\(\int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^3} \, dx\) [463]

Optimal result

Integrand size = 16, antiderivative size = 104 \[ \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^3} \, dx=-\frac {b \left (3 a^2+b^2\right ) x}{\left (a^2-b^2\right )^3}-\frac {a}{2 \left (a^2-b^2\right ) (b+a \coth (x))^2}+\frac {2 a b}{\left (a^2-b^2\right )^2 (b+a \coth (x))}+\frac {a \left (a^2+3 b^2\right ) \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3} \] Output:

-b*(3*a^2+b^2)*x/(a^2-b^2)^3-1/2*a/(a^2-b^2)/(b+a*coth(x))^2+2*a*b/(a^2-b^ 
2)^2/(b+a*coth(x))+a*(a^2+3*b^2)*ln(a*cosh(x)+b*sinh(x))/(a^2-b^2)^3
 

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.12 \[ \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^3} \, dx=-\frac {b \left (3 a^2+b^2\right ) x}{(a-b)^3 (a+b)^3}+\frac {\left (a^3+3 a b^2\right ) \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}+\frac {a^3}{2 (a-b)^2 (a+b)^2 (a \cosh (x)+b \sinh (x))^2}+\frac {3 a b \sinh (x)}{(a-b)^2 (a+b)^2 (a \cosh (x)+b \sinh (x))} \] Input:

Integrate[Sinh[x]^3/(a*Cosh[x] + b*Sinh[x])^3,x]
 

Output:

-((b*(3*a^2 + b^2)*x)/((a - b)^3*(a + b)^3)) + ((a^3 + 3*a*b^2)*Log[a*Cosh 
[x] + b*Sinh[x]])/(a^2 - b^2)^3 + a^3/(2*(a - b)^2*(a + b)^2*(a*Cosh[x] + 
b*Sinh[x])^2) + (3*a*b*Sinh[x])/((a - b)^2*(a + b)^2*(a*Cosh[x] + b*Sinh[x 
]))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.23 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.35, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {3042, 26, 3564, 3042, 3964, 26, 3042, 4012, 3042, 4014, 26, 3042, 4013}

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.

\(\displaystyle \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {i \sin (i x)^3}{(a \cos (i x)-i b \sin (i x))^3}dx\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \frac {\sin (i x)^3}{(a \cos (i x)-i b \sin (i x))^3}dx\)

\(\Big \downarrow \) 3564

\(\displaystyle i \int \frac {1}{(-i b-i a \coth (x))^3}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle i \int \frac {1}{\left (-i b-a \tan \left (i x+\frac {\pi }{2}\right )\right )^3}dx\)

\(\Big \downarrow \) 3964

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4012

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4014

\(\displaystyle i \left (\frac {i \left (-\frac {2 a b}{\left (a^2-b^2\right ) (a \coth (x)+b)}-\frac {-\frac {b x \left (3 a^2+b^2\right )}{a^2-b^2}+\frac {i a \left (a^2+3 b^2\right ) \int -\frac {i (a+b \coth (x))}{b+a \coth (x)}dx}{a^2-b^2}}{a^2-b^2}\right )}{a^2-b^2}+\frac {i a}{2 \left (a^2-b^2\right ) (a \coth (x)+b)^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (\frac {i \left (-\frac {\frac {a \left (a^2+3 b^2\right ) \int \frac {a+b \coth (x)}{b+a \coth (x)}dx}{a^2-b^2}-\frac {b x \left (3 a^2+b^2\right )}{a^2-b^2}}{a^2-b^2}-\frac {2 a b}{\left (a^2-b^2\right ) (a \coth (x)+b)}\right )}{a^2-b^2}+\frac {i a}{2 \left (a^2-b^2\right ) (a \coth (x)+b)^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle i \left (\frac {i \left (-\frac {2 a b}{\left (a^2-b^2\right ) (a \coth (x)+b)}-\frac {-\frac {b x \left (3 a^2+b^2\right )}{a^2-b^2}+\frac {a \left (a^2+3 b^2\right ) \int \frac {a-i b \tan \left (i x+\frac {\pi }{2}\right )}{b-i a \tan \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}}{a^2-b^2}\right )}{a^2-b^2}+\frac {i a}{2 \left (a^2-b^2\right ) (a \coth (x)+b)^2}\right )\)

