\(\int \frac {\text {csch}^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\) [209]

Optimal result

Integrand size = 24, antiderivative size = 164 \[ \int \frac {\text {csch}^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\frac {b^{3/4} \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/2} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {\text {arctanh}(\cosh (c+d x))}{2 a d}+\frac {b^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/2} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \] Output:

1/2*b^(3/4)*arctan(b^(1/4)*cosh(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/a^(3/2)/(a 
^(1/2)-b^(1/2))^(1/2)/d+1/2*arctanh(cosh(d*x+c))/a/d+1/2*b^(3/4)*arctanh(b 
^(1/4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/a^(3/2)/(a^(1/2)+b^(1/2))^(1/2 
)/d-1/2*coth(d*x+c)*csch(d*x+c)/a/d
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 1.63 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.70 \[ \int \frac {\text {csch}^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=-\frac {\text {csch}^2\left (\frac {1}{2} (c+d x)\right )-4 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+4 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+4 b \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-c \text {$\#$1}-d x \text {$\#$1}-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}+c \text {$\#$1}^3+d x \text {$\#$1}^3+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^3}{-b-8 a \text {$\#$1}^2+3 b \text {$\#$1}^2-3 b \text {$\#$1}^4+b \text {$\#$1}^6}\&\right ]+\text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d} \] Input:

Integrate[Csch[c + d*x]^3/(a - b*Sinh[c + d*x]^4),x]
 

Output:

-1/8*(Csch[(c + d*x)/2]^2 - 4*Log[Cosh[(c + d*x)/2]] + 4*Log[Sinh[(c + d*x 
)/2]] + 4*b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^ 
8 & , (-(c*#1) - d*x*#1 - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + C 
osh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1 + c*#1^3 + d*x*#1^3 + 2*Log 
[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + 
 d*x)/2]*#1]*#1^3)/(-b - 8*a*#1^2 + 3*b*#1^2 - 3*b*#1^4 + b*#1^6) & ] + Se 
ch[(c + d*x)/2]^2)/(a*d)
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3042, 26, 3694, 1484, 2009}

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.

\(\displaystyle \int \frac {\text {csch}^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {i}{\sin (i c+i d x)^3 \left (a-b \sin (i c+i d x)^4\right )}dx\)

\(\Big \downarrow \) 26

\(\displaystyle -i \int \frac {1}{\sin (i c+i d x)^3 \left (a-b \sin (i c+i d x)^4\right )}dx\)

\(\Big \downarrow \) 3694

\(\displaystyle \frac {\int \frac {1}{\left (1-\cosh ^2(c+d x)\right )^2 \left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )}d\cosh (c+d x)}{d}\)

\(\Big \downarrow \) 1484

\(\displaystyle \frac {\int \left (\frac {b}{a \left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )}-\frac {1}{2 a \left (\cosh ^2(c+d x)-1\right )}+\frac {1}{4 a (\cosh (c+d x)-1)^2}+\frac {1}{4 a (\cosh (c+d x)+1)^2}\right )d\cosh (c+d x)}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {b^{3/4} \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/2} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {b^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/2} \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\text {arctanh}(\cosh (c+d x))}{2 a}+\frac {1}{4 a (1-\cosh (c+d x))}-\frac {1}{4 a (\cosh (c+d x)+1)}}{d}\)

Input:

Int[Csch[c + d*x]^3/(a - b*Sinh[c + d*x]^4),x]
 

Output:

((b^(3/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^(3 
/2)*Sqrt[Sqrt[a] - Sqrt[b]]) + ArcTanh[Cosh[c + d*x]]/(2*a) + (b^(3/4)*Arc 
Tanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^(3/2)*Sqrt[Sqr 
t[a] + Sqrt[b]]) + 1/(4*a*(1 - Cosh[c + d*x])) - 1/(4*a*(1 + Cosh[c + d*x] 
)))/d
 

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 1484

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symb 
ol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; Fre 
eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 
 0] && IntegerQ[q]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3694

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^ 
(p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f 
Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4)^p, 
 x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 
 1)/2]
 

Maple [A] (verified)

