3.1.9 \(\int (c+d x)^3 \text {csch}^3(a+b x) \, dx\) [9]

3.1.9.1 Optimal result

Integrand size = 16, antiderivative size = 256 \[ \int (c+d x)^3 \text {csch}^3(a+b x) \, dx=-\frac {6 d^2 (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b^3}+\frac {(c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {(c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {3 d^3 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^4}+\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}+\frac {3 d^3 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^4}-\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}+\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3}+\frac {3 d^3 \operatorname {PolyLog}\left (4,-e^{a+b x}\right )}{b^4}-\frac {3 d^3 \operatorname {PolyLog}\left (4,e^{a+b x}\right )}{b^4} \]

output

-6*d^2*(d*x+c)*arctanh(exp(b*x+a))/b^3+(d*x+c)^3*arctanh(exp(b*x+a))/b-3/2 
*d*(d*x+c)^2*csch(b*x+a)/b^2-1/2*(d*x+c)^3*coth(b*x+a)*csch(b*x+a)/b-3*d^3 
*polylog(2,-exp(b*x+a))/b^4+3/2*d*(d*x+c)^2*polylog(2,-exp(b*x+a))/b^2+3*d 
^3*polylog(2,exp(b*x+a))/b^4-3/2*d*(d*x+c)^2*polylog(2,exp(b*x+a))/b^2-3*d 
^2*(d*x+c)*polylog(3,-exp(b*x+a))/b^3+3*d^2*(d*x+c)*polylog(3,exp(b*x+a))/ 
b^3+3*d^3*polylog(4,-exp(b*x+a))/b^4-3*d^3*polylog(4,exp(b*x+a))/b^4
 

3.1.9.2 Mathematica [A] (verified)

Time = 2.10 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.72 \[ \int (c+d x)^3 \text {csch}^3(a+b x) \, dx=-\frac {b^2 (c+d x)^2 (3 d+b (c+d x) \coth (a+b x)) \text {csch}(a+b x)+b^3 c^3 \log \left (1-e^{a+b x}\right )-6 b c d^2 \log \left (1-e^{a+b x}\right )+3 b^3 c^2 d x \log \left (1-e^{a+b x}\right )-6 b d^3 x \log \left (1-e^{a+b x}\right )+3 b^3 c d^2 x^2 \log \left (1-e^{a+b x}\right )+b^3 d^3 x^3 \log \left (1-e^{a+b x}\right )-b^3 c^3 \log \left (1+e^{a+b x}\right )+6 b c d^2 \log \left (1+e^{a+b x}\right )-3 b^3 c^2 d x \log \left (1+e^{a+b x}\right )+6 b d^3 x \log \left (1+e^{a+b x}\right )-3 b^3 c d^2 x^2 \log \left (1+e^{a+b x}\right )-b^3 d^3 x^3 \log \left (1+e^{a+b x}\right )-3 d \left (-2 d^2+b^2 (c+d x)^2\right ) \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+3 d \left (-2 d^2+b^2 (c+d x)^2\right ) \operatorname {PolyLog}\left (2,e^{a+b x}\right )+6 b c d^2 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )+6 b d^3 x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )-6 b c d^2 \operatorname {PolyLog}\left (3,e^{a+b x}\right )-6 b d^3 x \operatorname {PolyLog}\left (3,e^{a+b x}\right )-6 d^3 \operatorname {PolyLog}\left (4,-e^{a+b x}\right )+6 d^3 \operatorname {PolyLog}\left (4,e^{a+b x}\right )}{2 b^4} \]

input

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

output

-1/2*(b^2*(c + d*x)^2*(3*d + b*(c + d*x)*Coth[a + b*x])*Csch[a + b*x] + b^ 
3*c^3*Log[1 - E^(a + b*x)] - 6*b*c*d^2*Log[1 - E^(a + b*x)] + 3*b^3*c^2*d* 
x*Log[1 - E^(a + b*x)] - 6*b*d^3*x*Log[1 - E^(a + b*x)] + 3*b^3*c*d^2*x^2* 
Log[1 - E^(a + b*x)] + b^3*d^3*x^3*Log[1 - E^(a + b*x)] - b^3*c^3*Log[1 + 
E^(a + b*x)] + 6*b*c*d^2*Log[1 + E^(a + b*x)] - 3*b^3*c^2*d*x*Log[1 + E^(a 
 + b*x)] + 6*b*d^3*x*Log[1 + E^(a + b*x)] - 3*b^3*c*d^2*x^2*Log[1 + E^(a + 
 b*x)] - b^3*d^3*x^3*Log[1 + E^(a + b*x)] - 3*d*(-2*d^2 + b^2*(c + d*x)^2) 
*PolyLog[2, -E^(a + b*x)] + 3*d*(-2*d^2 + b^2*(c + d*x)^2)*PolyLog[2, E^(a 
 + b*x)] + 6*b*c*d^2*PolyLog[3, -E^(a + b*x)] + 6*b*d^3*x*PolyLog[3, -E^(a 
 + b*x)] - 6*b*c*d^2*PolyLog[3, E^(a + b*x)] - 6*b*d^3*x*PolyLog[3, E^(a + 
 b*x)] - 6*d^3*PolyLog[4, -E^(a + b*x)] + 6*d^3*PolyLog[4, E^(a + b*x)])/b 
^4
 

