\(\int \frac {\text {csch}^2(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\) [214]

Optimal result

Integrand size = 24, antiderivative size = 139 \[ \int \frac {\text {csch}^2(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{5/4} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{5/4} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {\coth (c+d x)}{a d} \] Output:

-1/2*b^(1/2)*arctanh((a^(1/2)-b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))/a^(5/4)/ 
(a^(1/2)-b^(1/2))^(1/2)/d+1/2*b^(1/2)*arctanh((a^(1/2)+b^(1/2))^(1/2)*tanh 
(d*x+c)/a^(1/4))/a^(5/4)/(a^(1/2)+b^(1/2))^(1/2)/d-coth(d*x+c)/a/d
 

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.03 \[ \int \frac {\text {csch}^2(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\frac {\frac {\sqrt {b} \arctan \left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}-2 \coth (c+d x)}{2 a d} \] Input:

Integrate[Csch[c + d*x]^2/(a - b*Sinh[c + d*x]^4),x]
 

Output:

((Sqrt[b]*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqr 
t[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + (Sqrt[b]*ArcTanh[((Sqrt[a] + Sqrt[b]) 
*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - 2* 
Coth[c + d*x])/(2*a*d)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3042, 25, 3696, 1610, 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.

\(\displaystyle \int \frac {\text {csch}^2(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3696

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

\(\Big \downarrow \) 1610

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{5/4} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{5/4} \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\coth (c+d x)}{a}}{d}\)

Input:

Int[Csch[c + d*x]^2/(a - b*Sinh[c + d*x]^4),x]
 

Output:

(-1/2*(Sqrt[b]*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/( 
a^(5/4)*Sqrt[Sqrt[a] - Sqrt[b]]) + (Sqrt[b]*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b 
]]*Tanh[c + d*x])/a^(1/4)])/(2*a^(5/4)*Sqrt[Sqrt[a] + Sqrt[b]]) - Coth[c + 
 d*x]/a)/d
 

Defintions of rubi rules used

rule 25

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

rule 1610

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e*x^2)^q/(a 
+ b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4 
*a*c, 0] && IntegerQ[q] && IntegerQ[m]
 

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 3696

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

Maple [C] (verified)

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

Time = 1.19 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-\textit {\_R}^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}\right )}{a}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(130\)
default	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-\textit {\_R}^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}\right )}{a}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(130\)
risch	\(-\frac {2}{a d \left ({\mathrm e}^{2 d x +2 c}-1\right )}+4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (65536 a^{6} d^{4}-65536 a^{5} b \,d^{4}\right ) \textit {\_Z}^{4}-512 a^{3} d^{2} \textit {\_Z}^{2} b +b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (-\frac {8192 d^{3} a^{5}}{b^{2}}+\frac {8192 d^{3} a^{4}}{b}\right ) \textit {\_R}^{3}+\left (\frac {512 d^{2} a^{4}}{b^{2}}-\frac {512 a^{3} d^{2}}{b}\right ) \textit {\_R}^{2}+\frac {64 d \,a^{2} \textit {\_R}}{b}-\frac {2 a}{b}-1\right )\right )\)	\(151\)

Input:

int(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4),x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/2/a*tanh(1/2*d*x+1/2*c)-b/a*sum((_R^4-_R^2)/(_R^7*a-3*_R^5*a+3*_R^ 
3*a-8*_R^3*b-_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(a*_Z^8-4*a*_Z^6+(6 
*a-16*b)*_Z^4-4*a*_Z^2+a))-1/2/a/tanh(1/2*d*x+1/2*c))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1305 vs. \(2 (99) = 198\).

