\(\int (c+d x)^2 \text {csch}^3(a+b x) \, dx\) [10]

Optimal result

Integrand size = 16, antiderivative size = 154 \[ \int (c+d x)^2 \text {csch}^3(a+b x) \, dx=\frac {(c+d x)^2 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {d^2 \text {arctanh}(\cosh (a+b x))}{b^3}-\frac {d (c+d x) \text {csch}(a+b x)}{b^2}-\frac {(c+d x)^2 \coth (a+b x) \text {csch}(a+b x)}{2 b}+\frac {d (c+d x) \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {d (c+d x) \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}-\frac {d^2 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}+\frac {d^2 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(420\) vs. \(2(154)=308\).

Time = 6.21 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.73 \[ \int (c+d x)^2 \text {csch}^3(a+b x) \, dx=-\frac {d (c+d x) \text {csch}(a)}{b^2}+\frac {\left (-c^2-2 c d x-d^2 x^2\right ) \text {csch}^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b}+\frac {-b^2 c^2 \log \left (1-e^{a+b x}\right )+2 d^2 \log \left (1-e^{a+b x}\right )-2 b^2 c d x \log \left (1-e^{a+b x}\right )-b^2 d^2 x^2 \log \left (1-e^{a+b x}\right )+b^2 c^2 \log \left (1+e^{a+b x}\right )-2 d^2 \log \left (1+e^{a+b x}\right )+2 b^2 c d x \log \left (1+e^{a+b x}\right )+b^2 d^2 x^2 \log \left (1+e^{a+b x}\right )+2 b d (c+d x) \operatorname {PolyLog}\left (2,-e^{a+b x}\right )-2 b d (c+d x) \operatorname {PolyLog}\left (2,e^{a+b x}\right )-2 d^2 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )+2 d^2 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{2 b^3}+\frac {\left (-c^2-2 c d x-d^2 x^2\right ) \text {sech}^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b}+\frac {\text {csch}\left (\frac {a}{2}\right ) \text {csch}\left (\frac {a}{2}+\frac {b x}{2}\right ) \left (c d \sinh \left (\frac {b x}{2}\right )+d^2 x \sinh \left (\frac {b x}{2}\right )\right )}{2 b^2}+\frac {\text {sech}\left (\frac {a}{2}\right ) \text {sech}\left (\frac {a}{2}+\frac {b x}{2}\right ) \left (c d \sinh \left (\frac {b x}{2}\right )+d^2 x \sinh \left (\frac {b x}{2}\right )\right )}{2 b^2} \] Input:

Integrate[(c + d*x)^2*Csch[a + b*x]^3,x]
 

Output:

-((d*(c + d*x)*Csch[a])/b^2) + ((-c^2 - 2*c*d*x - d^2*x^2)*Csch[a/2 + (b*x 
)/2]^2)/(8*b) + (-(b^2*c^2*Log[1 - E^(a + b*x)]) + 2*d^2*Log[1 - E^(a + b* 
x)] - 2*b^2*c*d*x*Log[1 - E^(a + b*x)] - b^2*d^2*x^2*Log[1 - E^(a + b*x)] 
+ b^2*c^2*Log[1 + E^(a + b*x)] - 2*d^2*Log[1 + E^(a + b*x)] + 2*b^2*c*d*x* 
Log[1 + E^(a + b*x)] + b^2*d^2*x^2*Log[1 + E^(a + b*x)] + 2*b*d*(c + d*x)* 
PolyLog[2, -E^(a + b*x)] - 2*b*d*(c + d*x)*PolyLog[2, E^(a + b*x)] - 2*d^2 
*PolyLog[3, -E^(a + b*x)] + 2*d^2*PolyLog[3, E^(a + b*x)])/(2*b^3) + ((-c^ 
2 - 2*c*d*x - d^2*x^2)*Sech[a/2 + (b*x)/2]^2)/(8*b) + (Csch[a/2]*Csch[a/2 
+ (b*x)/2]*(c*d*Sinh[(b*x)/2] + d^2*x*Sinh[(b*x)/2]))/(2*b^2) + (Sech[a/2] 
*Sech[a/2 + (b*x)/2]*(c*d*Sinh[(b*x)/2] + d^2*x*Sinh[(b*x)/2]))/(2*b^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.73 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.19, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {3042, 26, 4674, 26, 3042, 26, 4257, 4670, 3011, 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.