3.1.9.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.20 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.15, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {3042, 26, 4674, 26, 3042, 26, 4670, 2715, 2838, 3011, 7163, 2720, 7143}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4674

\(\displaystyle -i \left (-\frac {3 d^2 \int -i (c+d x) \text {csch}(a+b x)dx}{b^2}+\frac {1}{2} \int -i (c+d x)^3 \text {csch}(a+b x)dx-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -i \left (\frac {3 i d^2 \int (c+d x) \text {csch}(a+b x)dx}{b^2}-\frac {1}{2} i \int (c+d x)^3 \text {csch}(a+b x)dx-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -i \left (\frac {3 i d^2 \int i (c+d x) \csc (i a+i b x)dx}{b^2}-\frac {1}{2} i \int i (c+d x)^3 \csc (i a+i b x)dx-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -i \left (-\frac {3 d^2 \int (c+d x) \csc (i a+i b x)dx}{b^2}+\frac {1}{2} \int (c+d x)^3 \csc (i a+i b x)dx-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\)

\(\Big \downarrow \) 4670

\(\displaystyle -i \left (-\frac {3 d^2 \left (\frac {i d \int \log \left (1-e^{a+b x}\right )dx}{b}-\frac {i d \int \log \left (1+e^{a+b x}\right )dx}{b}+\frac {2 i (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}\right )}{b^2}+\frac {1}{2} \left (\frac {3 i d \int (c+d x)^2 \log \left (1-e^{a+b x}\right )dx}{b}-\frac {3 i d \int (c+d x)^2 \log \left (1+e^{a+b x}\right )dx}{b}+\frac {2 i (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle -i \left (-\frac {3 d^2 \left (\frac {i d \int e^{-a-b x} \log \left (1-e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {i d \int e^{-a-b x} \log \left (1+e^{a+b x}\right )de^{a+b x}}{b^2}+\frac {2 i (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}\right )}{b^2}+\frac {1}{2} \left (\frac {3 i d \int (c+d x)^2 \log \left (1-e^{a+b x}\right )dx}{b}-\frac {3 i d \int (c+d x)^2 \log \left (1+e^{a+b x}\right )dx}{b}+\frac {2 i (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle -i \left (\frac {1}{2} \left (\frac {3 i d \int (c+d x)^2 \log \left (1-e^{a+b x}\right )dx}{b}-\frac {3 i d \int (c+d x)^2 \log \left (1+e^{a+b x}\right )dx}{b}+\frac {2 i (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 d^2 \left (\frac {2 i (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle -i \left (\frac {1}{2} \left (-\frac {3 i d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{a+b x}\right )dx}{b}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,e^{a+b x}\right )dx}{b}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 d^2 \left (\frac {2 i (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle -i \left (\frac {1}{2} \left (-\frac {3 i d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}-\frac {d \int \operatorname {PolyLog}\left (3,-e^{a+b x}\right )dx}{b}\right )}{b}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}-\frac {d \int \operatorname {PolyLog}\left (3,e^{a+b x}\right )dx}{b}\right )}{b}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 d^2 \left (\frac {2 i (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle -i \left (\frac {1}{2} \left (-\frac {3 i d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}-\frac {d \int e^{-a-b x} \operatorname {PolyLog}\left (3,-e^{a+b x}\right )de^{a+b x}}{b^2}\right )}{b}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}-\frac {d \int e^{-a-b x} \operatorname {PolyLog}\left (3,e^{a+b x}\right )de^{a+b x}}{b^2}\right )}{b}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 d^2 \left (\frac {2 i (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle -i \left (-\frac {3 d^2 \left (\frac {2 i (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )}{b^2}+\frac {1}{2} \left (\frac {2 i (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {3 i d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}-\frac {d \operatorname {PolyLog}\left (4,-e^{a+b x}\right )}{b^2}\right )}{b}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}-\frac {d \operatorname {PolyLog}\left (4,e^{a+b x}\right )}{b^2}\right )}{b}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}\right )-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\)

