Integrand size = 13, antiderivative size = 55 \[ \int \frac {\text {csch}^3(x)}{a+a \cosh (x)} \, dx=\frac {3 \text {arctanh}(\cosh (x))}{8 a}+\frac {1}{8 (a-a \cosh (x))}-\frac {1}{4 (a+a \cosh (x))}-\frac {a^3}{8 \left (a^2+a^2 \cosh (x)\right )^2} \] Output:
3/8*arctanh(cosh(x))/a+1/(8*a-8*a*cosh(x))-1/(4*a+4*a*cosh(x))-1/8*a^3/(a^ 2+a^2*cosh(x))^2
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.09 \[ \int \frac {\text {csch}^3(x)}{a+a \cosh (x)} \, dx=-\frac {4+2 \coth ^2\left (\frac {x}{2}\right )-12 \cosh ^2\left (\frac {x}{2}\right ) \left (\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )+\text {sech}^2\left (\frac {x}{2}\right )}{16 a (1+\cosh (x))} \] Input:
Integrate[Csch[x]^3/(a + a*Cosh[x]),x]
Output:
-1/16*(4 + 2*Coth[x/2]^2 - 12*Cosh[x/2]^2*(Log[Cosh[x/2]] - Log[Sinh[x/2]] ) + Sech[x/2]^2)/(a*(1 + Cosh[x]))
Time = 0.32 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 26, 3146, 54, 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.
\(\displaystyle \int \frac {\text {csch}^3(x)}{a \cosh (x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i}{\cos \left (-\frac {\pi }{2}+i x\right )^3 \left (a-a \sin \left (-\frac {\pi }{2}+i x\right )\right )}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {1}{\cos \left (i x-\frac {\pi }{2}\right )^3 \left (a-a \sin \left (i x-\frac {\pi }{2}\right )\right )}dx\) |
\(\Big \downarrow \) 3146 |
\(\displaystyle a^3 \int \frac {1}{(a-a \cosh (x))^2 (\cosh (x) a+a)^3}d(a \cosh (x))\) |
\(\Big \downarrow \) 54 |
\(\displaystyle a^3 \int \left (\frac {1}{8 a^3 (a-a \cosh (x))^2}+\frac {1}{4 a^3 (\cosh (x) a+a)^2}+\frac {1}{4 a^2 (\cosh (x) a+a)^3}+\frac {3}{8 a^3 \left (a^2-a^2 \cosh ^2(x)\right )}\right )d(a \cosh (x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^3 \left (\frac {3 \text {arctanh}(\cosh (x))}{8 a^4}+\frac {1}{8 a^3 (a-a \cosh (x))}-\frac {1}{4 a^3 (a \cosh (x)+a)}-\frac {1}{8 a^2 (a \cosh (x)+a)^2}\right )\) |
Input:
Int[Csch[x]^3/(a + a*Cosh[x]),x]
Output:
a^3*((3*ArcTanh[Cosh[x]])/(8*a^4) + 1/(8*a^3*(a - a*Cosh[x])) - 1/(8*a^2*( a + a*Cosh[x])^2) - 1/(4*a^3*(a + a*Cosh[x])))
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x )^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
Time = 2.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.69
method | result | size |
default | \(\frac {-\frac {\tanh \left (\frac {x}{2}\right )^{4}}{4}+\frac {3 \tanh \left (\frac {x}{2}\right )^{2}}{2}-\frac {1}{2 \tanh \left (\frac {x}{2}\right )^{2}}-3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8 a}\) | \(38\) |
risch | \(-\frac {{\mathrm e}^{x} \left (3 \,{\mathrm e}^{4 x}+6 \,{\mathrm e}^{3 x}-2 \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{x}+3\right )}{4 \left ({\mathrm e}^{x}+1\right )^{4} a \left ({\mathrm e}^{x}-1\right )^{2}}-\frac {3 \ln \left ({\mathrm e}^{x}-1\right )}{8 a}+\frac {3 \ln \left ({\mathrm e}^{x}+1\right )}{8 a}\) | \(65\) |
Input:
int(csch(x)^3/(a+cosh(x)*a),x,method=_RETURNVERBOSE)
Output:
1/8/a*(-1/4*tanh(1/2*x)^4+3/2*tanh(1/2*x)^2-1/2/tanh(1/2*x)^2-3*ln(tanh(1/ 2*x)))
Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (48) = 96\).