\(\displaystyle \int (c+d x)^2 \text {csch}^3(a+b x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -i (c+d x)^2 \csc (i a+i b x)^3dx\)

\(\Big \downarrow \) 26

\(\displaystyle -i \int (c+d x)^2 \csc (i a+i b x)^3dx\)

\(\Big \downarrow \) 4674

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4257

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

Int[(c + d*x)^2*Csch[a + b*x]^3,x]
 

Output:

(-I)*(((-I)*d^2*ArcTanh[Cosh[a + b*x]])/b^3 - (I*d*(c + d*x)*Csch[a + b*x] 
)/b^2 - ((I/2)*(c + d*x)^2*Coth[a + b*x]*Csch[a + b*x])/b + (((2*I)*(c + d 
*x)^2*ArcTanh[E^(a + b*x)])/b - ((2*I)*d*(-(((c + d*x)*PolyLog[2, -E^(a + 
b*x)])/b) + (d*PolyLog[3, -E^(a + b*x)])/b^2))/b + ((2*I)*d*(-(((c + d*x)* 
PolyLog[2, E^(a + b*x)])/b) + (d*PolyLog[3, E^(a + b*x)])/b^2))/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 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 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 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(443\) vs. \(2(147)=294\).

Time = 0.35 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.88

method	result	size
risch	\(-\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a} b \,d^{2} x^{2}+2 \,{\mathrm e}^{2 b x +2 a} b c d x +{\mathrm e}^{2 b x +2 a} b \,c^{2}+b \,d^{2} x^{2}+2 \,{\mathrm e}^{2 b x +2 a} d^{2} x +2 b c d x +2 \,{\mathrm e}^{2 b x +2 a} c d +b \,c^{2}-2 d^{2} x -2 c d \right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}+\frac {a^{2} d^{2} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {c^{2} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}-\frac {d^{2} \ln \left (1+{\mathrm e}^{b x +a}\right ) a^{2}}{2 b^{3}}+\frac {d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{2 b^{3}}+\frac {d^{2} \ln \left (1+{\mathrm e}^{b x +a}\right ) x^{2}}{2 b}+\frac {d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{2 b}-\frac {d^{2} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {c d \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {c d \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {2 d^{2} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {d^{2} \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 a c d \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {c d \ln \left (1+{\mathrm e}^{b x +a}\right ) x}{b}+\frac {c d \ln \left (1+{\mathrm e}^{b x +a}\right ) a}{b^{2}}-\frac {c d \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}-\frac {c d \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}\)	\(444\)

Input:

int((d*x+c)^2*csch(b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-exp(b*x+a)*(exp(2*b*x+2*a)*b*d^2*x^2+2*exp(2*b*x+2*a)*b*c*d*x+exp(2*b*x+2 
*a)*b*c^2+b*d^2*x^2+2*exp(2*b*x+2*a)*d^2*x+2*b*c*d*x+2*exp(2*b*x+2*a)*c*d+ 
b*c^2-2*d^2*x-2*c*d)/b^2/(exp(2*b*x+2*a)-1)^2+1/b^3*a^2*d^2*arctanh(exp(b* 
x+a))+1/b*c^2*arctanh(exp(b*x+a))-1/2/b^3*d^2*ln(1+exp(b*x+a))*a^2+1/2/b^3 
*d^2*ln(1-exp(b*x+a))*a^2+1/2/b*d^2*ln(1+exp(b*x+a))*x^2+1/b^2*d^2*polylog 
(2,-exp(b*x+a))*x-1/2/b*d^2*ln(1-exp(b*x+a))*x^2-1/b^2*d^2*polylog(2,exp(b 
*x+a))*x+1/b^2*c*d*polylog(2,-exp(b*x+a))-1/b^2*c*d*polylog(2,exp(b*x+a))- 
2/b^3*d^2*arctanh(exp(b*x+a))-d^2*polylog(3,-exp(b*x+a))/b^3+d^2*polylog(3 
,exp(b*x+a))/b^3-2/b^2*a*c*d*arctanh(exp(b*x+a))+1/b*c*d*ln(1+exp(b*x+a))* 
x+1/b^2*c*d*ln(1+exp(b*x+a))*a-1/b*c*d*ln(1-exp(b*x+a))*x-1/b^2*c*d*ln(1-e 
xp(b*x+a))*a
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2218 vs. \(2 (145) = 290\).

Time = 0.11 (sec) , antiderivative size = 2218, normalized size of antiderivative = 14.40 \[ \int (c+d x)^2 \text {csch}^3(a+b x) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^2*csch(b*x+a)^3,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int (c+d x)^2 \text {csch}^3(a+b x) \, dx=\int \left (c + d x\right )^{2} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \] Input:

integrate((d*x+c)**2*csch(b*x+a)**3,x)
 

Output:

Integral((c + d*x)**2*csch(a + b*x)**3, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (145) = 290\).