Time = 0.14 (sec) , antiderivative size = 1305, normalized size of antiderivative = 9.39 \[ \int \frac {\text {csch}^2(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Too large to display} \] Input:

integrate(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4),x, algorithm="fricas")
 

Output:

-1/4*((a*d*cosh(d*x + c)^2 + 2*a*d*cosh(d*x + c)*sinh(d*x + c) + a*d*sinh( 
d*x + c)^2 - a*d)*sqrt(((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b 
^2)*d^4)) + b)/((a^3 - a^2*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 + 2*(a^4 - a^3*b)*d^2*sqrt(b^3 
/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2 + 2*((a^5 - a^4*b)*d^3*sqrt(b^3/(( 
a^7 - 2*a^6*b + a^5*b^2)*d^4)) - a^2*b*d)*sqrt(((a^3 - a^2*b)*d^2*sqrt(b^3 
/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))) - (a*d*cosh(d 
*x + c)^2 + 2*a*d*cosh(d*x + c)*sinh(d*x + c) + a*d*sinh(d*x + c)^2 - a*d) 
*sqrt(((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b)/(( 
a^3 - a^2*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 + 2*(a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b 
+ a^5*b^2)*d^4)) - b^2 - 2*((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a 
^5*b^2)*d^4)) - a^2*b*d)*sqrt(((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b 
+ a^5*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))) - (a*d*cosh(d*x + c)^2 + 2*a*d 
*cosh(d*x + c)*sinh(d*x + c) + a*d*sinh(d*x + c)^2 - a*d)*sqrt(-((a^3 - a^ 
2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b)/((a^3 - a^2*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 - 2*(a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) 
 - b^2 + 2*((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + 
a^2*b*d)*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\text {csch}^2(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Timed out} \] Input:

integrate(csch(d*x+c)**2/(a-b*sinh(d*x+c)**4),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4),x, algorithm="maxima")
 

Output:

-2/(a*d*e^(2*d*x + 2*c) - a*d) - 4*integrate((b*e^(6*d*x + 6*c) - 2*b*e^(4 
*d*x + 4*c) + b*e^(2*d*x + 2*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]

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

integrate(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4),x, algorithm="giac")
 

Output:

sage0*x
 

Mupad [B] (verification not implemented)

Time = 12.43 (sec) , antiderivative size = 2128, normalized size of antiderivative = 15.31 \[ \int \frac {\text {csch}^2(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Too large to display} \] Input:

int(1/(sinh(c + d*x)^2*(a - b*sinh(c + d*x)^4)),x)
 

Output:

log((((((4194304*d^2*(512*a^4 - 1184*a^3*b - 253*a*b^3 - b^4 + 930*a^2*b^2 
 + b^4*exp(2*c + 2*d*x) + 627*a*b^3*exp(2*c + 2*d*x) + 768*a^3*b*exp(2*c + 
 2*d*x) - 1392*a^2*b^2*exp(2*c + 2*d*x)))/(a^2*b^4*(a - b)^2) - (16777216* 
d^3*(((a^5*b^3)^(1/2) + a^3*b)/(a^5*d^2*(a - b)))^(1/2)*(40*a*b^2 - 35*b^3 
 + 512*a^3*exp(2*c + 2*d*x) + 64*b^3*exp(2*c + 2*d*x) + 326*a*b^2*exp(2*c 
+ 2*d*x) - 896*a^2*b*exp(2*c + 2*d*x)))/(b^5*(a - b)))*(((a^5*b^3)^(1/2) + 
 a^3*b)/(a^5*d^2*(a - b)))^(1/2))/4 - (2097152*d*(256*a^2*b - 256*a*b^2 - 
5*b^3 - 1024*a^3*exp(2*c + 2*d*x) + 6*b^3*exp(2*c + 2*d*x) + 756*a*b^2*exp 
(2*c + 2*d*x) + 256*a^2*b*exp(2*c + 2*d*x)))/(a^3*b^4*(a - b)))*(((a^5*b^3 
)^(1/2) + a^3*b)/(a^5*d^2*(a - b)))^(1/2))/4 - (524288*(185*a*b^2 - 464*a^ 
2*b + 256*a^3 + 24*b^3 - 1024*a^3*exp(2*c + 2*d*x) - 35*b^3*exp(2*c + 2*d* 
x) - 988*a*b^2*exp(2*c + 2*d*x) + 2048*a^2*b*exp(2*c + 2*d*x)))/(a^4*b^3*( 
a - b)^2))*(((a^5*b^3)^(1/2) + a^3*b)/(16*(a^6*d^2 - a^5*b*d^2)))^(1/2) - 
log((((((4194304*d^2*(512*a^4 - 1184*a^3*b - 253*a*b^3 - b^4 + 930*a^2*b^2 
 + b^4*exp(2*c + 2*d*x) + 627*a*b^3*exp(2*c + 2*d*x) + 768*a^3*b*exp(2*c + 
 2*d*x) - 1392*a^2*b^2*exp(2*c + 2*d*x)))/(a^2*b^4*(a - b)^2) + (16777216* 
d^3*(((a^5*b^3)^(1/2) + a^3*b)/(a^5*d^2*(a - b)))^(1/2)*(40*a*b^2 - 35*b^3 
 + 512*a^3*exp(2*c + 2*d*x) + 64*b^3*exp(2*c + 2*d*x) + 326*a*b^2*exp(2*c 
+ 2*d*x) - 896*a^2*b*exp(2*c + 2*d*x)))/(b^5*(a - b)))*(((a^5*b^3)^(1/2) + 
 a^3*b)/(a^5*d^2*(a - b)))^(1/2))/4 + (2097152*d*(256*a^2*b - 256*a*b^2...
 