input

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

output

(-I)*((((-3*I)/2)*d*(c + d*x)^2*Csch[a + b*x])/b^2 - ((I/2)*(c + d*x)^3*Co 
th[a + b*x]*Csch[a + b*x])/b - (3*d^2*(((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))/b^2 + (((2*I)*(c + d*x)^3*ArcTanh[E^(a + b*x)])/b - ((3*I)*d*(-(((c 
 + d*x)^2*PolyLog[2, -E^(a + b*x)])/b) + (2*d*(((c + d*x)*PolyLog[3, -E^(a 
 + b*x)])/b - (d*PolyLog[4, -E^(a + b*x)])/b^2))/b))/b + ((3*I)*d*(-(((c + 
 d*x)^2*PolyLog[2, E^(a + b*x)])/b) + (2*d*(((c + d*x)*PolyLog[3, E^(a + b 
*x)])/b - (d*PolyLog[4, E^(a + b*x)])/b^2))/b))/b)/2)
 

3.1.9.3.1 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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

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 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

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 4674

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

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7163

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. 
)*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a 
+ b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F]))   Int[(e + f*x) 
^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c 
, d, e, f, n, p}, x] && GtQ[m, 0]
 

3.1.9.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(875\) vs. \(2(238)=476\).

Time = 1.01 (sec) , antiderivative size = 876, normalized size of antiderivative = 3.42

method	result	size
risch	\(\text {Expression too large to display}\)	\(876\)

input

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

output

1/b*c^3*arctanh(exp(b*x+a))+6/b^4*d^3*a*arctanh(exp(b*x+a))+3/b^3*d^3*ln(1 
-exp(b*x+a))*x+3/b^4*d^3*ln(1-exp(b*x+a))*a-3/b^3*d^3*ln(exp(b*x+a)+1)*x-3 
/b^4*d^3*ln(exp(b*x+a)+1)*a-6/b^3*d^2*c*arctanh(exp(b*x+a))+3/2/b^2*d^3*po 
lylog(2,-exp(b*x+a))*x^2-3/b^3*d^3*polylog(3,-exp(b*x+a))*x-1/2/b*d^3*ln(1 
-exp(b*x+a))*x^3-1/2/b^4*d^3*ln(1-exp(b*x+a))*a^3-3/2/b^2*d^3*polylog(2,ex 
p(b*x+a))*x^2+3/b^3*d^3*polylog(3,exp(b*x+a))*x+1/2/b*d^3*ln(exp(b*x+a)+1) 
*x^3+1/2/b^4*d^3*ln(exp(b*x+a)+1)*a^3-1/b^4*d^3*a^3*arctanh(exp(b*x+a))-3/ 
b^3*c*d^2*polylog(3,-exp(b*x+a))-3/2/b^2*d*c^2*polylog(2,exp(b*x+a))+3/2/b 
^2*d*c^2*polylog(2,-exp(b*x+a))+3/b^3*c*d^2*polylog(3,exp(b*x+a))-3/b^2*d* 
a*c^2*arctanh(exp(b*x+a))+3/2/b*d*c^2*ln(exp(b*x+a)+1)*x+3/2/b^2*d*c^2*ln( 
exp(b*x+a)+1)*a+3/b^3*d^2*a^2*c*arctanh(exp(b*x+a))-3/2/b*c*d^2*ln(1-exp(b 
*x+a))*x^2+3/2/b^3*c*d^2*ln(1-exp(b*x+a))*a^2-3/b^2*c*d^2*polylog(2,exp(b* 
x+a))*x+3/2/b*c*d^2*ln(exp(b*x+a)+1)*x^2-3/2/b^3*c*d^2*ln(exp(b*x+a)+1)*a^ 
2+3/b^2*c*d^2*polylog(2,-exp(b*x+a))*x-3/2/b*d*c^2*ln(1-exp(b*x+a))*x-3/2/ 
b^2*d*c^2*ln(1-exp(b*x+a))*a-exp(b*x+a)*(exp(2*b*x+2*a)*b*d^3*x^3+3*exp(2* 
b*x+2*a)*b*c*d^2*x^2+3*exp(2*b*x+2*a)*b*c^2*d*x+b*d^3*x^3+3*exp(2*b*x+2*a) 
*d^3*x^2+exp(2*b*x+2*a)*b*c^3+3*b*c*d^2*x^2+6*exp(2*b*x+2*a)*c*d^2*x+3*b*c 
^2*d*x+3*exp(2*b*x+2*a)*c^2*d-3*d^3*x^2+b*c^3-6*c*d^2*x-3*d*c^2)/b^2/(exp( 
2*b*x+2*a)-1)^2-3*d^3*polylog(4,exp(b*x+a))/b^4+3*d^3*polylog(4,-exp(b*x+a 
))/b^4-3*d^3*polylog(2,-exp(b*x+a))/b^4+3*d^3*polylog(2,exp(b*x+a))/b^4
 

3.1.9.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4008 vs. \(2 (234) = 468\).

