\(\int \frac {\text {csch}^4(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\) [215]

Optimal result

Integrand size = 24, antiderivative size = 148 \[ \int \frac {\text {csch}^4(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} \sqrt {\sqrt {a}+\sqrt {b}} d}+\frac {\coth (c+d x)}{a d}-\frac {\coth ^3(c+d x)}{3 a d} \] Output:

1/2*b*arctanh((a^(1/2)-b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))/a^(7/4)/(a^(1/2 
)-b^(1/2))^(1/2)/d+1/2*b*arctanh((a^(1/2)+b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/ 
4))/a^(7/4)/(a^(1/2)+b^(1/2))^(1/2)/d+coth(d*x+c)/a/d-1/3*coth(d*x+c)^3/a/ 
d
 

Mathematica [A] (verified)

Time = 2.31 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.11 \[ \int \frac {\text {csch}^4(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\frac {-\frac {3 b \arctan \left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {3 b \text {arctanh}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}+4 \sqrt {a} \coth (c+d x)-2 \sqrt {a} \coth (c+d x) \text {csch}^2(c+d x)}{6 a^{3/2} d} \] Input:

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

Output:

((-3*b*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b 
]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + (3*b*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c 
 + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] + 4*Sqrt[a] 
*Coth[c + d*x] - 2*Sqrt[a]*Coth[c + d*x]*Csch[c + d*x]^2)/(6*a^(3/2)*d)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3696

\(\displaystyle \frac {\int \frac {\coth ^4(c+d x) \left (1-\tanh ^2(c+d x)\right )^3}{(a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a}d\tanh (c+d x)}{d}\)

\(\Big \downarrow \) 1610

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\coth ^3(c+d x)}{3 a}+\frac {\coth (c+d x)}{a}}{d}\)

Input:

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

Output:

((b*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*a^(7/4)*S 
qrt[Sqrt[a] - Sqrt[b]]) + (b*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x 
])/a^(1/4)])/(2*a^(7/4)*Sqrt[Sqrt[a] + Sqrt[b]]) + Coth[c + d*x]/a - Coth[ 
c + d*x]^3/(3*a))/d
 

Defintions of rubi rules used

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 = 2.01 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.14

method	result	size
derivativedivides	\(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+3 \textit {\_R}^{2}-1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}\right )}{4 a}-\frac {1}{24 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(168\)
default	\(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+3 \textit {\_R}^{2}-1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}\right )}{4 a}-\frac {1}{24 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(168\)
risch	\(-\frac {4 \left (3 \,{\mathrm e}^{2 d x +2 c}-1\right )}{3 a d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}+16 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16777216 a^{8} d^{4}-16777216 a^{7} b \,d^{4}\right ) \textit {\_Z}^{4}-8192 a^{4} b^{2} d^{2} \textit {\_Z}^{2}+b^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (-\frac {524288 d^{3} a^{7}}{b^{4}}+\frac {524288 d^{3} a^{6}}{b^{3}}\right ) \textit {\_R}^{3}+\left (\frac {8192 a^{5} d^{2}}{b^{3}}-\frac {8192 d^{2} a^{4}}{b^{2}}\right ) \textit {\_R}^{2}+\left (\frac {128 d \,a^{3}}{b^{2}}+\frac {128 a^{2} d}{b}\right ) \textit {\_R} -\frac {2 a}{b}-1\right )\right )\)	\(177\)

Input:

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

Output:

1/d*(-1/8/a*(1/3*tanh(1/2*d*x+1/2*c)^3-3*tanh(1/2*d*x+1/2*c))-1/4*b/a*sum( 
(_R^6-3*_R^4+3*_R^2-1)/(_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/ 
24/a/tanh(1/2*d*x+1/2*c)^3+3/8/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. 2206 vs. \(2 (110) = 220\).