\(\Big \downarrow \) 4013

\(\displaystyle i \left (\frac {i a}{2 \left (a^2-b^2\right ) (a \coth (x)+b)^2}+\frac {i \left (-\frac {2 a b}{\left (a^2-b^2\right ) (a \coth (x)+b)}-\frac {\frac {a \left (a^2+3 b^2\right ) \log (a \cosh (x)+b \sinh (x))}{a^2-b^2}-\frac {b x \left (3 a^2+b^2\right )}{a^2-b^2}}{a^2-b^2}\right )}{a^2-b^2}\right )\)

Input:

Int[Sinh[x]^3/(a*Cosh[x] + b*Sinh[x])^3,x]
 

Output:

I*(((I/2)*a)/((a^2 - b^2)*(b + a*Coth[x])^2) + (I*((-2*a*b)/((a^2 - b^2)*( 
b + a*Coth[x])) - (-((b*(3*a^2 + b^2)*x)/(a^2 - b^2)) + (a*(a^2 + 3*b^2)*L 
og[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2))/(a^2 - b^2)))/(a^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 3042

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

rule 3564

Int[sin[(c_.) + (d_.)*(x_)]^(m_)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin 
[(c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Int[(b + a*Cot[c + d*x])^n, x] /; 
 FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && NeQ[a^2 + b^2, 0 
]
 

rule 3964

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + 
b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) 
 Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, 
 b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
 

rule 4012

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ 
(f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2)   Int[(a + b*Tan[e + f*x] 
)^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a 
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 
]
 

rule 4013

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* 
(x_)]), x_Symbol] :> Simp[(c/(b*f))*Log[RemoveContent[a*Cos[e + f*x] + b*Si 
n[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && 
 NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
 

rule 4014

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. 
)*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a 
*d)/(a^2 + b^2)   Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] /; 
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && N 
eQ[a*c + b*d, 0]
 

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.54

method	result	size
default	\(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{\left (a +b \right )^{3}}+\frac {2 a \left (\frac {2 b a \left (a^{2}-b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (-a^{4}+6 b^{2} a^{2}-5 b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{2}+2 b a \left (a^{2}-b^{2}\right ) \tanh \left (\frac {x}{2}\right )}{\left (a +2 b \tanh \left (\frac {x}{2}\right )+a \tanh \left (\frac {x}{2}\right )^{2}\right )^{2}}+\frac {\left (a^{2}+3 b^{2}\right ) \ln \left (a +2 b \tanh \left (\frac {x}{2}\right )+a \tanh \left (\frac {x}{2}\right )^{2}\right )}{2}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}-\frac {\ln \left (1+\tanh \left (\frac {x}{2}\right )\right )}{\left (a -b \right )^{3}}\)	\(160\)
parallelrisch	\(\frac {2 a \,b^{2} \left (a^{2}+3 b^{2}\right ) \left (a +b \tanh \left (x \right )\right )^{2} \ln \left (a +b \tanh \left (x \right )\right )-2 a \,b^{2} \left (a^{2}+3 b^{2}\right ) \left (a +b \tanh \left (x \right )\right )^{2} \ln \left (1-\tanh \left (x \right )\right )+\left (-2 x \,b^{4} \left (a +b \right )^{2} \tanh \left (x \right )^{2}+2 \left (-2 b^{4} x +\left (-4 x +3\right ) a \,b^{3}-2 a^{2} \left (x +\frac {3}{2}\right ) b^{2}-a^{3} b +a^{4}\right ) a b \tanh \left (x \right )+\left (-2 b^{4} x +\left (-4 x +5\right ) a \,b^{3}-2 \left (x +\frac {5}{2}\right ) b^{2} a^{2}-a^{3} b +a^{4}\right ) a^{2}\right ) \left (a +b \right )}{2 \left (a -b \right )^{3} \left (a +b \right )^{3} b^{2} \left (a +b \tanh \left (x \right )\right )^{2}}\)	\(191\)
risch	\(\frac {x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {2 a^{3} x}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}-\frac {6 a x \,b^{2}}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}+\frac {2 a^{2} \left (a^{2} {\mathrm e}^{2 x}-2 a b \,{\mathrm e}^{2 x}-3 b^{2} {\mathrm e}^{2 x}-3 a b +3 b^{2}\right )}{\left (a -b \right )^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left ({\mathrm e}^{2 x} a +b \,{\mathrm e}^{2 x}+a -b \right )^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}+\frac {3 a \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right ) b^{2}}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}\)	\(277\)