Time = 0.08 (sec) , antiderivative size = 631, normalized size of antiderivative = 11.47 \[ \int \frac {\text {csch}^3(x)}{a+a \cosh (x)} \, dx=\text {Too large to display} \] Input:
integrate(csch(x)^3/(a+a*cosh(x)),x, algorithm="fricas")
Output:
-1/8*(6*cosh(x)^5 + 6*(5*cosh(x) + 2)*sinh(x)^4 + 6*sinh(x)^5 + 12*cosh(x) ^4 + 4*(15*cosh(x)^2 + 12*cosh(x) - 1)*sinh(x)^3 - 4*cosh(x)^3 + 12*(5*cos h(x)^3 + 6*cosh(x)^2 - cosh(x) + 1)*sinh(x)^2 + 12*cosh(x)^2 - 3*(cosh(x)^ 6 + 2*(3*cosh(x) + 1)*sinh(x)^5 + sinh(x)^6 + 2*cosh(x)^5 + (15*cosh(x)^2 + 10*cosh(x) - 1)*sinh(x)^4 - cosh(x)^4 + 4*(5*cosh(x)^3 + 5*cosh(x)^2 - c osh(x) - 1)*sinh(x)^3 - 4*cosh(x)^3 + (15*cosh(x)^4 + 20*cosh(x)^3 - 6*cos h(x)^2 - 12*cosh(x) - 1)*sinh(x)^2 - cosh(x)^2 + 2*(3*cosh(x)^5 + 5*cosh(x )^4 - 2*cosh(x)^3 - 6*cosh(x)^2 - cosh(x) + 1)*sinh(x) + 2*cosh(x) + 1)*lo g(cosh(x) + sinh(x) + 1) + 3*(cosh(x)^6 + 2*(3*cosh(x) + 1)*sinh(x)^5 + si nh(x)^6 + 2*cosh(x)^5 + (15*cosh(x)^2 + 10*cosh(x) - 1)*sinh(x)^4 - cosh(x )^4 + 4*(5*cosh(x)^3 + 5*cosh(x)^2 - cosh(x) - 1)*sinh(x)^3 - 4*cosh(x)^3 + (15*cosh(x)^4 + 20*cosh(x)^3 - 6*cosh(x)^2 - 12*cosh(x) - 1)*sinh(x)^2 - cosh(x)^2 + 2*(3*cosh(x)^5 + 5*cosh(x)^4 - 2*cosh(x)^3 - 6*cosh(x)^2 - co sh(x) + 1)*sinh(x) + 2*cosh(x) + 1)*log(cosh(x) + sinh(x) - 1) + 6*(5*cosh (x)^4 + 8*cosh(x)^3 - 2*cosh(x)^2 + 4*cosh(x) + 1)*sinh(x) + 6*cosh(x))/(a *cosh(x)^6 + a*sinh(x)^6 + 2*a*cosh(x)^5 + 2*(3*a*cosh(x) + a)*sinh(x)^5 - a*cosh(x)^4 + (15*a*cosh(x)^2 + 10*a*cosh(x) - a)*sinh(x)^4 - 4*a*cosh(x) ^3 + 4*(5*a*cosh(x)^3 + 5*a*cosh(x)^2 - a*cosh(x) - a)*sinh(x)^3 - a*cosh( x)^2 + (15*a*cosh(x)^4 + 20*a*cosh(x)^3 - 6*a*cosh(x)^2 - 12*a*cosh(x) - a )*sinh(x)^2 + 2*a*cosh(x) + 2*(3*a*cosh(x)^5 + 5*a*cosh(x)^4 - 2*a*cosh...
\[ \int \frac {\text {csch}^3(x)}{a+a \cosh (x)} \, dx=\frac {\int \frac {\operatorname {csch}^{3}{\left (x \right )}}{\cosh {\left (x \right )} + 1}\, dx}{a} \] Input:
integrate(csch(x)**3/(a+a*cosh(x)),x)
Output:
Integral(csch(x)**3/(cosh(x) + 1), x)/a
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (48) = 96\).
Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.87 \[ \int \frac {\text {csch}^3(x)}{a+a \cosh (x)} \, dx=-\frac {3 \, e^{\left (-x\right )} + 6 \, e^{\left (-2 \, x\right )} - 2 \, e^{\left (-3 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 3 \, e^{\left (-5 \, x\right )}}{4 \, {\left (2 \, a e^{\left (-x\right )} - a e^{\left (-2 \, x\right )} - 4 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + 2 \, a e^{\left (-5 \, x\right )} + a e^{\left (-6 \, x\right )} + a\right )}} + \frac {3 \, \log \left (e^{\left (-x\right )} + 1\right )}{8 \, a} - \frac {3 \, \log \left (e^{\left (-x\right )} - 1\right )}{8 \, a} \] Input:
integrate(csch(x)^3/(a+a*cosh(x)),x, algorithm="maxima")
Output:
-1/4*(3*e^(-x) + 6*e^(-2*x) - 2*e^(-3*x) + 6*e^(-4*x) + 3*e^(-5*x))/(2*a*e ^(-x) - a*e^(-2*x) - 4*a*e^(-3*x) - a*e^(-4*x) + 2*a*e^(-5*x) + a*e^(-6*x) + a) + 3/8*log(e^(-x) + 1)/a - 3/8*log(e^(-x) - 1)/a
Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.71 \[ \int \frac {\text {csch}^3(x)}{a+a \cosh (x)} \, dx=\frac {3 \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, a} - \frac {3 \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, a} + \frac {3 \, e^{\left (-x\right )} + 3 \, e^{x} - 10}{16 \, a {\left (e^{\left (-x\right )} + e^{x} - 2\right )}} - \frac {9 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 52 \, e^{\left (-x\right )} + 52 \, e^{x} + 84}{32 \, a {\left (e^{\left (-x\right )} + e^{x} + 2\right )}^{2}} \] Input:
integrate(csch(x)^3/(a+a*cosh(x)),x, algorithm="giac")
Output:
3/16*log(e^(-x) + e^x + 2)/a - 3/16*log(e^(-x) + e^x - 2)/a + 1/16*(3*e^(- x) + 3*e^x - 10)/(a*(e^(-x) + e^x - 2)) - 1/32*(9*(e^(-x) + e^x)^2 + 52*e^ (-x) + 52*e^x + 84)/(a*(e^(-x) + e^x + 2)^2)
Time = 2.02 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.07 \[ \int \frac {\text {csch}^3(x)}{a+a \cosh (x)} \, dx=\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {-a^2}}{a}\right )}{4\,\sqrt {-a^2}}-\frac {1}{2\,a\,\left (6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^x-1\right )}-\frac {1}{2\,a\,\left ({\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {1}{a\,\left (3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1\right )} \] Input:
int(1/(sinh(x)^3*(a + a*cosh(x))),x)
Output:
(3*atan((exp(x)*(-a^2)^(1/2))/a))/(4*(-a^2)^(1/2)) - 1/(2*a*(6*exp(2*x) + 4*exp(3*x) + exp(4*x) + 4*exp(x) + 1)) - 1/(4*a*(exp(x) - 1)) - 1/(2*a*(ex p(x) + 1)) - 1/(4*a*(exp(2*x) - 2*exp(x) + 1)) + 1/(a*(3*exp(2*x) + exp(3* x) + 3*exp(x) + 1))
Time = 0.26 (sec) , antiderivative size = 245, normalized size of antiderivative = 4.45 \[ \int \frac {\text {csch}^3(x)}{a+a \cosh (x)} \, dx=\frac {-3 e^{6 x} \mathrm {log}\left (e^{x}-1\right )+3 e^{6 x} \mathrm {log}\left (e^{x}+1\right )+3 e^{6 x}-6 e^{5 x} \mathrm {log}\left (e^{x}-1\right )+6 e^{5 x} \mathrm {log}\left (e^{x}+1\right )+3 e^{4 x} \mathrm {log}\left (e^{x}-1\right )-3 e^{4 x} \mathrm {log}\left (e^{x}+1\right )-15 e^{4 x}+12 e^{3 x} \mathrm {log}\left (e^{x}-1\right )-12 e^{3 x} \mathrm {log}\left (e^{x}+1\right )-8 e^{3 x}+3 e^{2 x} \mathrm {log}\left (e^{x}-1\right )-3 e^{2 x} \mathrm {log}\left (e^{x}+1\right )-15 e^{2 x}-6 e^{x} \mathrm {log}\left (e^{x}-1\right )+6 e^{x} \mathrm {log}\left (e^{x}+1\right )-3 \,\mathrm {log}\left (e^{x}-1\right )+3 \,\mathrm {log}\left (e^{x}+1\right )+3}{8 a \left (e^{6 x}+2 e^{5 x}-e^{4 x}-4 e^{3 x}-e^{2 x}+2 e^{x}+1\right )} \] Input:
int(csch(x)^3/(a+a*cosh(x)),x)
Output:
( - 3*e**(6*x)*log(e**x - 1) + 3*e**(6*x)*log(e**x + 1) + 3*e**(6*x) - 6*e **(5*x)*log(e**x - 1) + 6*e**(5*x)*log(e**x + 1) + 3*e**(4*x)*log(e**x - 1 ) - 3*e**(4*x)*log(e**x + 1) - 15*e**(4*x) + 12*e**(3*x)*log(e**x - 1) - 1 2*e**(3*x)*log(e**x + 1) - 8*e**(3*x) + 3*e**(2*x)*log(e**x - 1) - 3*e**(2 *x)*log(e**x + 1) - 15*e**(2*x) - 6*e**x*log(e**x - 1) + 6*e**x*log(e**x + 1) - 3*log(e**x - 1) + 3*log(e**x + 1) + 3)/(8*a*(e**(6*x) + 2*e**(5*x) - e**(4*x) - 4*e**(3*x) - e**(2*x) + 2*e**x + 1))