Time = 0.15 (sec) , antiderivative size = 2206, normalized size of antiderivative = 14.91 \[ \int \frac {\text {csch}^4(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

1/12*(3*(a*d*cosh(d*x + c)^6 + 6*a*d*cosh(d*x + c)*sinh(d*x + c)^5 + a*d*s 
inh(d*x + c)^6 - 3*a*d*cosh(d*x + c)^4 + 3*(5*a*d*cosh(d*x + c)^2 - a*d)*s 
inh(d*x + c)^4 + 3*a*d*cosh(d*x + c)^2 + 4*(5*a*d*cosh(d*x + c)^3 - 3*a*d* 
cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*a*d*cosh(d*x + c)^4 - 6*a*d*cosh(d*x 
 + c)^2 + a*d)*sinh(d*x + c)^2 - a*d + 6*(a*d*cosh(d*x + c)^5 - 2*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*sq 
rt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))*log(b^ 
4*cosh(d*x + c)^2 + 2*b^4*cosh(d*x + c)*sinh(d*x + c) + b^4*sinh(d*x + c)^ 
2 - b^4 + 2*(a^5*b - a^4*b^2)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4) 
) + 2*(a^2*b^3*d - (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d 
^4)))*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + 
b^2)/((a^4 - a^3*b)*d^2))) - 3*(a*d*cosh(d*x + c)^6 + 6*a*d*cosh(d*x + c)* 
sinh(d*x + c)^5 + a*d*sinh(d*x + c)^6 - 3*a*d*cosh(d*x + c)^4 + 3*(5*a*d*c 
osh(d*x + c)^2 - a*d)*sinh(d*x + c)^4 + 3*a*d*cosh(d*x + c)^2 + 4*(5*a*d*c 
osh(d*x + c)^3 - 3*a*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*a*d*cosh(d*x 
+ c)^4 - 6*a*d*cosh(d*x + c)^2 + a*d)*sinh(d*x + c)^2 - a*d + 6*(a*d*cosh( 
d*x + c)^5 - 2*a*d*cosh(d*x + c)^3 + a*d*cosh(d*x + c))*sinh(d*x + c))*sqr 
t(((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^2)/((a^ 
4 - a^3*b)*d^2))*log(b^4*cosh(d*x + c)^2 + 2*b^4*cosh(d*x + c)*sinh(d*x + 
c) + b^4*sinh(d*x + c)^2 - b^4 + 2*(a^5*b - a^4*b^2)*d^2*sqrt(b^5/((a^9...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-16*b*integrate(e^(4*d*x + 4*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) - 4/3*(3*e^(2*d*x + 2*c) - 1)/(a*d*e^(6*d*x + 6*c) - 3*a*d*e^(4* 
d*x + 4*c) + 3*a*d*e^(2*d*x + 2*c) - a*d)
 

Giac [F]

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

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

Output:

sage0*x
 

Mupad [B] (verification not implemented)

Time = 13.50 (sec) , antiderivative size = 2178, normalized size of antiderivative = 14.72 \[ \int \frac {\text {csch}^4(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Too large to display} \] Input:

int(1/(sinh(c + d*x)^4*(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^4*b^2*(a - b)^2) + (8388608*d 
^3*(((a^7*b^5)^(1/2) + a^4*b^2)/(a^7*d^2*(a - b)))^(1/2)*(181*a*b^2 - 432* 
a^2*b + 256*a^3 + 5*b^3 - 512*a^3*exp(2*c + 2*d*x) - 6*b^3*exp(2*c + 2*d*x 
) - 622*a*b^2*exp(2*c + 2*d*x) + 1152*a^2*b*exp(2*c + 2*d*x)))/(a^2*b^3*(a 
 - b)))*(((a^7*b^5)^(1/2) + a^4*b^2)/(a^7*d^2*(a - b)))^(1/2))/4 + (209715 
2*d*(176*a*b - 256*a^2 + 75*b^2 + 1536*a^2*exp(2*c + 2*d*x) - 134*b^2*exp( 
2*c + 2*d*x) - 1408*a*b*exp(2*c + 2*d*x)))/(a^5*b*(a - b)))*(((a^7*b^5)^(1 
/2) + a^4*b^2)/(a^7*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^7*(a - b) 
^2))*(((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^8*d^2 - a^7*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^4*b^2*(a - b)^2) - (8388608*d^3*( 
((a^7*b^5)^(1/2) + a^4*b^2)/(a^7*d^2*(a - b)))^(1/2)*(181*a*b^2 - 432*a^2* 
b + 256*a^3 + 5*b^3 - 512*a^3*exp(2*c + 2*d*x) - 6*b^3*exp(2*c + 2*d*x) - 
622*a*b^2*exp(2*c + 2*d*x) + 1152*a^2*b*exp(2*c + 2*d*x)))/(a^2*b^3*(a - b 
)))*(((a^7*b^5)^(1/2) + a^4*b^2)/(a^7*d^2*(a - b)))^(1/2))/4 - (2097152...
 