Input:

int(sinh(x)^3/(a*cosh(x)+b*sinh(x))^3,x,method=_RETURNVERBOSE)
 

Output:

-1/(a+b)^3*ln(tanh(1/2*x)-1)+2*a/(a-b)^3/(a+b)^3*((2*b*a*(a^2-b^2)*tanh(1/ 
2*x)^3+(-a^4+6*a^2*b^2-5*b^4)*tanh(1/2*x)^2+2*b*a*(a^2-b^2)*tanh(1/2*x))/( 
a+2*b*tanh(1/2*x)+a*tanh(1/2*x)^2)^2+1/2*(a^2+3*b^2)*ln(a+2*b*tanh(1/2*x)+ 
a*tanh(1/2*x)^2))-1/(a-b)^3*ln(1+tanh(1/2*x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1268 vs. \(2 (102) = 204\).

Time = 0.12 (sec) , antiderivative size = 1268, normalized size of antiderivative = 12.19 \[ \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^3} \, dx=\text {Too large to display} \] Input:

integrate(sinh(x)^3/(a*cosh(x)+b*sinh(x))^3,x, algorithm="fricas")
 

Output:

-((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*x*cosh(x)^4 + 
4*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*x*cosh(x)*sinh 
(x)^3 + (a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*x*sinh(x 
)^4 + 6*a^4*b - 12*a^3*b^2 + 6*a^2*b^3 - 2*(a^5 - 3*a^4*b - a^3*b^2 + 3*a^ 
2*b^3 - (a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*x)*cosh(x) 
^2 - 2*(a^5 - 3*a^4*b - a^3*b^2 + 3*a^2*b^3 - 3*(a^5 + 5*a^4*b + 10*a^3*b^ 
2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*x*cosh(x)^2 - (a^5 + 3*a^4*b + 2*a^3*b^2 - 
 2*a^2*b^3 - 3*a*b^4 - b^5)*x)*sinh(x)^2 + (a^5 + a^4*b - 2*a^3*b^2 - 2*a^ 
2*b^3 + a*b^4 + b^5)*x - (a^5 - 2*a^4*b + 4*a^3*b^2 - 6*a^2*b^3 + 3*a*b^4 
+ (a^5 + 2*a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 3*a*b^4)*cosh(x)^4 + 4*(a^5 + 2 
*a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 3*a*b^4)*cosh(x)*sinh(x)^3 + (a^5 + 2*a^4 
*b + 4*a^3*b^2 + 6*a^2*b^3 + 3*a*b^4)*sinh(x)^4 + 2*(a^5 + 2*a^3*b^2 - 3*a 
*b^4)*cosh(x)^2 + 2*(a^5 + 2*a^3*b^2 - 3*a*b^4 + 3*(a^5 + 2*a^4*b + 4*a^3* 
b^2 + 6*a^2*b^3 + 3*a*b^4)*cosh(x)^2)*sinh(x)^2 + 4*((a^5 + 2*a^4*b + 4*a^ 
3*b^2 + 6*a^2*b^3 + 3*a*b^4)*cosh(x)^3 + (a^5 + 2*a^3*b^2 - 3*a*b^4)*cosh( 
x))*sinh(x))*log(2*(a*cosh(x) + b*sinh(x))/(cosh(x) - sinh(x))) + 4*((a^5 
+ 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*x*cosh(x)^3 - (a^5 - 
3*a^4*b - a^3*b^2 + 3*a^2*b^3 - (a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3 
*a*b^4 - b^5)*x)*cosh(x))*sinh(x))/(a^8 - 2*a^7*b - 2*a^6*b^2 + 6*a^5*b^3 
- 6*a^3*b^5 + 2*a^2*b^6 + 2*a*b^7 - b^8 + (a^8 + 2*a^7*b - 2*a^6*b^2 - ...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3825 vs. \(2 (88) = 176\).

