\(\int (c+d x)^2 \text {csch}^2(a+b x) \, dx\) [29]

Optimal result

Integrand size = 16, antiderivative size = 74 \[ \int (c+d x)^2 \text {csch}^2(a+b x) \, dx=-\frac {(c+d x)^2}{b}-\frac {(c+d x)^2 \coth (a+b x)}{b}+\frac {2 d (c+d x) \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac {d^2 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.85 \[ \int (c+d x)^2 \text {csch}^2(a+b x) \, dx=\frac {-\frac {2 b (c+d x) \left (b (c+d x)-d \left (-1+e^{2 a}\right ) \log \left (1-e^{-a-b x}\right )-d \left (-1+e^{2 a}\right ) \log \left (1+e^{-a-b x}\right )\right )}{-1+e^{2 a}}-2 d^2 \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )-2 d^2 \operatorname {PolyLog}\left (2,e^{-a-b x}\right )+b^2 (c+d x)^2 \text {csch}(a) \text {csch}(a+b x) \sinh (b x)}{b^3} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.43, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3042, 25, 4672, 26, 3042, 26, 4201, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4672

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {(c+d x)^2 \coth (a+b x)}{b}-\frac {2 i d \left (2 i \left (\frac {(c+d x) \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {d \int e^{-2 a-2 b x+i \pi } \log \left (1+e^{2 a+2 b x-i \pi }\right )de^{2 a+2 b x-i \pi }}{4 b^2}\right )-\frac {i (c+d x)^2}{2 d}\right )}{b}\)

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4201

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

rule 4672

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp 
[(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1) 
*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(239\) vs. \(2(74)=148\).

Time = 0.10 (sec) , antiderivative size = 240, normalized size of antiderivative = 3.24

method	result	size
risch	\(-\frac {2 \left (x^{2} d^{2}+2 c d x +c^{2}\right )}{\left ({\mathrm e}^{2 b x +2 a}-1\right ) b}+\frac {2 d c \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}}-\frac {4 d c \ln \left ({\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 d c \ln \left ({\mathrm e}^{b x +a}+1\right )}{b^{2}}-\frac {2 d^{2} x^{2}}{b}-\frac {4 d^{2} a x}{b^{2}}-\frac {2 d^{2} a^{2}}{b^{3}}+\frac {2 d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {2 d^{2} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 d^{2} \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{2}}+\frac {2 d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 d^{2} a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}+\frac {4 d^{2} a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{3}}\)	\(240\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 623 vs. \(2 (73) = 146\).

Time = 0.10 (sec) , antiderivative size = 623, normalized size of antiderivative = 8.42 \[ \int (c+d x)^2 \text {csch}^2(a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

-2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + 2*a*b*c*d 
 - a^2*d^2)*cosh(b*x + a)^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + 2*a*b*c*d - a 
^2*d^2)*cosh(b*x + a)*sinh(b*x + a) + (b^2*d^2*x^2 + 2*b^2*c*d*x + 2*a*b*c 
*d - a^2*d^2)*sinh(b*x + a)^2 - (d^2*cosh(b*x + a)^2 + 2*d^2*cosh(b*x + a) 
*sinh(b*x + a) + d^2*sinh(b*x + a)^2 - d^2)*dilog(cosh(b*x + a) + sinh(b*x 
 + a)) - (d^2*cosh(b*x + a)^2 + 2*d^2*cosh(b*x + a)*sinh(b*x + a) + d^2*si 
nh(b*x + a)^2 - d^2)*dilog(-cosh(b*x + a) - sinh(b*x + a)) + (b*d^2*x + b* 
c*d - (b*d^2*x + b*c*d)*cosh(b*x + a)^2 - 2*(b*d^2*x + b*c*d)*cosh(b*x + a 
)*sinh(b*x + a) - (b*d^2*x + b*c*d)*sinh(b*x + a)^2)*log(cosh(b*x + a) + s 
inh(b*x + a) + 1) + (b*c*d - a*d^2 - (b*c*d - a*d^2)*cosh(b*x + a)^2 - 2*( 
b*c*d - a*d^2)*cosh(b*x + a)*sinh(b*x + a) - (b*c*d - a*d^2)*sinh(b*x + a) 
^2)*log(cosh(b*x + a) + sinh(b*x + a) - 1) + (b*d^2*x + a*d^2 - (b*d^2*x + 
 a*d^2)*cosh(b*x + a)^2 - 2*(b*d^2*x + a*d^2)*cosh(b*x + a)*sinh(b*x + a) 
- (b*d^2*x + a*d^2)*sinh(b*x + a)^2)*log(-cosh(b*x + a) - sinh(b*x + a) + 
1))/(b^3*cosh(b*x + a)^2 + 2*b^3*cosh(b*x + a)*sinh(b*x + a) + b^3*sinh(b* 
x + a)^2 - b^3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 4*e**(2*a + 2*b*x)*int(x/(e**(4*a + 4*b*x) - 2*e**(2*a + 2*b*x) + 1),x 
)*b**2*d**2 + 2*e**(2*a + 2*b*x)*log(e**(a + b*x) - 1)*b*c*d + e**(2*a + 2 
*b*x)*log(e**(a + b*x) - 1)*d**2 + 2*e**(2*a + 2*b*x)*log(e**(a + b*x) + 1 
)*b*c*d + e**(2*a + 2*b*x)*log(e**(a + b*x) + 1)*d**2 - 2*e**(2*a + 2*b*x) 
*b**2*c**2 - 4*e**(2*a + 2*b*x)*b**2*c*d*x - 2*e**(2*a + 2*b*x)*b*d**2*x + 
 4*int(x/(e**(4*a + 4*b*x) - 2*e**(2*a + 2*b*x) + 1),x)*b**2*d**2 - 2*log( 
e**(a + b*x) - 1)*b*c*d - log(e**(a + b*x) - 1)*d**2 - 2*log(e**(a + b*x) 
+ 1)*b*c*d - log(e**(a + b*x) + 1)*d**2 - 2*b**2*d**2*x**2)/(b**3*(e**(2*a 
 + 2*b*x) - 1))