Reduce [F]

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

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

Output:

(8*( - 1179648*e**(10*c + 6*d*x)*int(e**(4*d*x)/(48*e**(16*c + 16*d*x)*a*b 
 + e**(16*c + 16*d*x)*b**2 - 384*e**(14*c + 14*d*x)*a*b - 8*e**(14*c + 14* 
d*x)*b**2 - 768*e**(12*c + 12*d*x)*a**2 + 1328*e**(12*c + 12*d*x)*a*b + 28 
*e**(12*c + 12*d*x)*b**2 + 3072*e**(10*c + 10*d*x)*a**2 - 2624*e**(10*c + 
10*d*x)*a*b - 56*e**(10*c + 10*d*x)*b**2 - 4608*e**(8*c + 8*d*x)*a**2 + 32 
64*e**(8*c + 8*d*x)*a*b + 70*e**(8*c + 8*d*x)*b**2 + 3072*e**(6*c + 6*d*x) 
*a**2 - 2624*e**(6*c + 6*d*x)*a*b - 56*e**(6*c + 6*d*x)*b**2 - 768*e**(4*c 
 + 4*d*x)*a**2 + 1328*e**(4*c + 4*d*x)*a*b + 28*e**(4*c + 4*d*x)*b**2 - 38 
4*e**(2*c + 2*d*x)*a*b - 8*e**(2*c + 2*d*x)*b**2 + 48*a*b + b**2),x)*a**4* 
d + 638976*e**(10*c + 6*d*x)*int(e**(4*d*x)/(48*e**(16*c + 16*d*x)*a*b + e 
**(16*c + 16*d*x)*b**2 - 384*e**(14*c + 14*d*x)*a*b - 8*e**(14*c + 14*d*x) 
*b**2 - 768*e**(12*c + 12*d*x)*a**2 + 1328*e**(12*c + 12*d*x)*a*b + 28*e** 
(12*c + 12*d*x)*b**2 + 3072*e**(10*c + 10*d*x)*a**2 - 2624*e**(10*c + 10*d 
*x)*a*b - 56*e**(10*c + 10*d*x)*b**2 - 4608*e**(8*c + 8*d*x)*a**2 + 3264*e 
**(8*c + 8*d*x)*a*b + 70*e**(8*c + 8*d*x)*b**2 + 3072*e**(6*c + 6*d*x)*a** 
2 - 2624*e**(6*c + 6*d*x)*a*b - 56*e**(6*c + 6*d*x)*b**2 - 768*e**(4*c + 4 
*d*x)*a**2 + 1328*e**(4*c + 4*d*x)*a*b + 28*e**(4*c + 4*d*x)*b**2 - 384*e* 
*(2*c + 2*d*x)*a*b - 8*e**(2*c + 2*d*x)*b**2 + 48*a*b + b**2),x)*a**3*b*d 
- 13824*e**(10*c + 6*d*x)*int(e**(4*d*x)/(48*e**(16*c + 16*d*x)*a*b + e**( 
16*c + 16*d*x)*b**2 - 384*e**(14*c + 14*d*x)*a*b - 8*e**(14*c + 14*d*x)...