Time = 1.66 (sec) , antiderivative size = 3825, normalized size of antiderivative = 36.78 \[ \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^3} \, dx=\text {Too large to display} \] Input:

integrate(sinh(x)**3/(a*cosh(x)+b*sinh(x))**3,x)
 

Output:

Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0)), ((log(cosh(x)) - sinh(x)**2/(2*cos 
h(x)**2))/a**3, Eq(b, 0)), (-3*x*sinh(x)**3/(-24*b**3*sinh(x)**3 + 72*b**3 
*sinh(x)**2*cosh(x) - 72*b**3*sinh(x)*cosh(x)**2 + 24*b**3*cosh(x)**3) + 9 
*x*sinh(x)**2*cosh(x)/(-24*b**3*sinh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) - 
72*b**3*sinh(x)*cosh(x)**2 + 24*b**3*cosh(x)**3) - 9*x*sinh(x)*cosh(x)**2/ 
(-24*b**3*sinh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) - 72*b**3*sinh(x)*cosh(x 
)**2 + 24*b**3*cosh(x)**3) + 3*x*cosh(x)**3/(-24*b**3*sinh(x)**3 + 72*b**3 
*sinh(x)**2*cosh(x) - 72*b**3*sinh(x)*cosh(x)**2 + 24*b**3*cosh(x)**3) - 9 
*sinh(x)**3/(-24*b**3*sinh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) - 72*b**3*si 
nh(x)*cosh(x)**2 + 24*b**3*cosh(x)**3) + 6*sinh(x)**2*cosh(x)/(-24*b**3*si 
nh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) - 72*b**3*sinh(x)*cosh(x)**2 + 24*b* 
*3*cosh(x)**3) - cosh(x)**3/(-24*b**3*sinh(x)**3 + 72*b**3*sinh(x)**2*cosh 
(x) - 72*b**3*sinh(x)*cosh(x)**2 + 24*b**3*cosh(x)**3), Eq(a, -b)), (3*x*s 
inh(x)**3/(24*b**3*sinh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) + 72*b**3*sinh( 
x)*cosh(x)**2 + 24*b**3*cosh(x)**3) + 9*x*sinh(x)**2*cosh(x)/(24*b**3*sinh 
(x)**3 + 72*b**3*sinh(x)**2*cosh(x) + 72*b**3*sinh(x)*cosh(x)**2 + 24*b**3 
*cosh(x)**3) + 9*x*sinh(x)*cosh(x)**2/(24*b**3*sinh(x)**3 + 72*b**3*sinh(x 
)**2*cosh(x) + 72*b**3*sinh(x)*cosh(x)**2 + 24*b**3*cosh(x)**3) + 3*x*cosh 
(x)**3/(24*b**3*sinh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) + 72*b**3*sinh(x)* 
cosh(x)**2 + 24*b**3*cosh(x)**3) - 9*sinh(x)**3/(24*b**3*sinh(x)**3 + 7...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (102) = 204\).

Time = 0.06 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.78 \[ \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^3} \, dx=\frac {{\left (a^{3} + 3 \, a b^{2}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {2 \, {\left (3 \, a^{3} b + 3 \, a^{2} b^{2} + {\left (a^{4} + 2 \, a^{3} b - 3 \, a^{2} b^{2}\right )} e^{\left (-2 \, x\right )}\right )}}{a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6} - b^{7} + 2 \, {\left (a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - 3 \, a^{2} b^{5} - a b^{6} + b^{7}\right )} e^{\left (-2 \, x\right )} + {\left (a^{7} - 3 \, a^{6} b + a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{3} b^{4} - a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right )} e^{\left (-4 \, x\right )}} + \frac {x}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} \] Input:

integrate(sinh(x)^3/(a*cosh(x)+b*sinh(x))^3,x, algorithm="maxima")
 

Output:

(a^3 + 3*a*b^2)*log(-(a - b)*e^(-2*x) - a - b)/(a^6 - 3*a^4*b^2 + 3*a^2*b^ 
4 - b^6) + 2*(3*a^3*b + 3*a^2*b^2 + (a^4 + 2*a^3*b - 3*a^2*b^2)*e^(-2*x))/ 
(a^7 + a^6*b - 3*a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 + 3*a^2*b^5 - a*b^6 - b^7 
 + 2*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 
+ b^7)*e^(-2*x) + (a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b 
^5 + 3*a*b^6 - b^7)*e^(-4*x)) + x/(a^3 + 3*a^2*b + 3*a*b^2 + b^3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (102) = 204\).