Time = 1.69 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+2 b \left (\frac {\arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b +\sqrt {a b}\, a}}-\frac {\arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b -\sqrt {a b}\, a}}\right )}{d}\)	\(181\)
default	\(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+2 b \left (\frac {\arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b +\sqrt {a b}\, a}}-\frac {\arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b -\sqrt {a b}\, a}}\right )}{d}\)	\(181\)
risch	\(-\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{2 a d}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a d}+8 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (1048576 a^{7} d^{4}-1048576 a^{6} b \,d^{4}\right ) \textit {\_Z}^{4}+2048 a^{3} b^{2} d^{2} \textit {\_Z}^{2}-b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (\frac {65536 d^{3} a^{5}}{b^{2}}-\frac {65536 d^{3} a^{4}}{b}\right ) \textit {\_R}^{3}+\left (\frac {64 a^{2} d}{b}+64 a d \right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\)	\(187\)

Input:

int(csch(d*x+c)^3/(a-b*sinh(d*x+c)^4),x,method=_RETURNVERBOSE)
 

Output:

1/d*(1/8*tanh(1/2*d*x+1/2*c)^2/a-1/8/a/tanh(1/2*d*x+1/2*c)^2-1/2/a*ln(tanh 
(1/2*d*x+1/2*c))+2*b*(1/4/a/(-a*b+(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(2*tanh( 
1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b+(a*b)^(1/2)*a)^(1/2))-1/4/a/(- 
a*b-(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(-2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1 
/2)+2*a)/(-a*b-(a*b)^(1/2)*a)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1954 vs. \(2 (120) = 240\).

Time = 0.17 (sec) , antiderivative size = 1954, normalized size of antiderivative = 11.91 \[ \int \frac {\text {csch}^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Too large to display} \] Input:

integrate(csch(d*x+c)^3/(a-b*sinh(d*x+c)^4),x, algorithm="fricas")
 

Output:

-1/4*(4*cosh(d*x + c)^3 + 12*cosh(d*x + c)*sinh(d*x + c)^2 + 4*sinh(d*x + 
c)^3 - (a*d*cosh(d*x + c)^4 + 4*a*d*cosh(d*x + c)*sinh(d*x + c)^3 + a*d*si 
nh(d*x + c)^4 - 2*a*d*cosh(d*x + c)^2 + 2*(3*a*d*cosh(d*x + c)^2 - a*d)*si 
nh(d*x + c)^2 + a*d + 4*(a*d*cosh(d*x + c)^3 - a*d*cosh(d*x + c))*sinh(d*x 
 + c))*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) 
+ b^2)/((a^4 - a^3*b)*d^2))*log(b^2*cosh(d*x + c)^2 + 2*b^2*cosh(d*x + c)* 
sinh(d*x + c) + b^2*sinh(d*x + c)^2 + b^2 + 2*(a^2*b*d*cosh(d*x + c) + a^2 
*b*d*sinh(d*x + c) - ((a^5 - a^4*b)*d^3*cosh(d*x + c) + (a^5 - a^4*b)*d^3* 
sinh(d*x + c))*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)))*sqrt(-((a^4 - a^ 
3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d 
^2))) + (a*d*cosh(d*x + c)^4 + 4*a*d*cosh(d*x + c)*sinh(d*x + c)^3 + a*d*s 
inh(d*x + c)^4 - 2*a*d*cosh(d*x + c)^2 + 2*(3*a*d*cosh(d*x + c)^2 - a*d)*s 
inh(d*x + c)^2 + a*d + 4*(a*d*cosh(d*x + c)^3 - a*d*cosh(d*x + c))*sinh(d* 
x + c))*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) 
 + b^2)/((a^4 - a^3*b)*d^2))*log(b^2*cosh(d*x + c)^2 + 2*b^2*cosh(d*x + c) 
*sinh(d*x + c) + b^2*sinh(d*x + c)^2 + b^2 - 2*(a^2*b*d*cosh(d*x + c) + a^ 
2*b*d*sinh(d*x + c) - ((a^5 - a^4*b)*d^3*cosh(d*x + c) + (a^5 - a^4*b)*d^3 
*sinh(d*x + c))*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)))*sqrt(-((a^4 - a 
^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)* 
d^2))) - (a*d*cosh(d*x + c)^4 + 4*a*d*cosh(d*x + c)*sinh(d*x + c)^3 + a...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\text {csch}^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Timed out} \] Input:

integrate(csch(d*x+c)**3/(a-b*sinh(d*x+c)**4),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(csch(d*x+c)^3/(a-b*sinh(d*x+c)^4),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(csch(d*x+c)^3/(a-b*sinh(d*x+c)^4),x, algorithm="giac")
 

