x+c)*csch(b*x+a)**3,x)
 

Output:

Integral((c + d*x)*csch(a + b*x)**3, x)
 

Maxima [F]

\[ \int (c+d x) \text {csch}^3(a+b x) \, dx=\int { {\left (d x + c\right )} \operatorname {csch}\left (b x + a\right )^{3} \,d x } \] Input:

integrate((d*x+c)*csch(b*x+a)^3,x, algorithm="maxima")
 

Output:

-d*(((b*x*e^(3*a) + e^(3*a))*e^(3*b*x) + (b*x*e^a - e^a)*e^(b*x))/(b^2*e^( 
4*b*x + 4*a) - 2*b^2*e^(2*b*x + 2*a) + b^2) + 8*integrate(1/16*x/(e^(b*x + 
 a) + 1), x) + 8*integrate(1/16*x/(e^(b*x + a) - 1), x)) + 1/2*c*(log(e^(- 
b*x - a) + 1)/b - log(e^(-b*x - a) - 1)/b + 2*(e^(-b*x - a) + e^(-3*b*x - 
3*a))/(b*(2*e^(-2*b*x - 2*a) - e^(-4*b*x - 4*a) - 1)))
 

Giac [F]

\[ \int (c+d x) \text {csch}^3(a+b x) \, dx=\int { {\left (d x + c\right )} \operatorname {csch}\left (b x + a\right )^{3} \,d x } \] Input:

integrate((d*x+c)*csch(b*x+a)^3,x, algorithm="giac")
 

Output:

integrate((d*x + c)*csch(b*x + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x) \text {csch}^3(a+b x) \, dx=\int \frac {c+d\,x}{{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \] Input:

int((c + d*x)/sinh(a + b*x)^3,x)
 

Output:

int((c + d*x)/sinh(a + b*x)^3, x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int (c+d x) \text {csch}^3(a+b x) \, dx=\frac {-16 e^{4 b x +5 a} \left (\int \frac {e^{b x} x}{e^{6 b x +6 a}-3 e^{4 b x +4 a}+3 e^{2 b x +2 a}-1}d x \right ) b^{2} d -3 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}-1\right ) b c -4 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}-1\right ) d +3 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}+1\right ) b c +4 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}+1\right ) d -6 e^{3 b x +3 a} b c -8 e^{3 b x +3 a} d +32 e^{2 b x +3 a} \left (\int \frac {e^{b x} x}{e^{6 b x +6 a}-3 e^{4 b x +4 a}+3 e^{2 b x +2 a}-1}d x \right ) b^{2} d +6 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}-1\right ) b c +8 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}-1\right ) d -6 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}+1\right ) b c -8 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}+1\right ) d -6 e^{b x +a} b c -16 e^{b x +a} b d x +8 e^{b x +a} d -16 e^{a} \left (\int \frac {e^{b x} x}{e^{6 b x +6 a}-3 e^{4 b x +4 a}+3 e^{2 b x +2 a}-1}d x \right ) b^{2} d -3 \,\mathrm {log}\left (e^{b x +a}-1\right ) b c -4 \,\mathrm {log}\left (e^{b x +a}-1\right ) d +3 \,\mathrm {log}\left (e^{b x +a}+1\right ) b c +4 \,\mathrm {log}\left (e^{b x +a}+1\right ) d}{6 b^{2} \left (e^{4 b x +4 a}-2 e^{2 b x +2 a}+1\right )} \] Input:

int((d*x+c)*csch(b*x+a)^3,x)
 

Output:

( - 16*e**(5*a + 4*b*x)*int((e**(b*x)*x)/(e**(6*a + 6*b*x) - 3*e**(4*a + 4 
*b*x) + 3*e**(2*a + 2*b*x) - 1),x)*b**2*d - 3*e**(4*a + 4*b*x)*log(e**(a + 
 b*x) - 1)*b*c - 4*e**(4*a + 4*b*x)*log(e**(a + b*x) - 1)*d + 3*e**(4*a + 
4*b*x)*log(e**(a + b*x) + 1)*b*c + 4*e**(4*a + 4*b*x)*log(e**(a + b*x) + 1 
)*d - 6*e**(3*a + 3*b*x)*b*c - 8*e**(3*a + 3*b*x)*d + 32*e**(3*a + 2*b*x)* 
int((e**(b*x)*x)/(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b* 
x) - 1),x)*b**2*d + 6*e**(2*a + 2*b*x)*log(e**(a + b*x) - 1)*b*c + 8*e**(2 
*a + 2*b*x)*log(e**(a + b*x) - 1)*d - 6*e**(2*a + 2*b*x)*log(e**(a + b*x) 
+ 1)*b*c - 8*e**(2*a + 2*b*x)*log(e**(a + b*x) + 1)*d - 6*e**(a + b*x)*b*c 
 - 16*e**(a + b*x)*b*d*x + 8*e**(a + b*x)*d - 16*e**a*int((e**(b*x)*x)/(e* 
*(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) - 1),x)*b**2*d - 
3*log(e**(a + b*x) - 1)*b*c - 4*log(e**(a + b*x) - 1)*d + 3*log(e**(a + b* 
x) + 1)*b*c + 4*log(e**(a + b*x) + 1)*d)/(6*b**2*(e**(4*a + 4*b*x) - 2*e** 
(2*a + 2*b*x) + 1))