Reduce [F]

\[ \int \frac {\text {csch}^2(c+d x)}{a-b \sinh ^4(c+d x)} \, dx =\text {Too large to display} \] Input:

int(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4),x)
 

Output:

(1024*e**(6*c + 2*d*x)*int(e**(4*d*x)/(e**(12*c + 12*d*x)*b - 6*e**(10*c + 
 10*d*x)*b - 16*e**(8*c + 8*d*x)*a + 15*e**(8*c + 8*d*x)*b + 32*e**(6*c + 
6*d*x)*a - 20*e**(6*c + 6*d*x)*b - 16*e**(4*c + 4*d*x)*a + 15*e**(4*c + 4* 
d*x)*b - 6*e**(2*c + 2*d*x)*b + b),x)*a**2*d - 448*e**(6*c + 2*d*x)*int(e* 
*(4*d*x)/(e**(12*c + 12*d*x)*b - 6*e**(10*c + 10*d*x)*b - 16*e**(8*c + 8*d 
*x)*a + 15*e**(8*c + 8*d*x)*b + 32*e**(6*c + 6*d*x)*a - 20*e**(6*c + 6*d*x 
)*b - 16*e**(4*c + 4*d*x)*a + 15*e**(4*c + 4*d*x)*b - 6*e**(2*c + 2*d*x)*b 
 + b),x)*a*b*d + 256*e**(4*c + 2*d*x)*int(e**(2*d*x)/(e**(12*c + 12*d*x)*b 
 - 6*e**(10*c + 10*d*x)*b - 16*e**(8*c + 8*d*x)*a + 15*e**(8*c + 8*d*x)*b 
+ 32*e**(6*c + 6*d*x)*a - 20*e**(6*c + 6*d*x)*b - 16*e**(4*c + 4*d*x)*a + 
15*e**(4*c + 4*d*x)*b - 6*e**(2*c + 2*d*x)*b + b),x)*a*b*d - 64*e**(2*c + 
2*d*x)*int(1/(e**(12*c + 12*d*x)*b - 6*e**(10*c + 10*d*x)*b - 16*e**(8*c + 
 8*d*x)*a + 15*e**(8*c + 8*d*x)*b + 32*e**(6*c + 6*d*x)*a - 20*e**(6*c + 6 
*d*x)*b - 16*e**(4*c + 4*d*x)*a + 15*e**(4*c + 4*d*x)*b - 6*e**(2*c + 2*d* 
x)*b + b),x)*a*b*d - 32*e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*a + 4*e**(2 
*c + 2*d*x)*log(e**(c + d*x) - 1)*b - 32*e**(2*c + 2*d*x)*log(e**(c + d*x) 
 + 1)*a + 4*e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)*b - e**(2*c + 2*d*x)*lo 
g(e**(8*c + 8*d*x)*b - 4*e**(6*c + 6*d*x)*b - 16*e**(4*c + 4*d*x)*a + 6*e* 
*(4*c + 4*d*x)*b - 4*e**(2*c + 2*d*x)*b + b)*b + 64*e**(2*c + 2*d*x)*a*d*x 
 - 32*e**(2*c + 2*d*x)*a - 1024*e**(4*c)*int(e**(4*d*x)/(e**(12*c + 12*...