Output:

sage0*x
 

Mupad [B] (verification not implemented)

Time = 13.07 (sec) , antiderivative size = 1517, normalized size of antiderivative = 9.25 \[ \int \frac {\text {csch}^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Too large to display} \] Input:

int(1/(sinh(c + d*x)^3*(a - b*sinh(c + d*x)^4)),x)
 

Output:

atan((exp(d*x)*exp(c)*(256*a^6*(-a^2*d^2)^(1/2) + b^6*(-a^2*d^2)^(1/2) + 9 
6*a^2*b^4*(-a^2*d^2)^(1/2) - 288*a^3*b^3*(-a^2*d^2)^(1/2) + 512*a^4*b^2*(- 
a^2*d^2)^(1/2) - 16*a*b^5*(-a^2*d^2)^(1/2) - 512*a^5*b*(-a^2*d^2)^(1/2)))/ 
(256*a^7*d - 16*a^2*b^5*d + 96*a^3*b^4*d - 288*a^4*b^3*d + 512*a^5*b^2*d + 
 a*b^6*d - 512*a^6*b*d))/(-a^2*d^2)^(1/2) - log((((((8589934592*d^3*exp(c 
+ d*x)*(8*a^2 - 7*a*b + 3*b^2)*(-((a^7*b^3)^(1/2) + a^3*b^2)/(a^6*d^2*(a - 
 b)))^(1/2))/(b^5*(a - b)^2) - (4294967296*d^2*(exp(2*c + 2*d*x) + 1)*(2*a 
*b^2 - 7*a^2*b + 12*a^3 + b^3))/(a^2*b^4*(a - b)^3))*(-((a^7*b^3)^(1/2) + 
a^3*b^2)/(a^6*d^2*(a - b)))^(1/2))/4 - (4294967296*d*exp(c + d*x)*(2*a^2 - 
 2*a*b + b^2))/(a^3*b^4*(a - b)^2))*(-((a^7*b^3)^(1/2) + a^3*b^2)/(a^6*d^2 
*(a - b)))^(1/2))/4 + (268435456*(exp(2*c + 2*d*x) + 1)*(4*a^3 - a*b^2 + b 
^3))/(a^5*b^3*(a - b)^3))*(-((a^7*b^3)^(1/2) + a^3*b^2)/(16*(a^7*d^2 - a^6 
*b*d^2)))^(1/2) + log((268435456*(exp(2*c + 2*d*x) + 1)*(4*a^3 - a*b^2 + b 
^3))/(a^5*b^3*(a - b)^3) - (((((8589934592*d^3*exp(c + d*x)*(8*a^2 - 7*a*b 
 + 3*b^2)*(-((a^7*b^3)^(1/2) + a^3*b^2)/(a^6*d^2*(a - b)))^(1/2))/(b^5*(a 
- b)^2) + (4294967296*d^2*(exp(2*c + 2*d*x) + 1)*(2*a*b^2 - 7*a^2*b + 12*a 
^3 + b^3))/(a^2*b^4*(a - b)^3))*(-((a^7*b^3)^(1/2) + a^3*b^2)/(a^6*d^2*(a 
- b)))^(1/2))/4 - (4294967296*d*exp(c + d*x)*(2*a^2 - 2*a*b + b^2))/(a^3*b 
^4*(a - b)^2))*(-((a^7*b^3)^(1/2) + a^3*b^2)/(a^6*d^2*(a - b)))^(1/2))/4)* 
(-((a^7*b^3)^(1/2) + a^3*b^2)/(16*(a^7*d^2 - a^6*b*d^2)))^(1/2) - log((...
 

Reduce [F]

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

int(csch(d*x+c)^3/(a-b*sinh(d*x+c)^4),x)
 

Output:

 - int(csch(c + d*x)**3/(sinh(c + d*x)**4*b - a),x)