Time = 0.19 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.55 \[ \int (c+d x)^2 \text {csch}^3(a+b x) \, dx=\frac {1}{2} \, c^{2} {\left (\frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} - \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {2 \, {\left (e^{\left (-b x - a\right )} + e^{\left (-3 \, b x - 3 \, a\right )}\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )}}\right )} + \frac {{\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )} c d}{b^{2}} - \frac {{\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )} c d}{b^{2}} - \frac {{\left (b d^{2} x^{2} e^{\left (3 \, a\right )} + 2 \, c d e^{\left (3 \, a\right )} + 2 \, {\left (b c d + d^{2}\right )} x e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} + {\left (b d^{2} x^{2} e^{a} - 2 \, c d e^{a} + 2 \, {\left (b c d - d^{2}\right )} x e^{a}\right )} e^{\left (b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} + \frac {{\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )} d^{2}}{2 \, b^{3}} - \frac {{\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )} d^{2}}{2 \, b^{3}} - \frac {d^{2} \log \left (e^{\left (b x + a\right )} + 1\right )}{b^{3}} + \frac {d^{2} \log \left (e^{\left (b x + a\right )} - 1\right )}{b^{3}} \] Input:

integrate((d*x+c)^2*csch(b*x+a)^3,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int (c+d x)^2 \text {csch}^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \operatorname {csch}\left (b x + a\right )^{3} \,d x } \] Input:

integrate((d*x+c)^2*csch(b*x+a)^3,x, algorithm="giac")
 

Output:

integrate((d*x + c)^2*csch(b*x + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^2 \text {csch}^3(a+b x) \, dx=\int \frac {{\left (c+d\,x\right )}^2}{{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \] Input:

int((c + d*x)^2/sinh(a + b*x)^3,x)
 

Output:

int((c + d*x)^2/sinh(a + b*x)^3, x)
 

Reduce [F]

\[ \int (c+d x)^2 \text {csch}^3(a+b x) \, dx =\text {Too large to display} \] Input:

int((d*x+c)^2*csch(b*x+a)^3,x)
 

Output:

( - 48*e**(5*a + 4*b*x)*int((e**(b*x)*x**2)/(e**(6*a + 6*b*x) - 3*e**(4*a 
+ 4*b*x) + 3*e**(2*a + 2*b*x) - 1),x)*b**3*d**2 - 96*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**3*c*d - 128*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**2 - 9*e**(4*a + 
4*b*x)*log(e**(a + b*x) - 1)*b**2*c**2 - 24*e**(4*a + 4*b*x)*log(e**(a + b 
*x) - 1)*b*c*d - 8*e**(4*a + 4*b*x)*log(e**(a + b*x) - 1)*d**2 + 9*e**(4*a 
 + 4*b*x)*log(e**(a + b*x) + 1)*b**2*c**2 + 24*e**(4*a + 4*b*x)*log(e**(a 
+ b*x) + 1)*b*c*d + 8*e**(4*a + 4*b*x)*log(e**(a + b*x) + 1)*d**2 - 18*e** 
(3*a + 3*b*x)*b**2*c**2 - 48*e**(3*a + 3*b*x)*b*c*d - 16*e**(3*a + 3*b*x)* 
d**2 + 96*e**(3*a + 2*b*x)*int((e**(b*x)*x**2)/(e**(6*a + 6*b*x) - 3*e**(4 
*a + 4*b*x) + 3*e**(2*a + 2*b*x) - 1),x)*b**3*d**2 + 192*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**3*c*d + 256*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**2 + 18*e**(2 
*a + 2*b*x)*log(e**(a + b*x) - 1)*b**2*c**2 + 48*e**(2*a + 2*b*x)*log(e**( 
a + b*x) - 1)*b*c*d + 16*e**(2*a + 2*b*x)*log(e**(a + b*x) - 1)*d**2 - 18* 
e**(2*a + 2*b*x)*log(e**(a + b*x) + 1)*b**2*c**2 - 48*e**(2*a + 2*b*x)*log 
(e**(a + b*x) + 1)*b*c*d - 16*e**(2*a + 2*b*x)*log(e**(a + b*x) + 1)*d**2 
- 18*e**(a + b*x)*b**2*c**2 - 96*e**(a + b*x)*b**2*c*d*x - 48*e**(a + b...