Time = 0.14 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.41 \[ \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^3} \, dx=\frac {{\left (a^{3} + 3 \, a b^{2}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac {x}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {3 \, a^{4} e^{\left (4 \, x\right )} + 3 \, a^{3} b e^{\left (4 \, x\right )} + 9 \, a^{2} b^{2} e^{\left (4 \, x\right )} + 9 \, a b^{3} e^{\left (4 \, x\right )} + 2 \, a^{4} e^{\left (2 \, x\right )} + 10 \, a^{3} b e^{\left (2 \, x\right )} + 6 \, a^{2} b^{2} e^{\left (2 \, x\right )} - 18 \, a b^{3} e^{\left (2 \, x\right )} + 3 \, a^{4} + 3 \, a^{3} b - 15 \, a^{2} b^{2} + 9 \, a b^{3}}{2 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}^{2}} \] Input:

integrate(sinh(x)^3/(a*cosh(x)+b*sinh(x))^3,x, algorithm="giac")
 

Output:

(a^3 + 3*a*b^2)*log(abs(a*e^(2*x) + b*e^(2*x) + a - b))/(a^6 - 3*a^4*b^2 + 
 3*a^2*b^4 - b^6) - x/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) - 1/2*(3*a^4*e^(4*x) 
 + 3*a^3*b*e^(4*x) + 9*a^2*b^2*e^(4*x) + 9*a*b^3*e^(4*x) + 2*a^4*e^(2*x) + 
 10*a^3*b*e^(2*x) + 6*a^2*b^2*e^(2*x) - 18*a*b^3*e^(2*x) + 3*a^4 + 3*a^3*b 
 - 15*a^2*b^2 + 9*a*b^3)/((a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b 
^5)*(a*e^(2*x) + b*e^(2*x) + a - b)^2)
 

Mupad [B] (verification not implemented)

Time = 0.98 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.53 \[ \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^3} \, dx=\frac {\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,\left (a^3+3\,a\,b^2\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}-\frac {x}{{\left (a-b\right )}^3}-\frac {2\,\left (3\,a^2\,b-a^3\right )}{{\left (a+b\right )}^3\,{\left (a-b\right )}^2\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )}-\frac {2\,a^3}{{\left (a+b\right )}^3\,\left (a-b\right )\,\left ({\mathrm {e}}^{4\,x}\,{\left (a+b\right )}^2+{\left (a-b\right )}^2+2\,{\mathrm {e}}^{2\,x}\,\left (a+b\right )\,\left (a-b\right )\right )} \] Input:

int(sinh(x)^3/(a*cosh(x) + b*sinh(x))^3,x)
 

Output:

(log(a - b + a*exp(2*x) + b*exp(2*x))*(3*a*b^2 + a^3))/(a^6 - b^6 + 3*a^2* 
b^4 - 3*a^4*b^2) - x/(a - b)^3 - (2*(3*a^2*b - a^3))/((a + b)^3*(a - b)^2* 
(a - b + exp(2*x)*(a + b))) - (2*a^3)/((a + b)^3*(a - b)*(exp(4*x)*(a + b) 
^2 + (a - b)^2 + 2*exp(2*x)*(a + b)*(a - b)))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 387, normalized size of antiderivative = 3.72 \[ \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^3} \, dx=\frac {2 \cosh \left (x \right )^{2} \mathrm {log}\left (\cosh \left (x \right ) a +\sinh \left (x \right ) b \right ) a^{5}+6 \cosh \left (x \right )^{2} \mathrm {log}\left (\cosh \left (x \right ) a +\sinh \left (x \right ) b \right ) a^{3} b^{2}-2 \cosh \left (x \right )^{2} a^{5}-6 \cosh \left (x \right )^{2} a^{4} b x +2 \cosh \left (x \right )^{2} a^{3} b^{2}-2 \cosh \left (x \right )^{2} a^{2} b^{3} x +4 \cosh \left (x \right ) \mathrm {log}\left (\cosh \left (x \right ) a +\sinh \left (x \right ) b \right ) \sinh \left (x \right ) a^{4} b +12 \cosh \left (x \right ) \mathrm {log}\left (\cosh \left (x \right ) a +\sinh \left (x \right ) b \right ) \sinh \left (x \right ) a^{2} b^{3}-12 \cosh \left (x \right ) \sinh \left (x \right ) a^{3} b^{2} x -4 \cosh \left (x \right ) \sinh \left (x \right ) a \,b^{4} x +2 \,\mathrm {log}\left (\cosh \left (x \right ) a +\sinh \left (x \right ) b \right ) \sinh \left (x \right )^{2} a^{3} b^{2}+6 \,\mathrm {log}\left (\cosh \left (x \right ) a +\sinh \left (x \right ) b \right ) \sinh \left (x \right )^{2} a \,b^{4}-\sinh \left (x \right )^{2} a^{5}+4 \sinh \left (x \right )^{2} a^{3} b^{2}-6 \sinh \left (x \right )^{2} a^{2} b^{3} x -3 \sinh \left (x \right )^{2} a \,b^{4}-2 \sinh \left (x \right )^{2} b^{5} x}{2 \cosh \left (x \right )^{2} a^{8}-6 \cosh \left (x \right )^{2} a^{6} b^{2}+6 \cosh \left (x \right )^{2} a^{4} b^{4}-2 \cosh \left (x \right )^{2} a^{2} b^{6}+4 \cosh \left (x \right ) \sinh \left (x \right ) a^{7} b -12 \cosh \left (x \right ) \sinh \left (x \right ) a^{5} b^{3}+12 \cosh \left (x \right ) \sinh \left (x \right ) a^{3} b^{5}-4 \cosh \left (x \right ) \sinh \left (x \right ) a \,b^{7}+2 \sinh \left (x \right )^{2} a^{6} b^{2}-6 \sinh \left (x \right )^{2} a^{4} b^{4}+6 \sinh \left (x \right )^{2} a^{2} b^{6}-2 \sinh \left (x \right )^{2} b^{8}} \] Input:

int(sinh(x)^3/(a*cosh(x)+b*sinh(x))^3,x)
 

Output:

(2*cosh(x)**2*log(cosh(x)*a + sinh(x)*b)*a**5 + 6*cosh(x)**2*log(cosh(x)*a 
 + sinh(x)*b)*a**3*b**2 - 2*cosh(x)**2*a**5 - 6*cosh(x)**2*a**4*b*x + 2*co 
sh(x)**2*a**3*b**2 - 2*cosh(x)**2*a**2*b**3*x + 4*cosh(x)*log(cosh(x)*a + 
sinh(x)*b)*sinh(x)*a**4*b + 12*cosh(x)*log(cosh(x)*a + sinh(x)*b)*sinh(x)* 
a**2*b**3 - 12*cosh(x)*sinh(x)*a**3*b**2*x - 4*cosh(x)*sinh(x)*a*b**4*x + 
2*log(cosh(x)*a + sinh(x)*b)*sinh(x)**2*a**3*b**2 + 6*log(cosh(x)*a + sinh 
(x)*b)*sinh(x)**2*a*b**4 - sinh(x)**2*a**5 + 4*sinh(x)**2*a**3*b**2 - 6*si 
nh(x)**2*a**2*b**3*x - 3*sinh(x)**2*a*b**4 - 2*sinh(x)**2*b**5*x)/(2*(cosh 
(x)**2*a**8 - 3*cosh(x)**2*a**6*b**2 + 3*cosh(x)**2*a**4*b**4 - cosh(x)**2 
*a**2*b**6 + 2*cosh(x)*sinh(x)*a**7*b - 6*cosh(x)*sinh(x)*a**5*b**3 + 6*co 
sh(x)*sinh(x)*a**3*b**5 - 2*cosh(x)*sinh(x)*a*b**7 + sinh(x)**2*a**6*b**2 
- 3*sinh(x)**2*a**4*b**4 + 3*sinh(x)**2*a**2*b**6 - sinh(x)**2*b**8))