Time = 0.31 (sec) , antiderivative size = 4008, normalized size of antiderivative = 15.66 \[ \int (c+d x)^3 \text {csch}^3(a+b x) \, dx=\text {Too large to display} \]

input

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

output

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

3.1.9.6 Sympy [F]

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

input

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

output

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

3.1.9.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 605 vs. \(2 (234) = 468\).

Time = 0.37 (sec) , antiderivative size = 605, normalized size of antiderivative = 2.36 \[ \int (c+d x)^3 \text {csch}^3(a+b x) \, dx=\frac {1}{2} \, c^{3} {\left (\frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} - \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {2 \, {\left (e^{\left (-b x - a\right )} + e^{\left (-3 \, b x - 3 \, a\right )}\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )}}\right )} + \frac {3 \, {\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )} c d^{2}}{2 \, b^{3}} - \frac {3 \, {\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )} c d^{2}}{2 \, b^{3}} - \frac {3 \, c d^{2} \log \left (e^{\left (b x + a\right )} + 1\right )}{b^{3}} + \frac {3 \, c d^{2} \log \left (e^{\left (b x + a\right )} - 1\right )}{b^{3}} - \frac {{\left (b d^{3} x^{3} e^{\left (3 \, a\right )} + 3 \, c^{2} d e^{\left (3 \, a\right )} + 3 \, {\left (b c d^{2} + d^{3}\right )} x^{2} e^{\left (3 \, a\right )} + 3 \, {\left (b c^{2} d + 2 \, c d^{2}\right )} x e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} + {\left (b d^{3} x^{3} e^{a} - 3 \, c^{2} d e^{a} + 3 \, {\left (b c d^{2} - d^{3}\right )} x^{2} e^{a} + 3 \, {\left (b c^{2} d - 2 \, c d^{2}\right )} x e^{a}\right )} e^{\left (b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} + \frac {{\left (b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})\right )} d^{3}}{2 \, b^{4}} - \frac {{\left (b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})\right )} d^{3}}{2 \, b^{4}} + \frac {3 \, {\left (b^{2} c^{2} d - 2 \, d^{3}\right )} {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{2 \, b^{4}} - \frac {3 \, {\left (b^{2} c^{2} d - 2 \, d^{3}\right )} {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{2 \, b^{4}} \]

input

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

output

1/2*c^3*(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))) + 
3/2*(b^2*x^2*log(e^(b*x + a) + 1) + 2*b*x*dilog(-e^(b*x + a)) - 2*polylog( 
3, -e^(b*x + a)))*c*d^2/b^3 - 3/2*(b^2*x^2*log(-e^(b*x + a) + 1) + 2*b*x*d 
ilog(e^(b*x + a)) - 2*polylog(3, e^(b*x + a)))*c*d^2/b^3 - 3*c*d^2*log(e^( 
b*x + a) + 1)/b^3 + 3*c*d^2*log(e^(b*x + a) - 1)/b^3 - ((b*d^3*x^3*e^(3*a) 
 + 3*c^2*d*e^(3*a) + 3*(b*c*d^2 + d^3)*x^2*e^(3*a) + 3*(b*c^2*d + 2*c*d^2) 
*x*e^(3*a))*e^(3*b*x) + (b*d^3*x^3*e^a - 3*c^2*d*e^a + 3*(b*c*d^2 - d^3)*x 
^2*e^a + 3*(b*c^2*d - 2*c*d^2)*x*e^a)*e^(b*x))/(b^2*e^(4*b*x + 4*a) - 2*b^ 
2*e^(2*b*x + 2*a) + b^2) + 1/2*(b^3*x^3*log(e^(b*x + a) + 1) + 3*b^2*x^2*d 
ilog(-e^(b*x + a)) - 6*b*x*polylog(3, -e^(b*x + a)) + 6*polylog(4, -e^(b*x 
 + a)))*d^3/b^4 - 1/2*(b^3*x^3*log(-e^(b*x + a) + 1) + 3*b^2*x^2*dilog(e^( 
b*x + a)) - 6*b*x*polylog(3, e^(b*x + a)) + 6*polylog(4, e^(b*x + a)))*d^3 
/b^4 + 3/2*(b^2*c^2*d - 2*d^3)*(b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + 
 a)))/b^4 - 3/2*(b^2*c^2*d - 2*d^3)*(b*x*log(-e^(b*x + a) + 1) + dilog(e^( 
b*x + a)))/b^4
 

3.1.9.8 Giac [F]

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

input

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

output

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

3.1.9.9 Mupad [F(-1)]

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

input

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

output

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