Integrand size = 13, antiderivative size = 61 \[ \int \frac {\sinh ^3(x)}{a+b \text {sech}(x)} \, dx=-\frac {\left (a^2-b^2\right ) \cosh (x)}{a^3}-\frac {b \cosh ^2(x)}{2 a^2}+\frac {\cosh ^3(x)}{3 a}+\frac {b \left (a^2-b^2\right ) \log (b+a \cosh (x))}{a^4} \]
Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.08 \[ \int \frac {\sinh ^3(x)}{a+b \text {sech}(x)} \, dx=\frac {\left (-9 a^3+12 a b^2\right ) \cosh (x)-3 a^2 b \cosh (2 x)+a^3 \cosh (3 x)+12 a^2 b \log (b+a \cosh (x))-12 b^3 \log (b+a \cosh (x))}{12 a^4} \]
((-9*a^3 + 12*a*b^2)*Cosh[x] - 3*a^2*b*Cosh[2*x] + a^3*Cosh[3*x] + 12*a^2* b*Log[b + a*Cosh[x]] - 12*b^3*Log[b + a*Cosh[x]])/(12*a^4)
Time = 0.39 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {3042, 26, 4360, 26, 25, 3042, 26, 3316, 27, 522, 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.
\(\displaystyle \int \frac {\sinh ^3(x)}{a+b \text {sech}(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \cos \left (-\frac {\pi }{2}+i x\right )^3}{a-b \csc \left (-\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\cos \left (i x-\frac {\pi }{2}\right )^3}{a-b \csc \left (i x-\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle i \int \frac {i \cosh (x) \sinh ^3(x)}{-b-a \cosh (x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\int -\frac {\cosh (x) \sinh ^3(x)}{b+a \cosh (x)}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\sinh ^3(x) \cosh (x)}{a \cosh (x)+b}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \sin \left (\frac {\pi }{2}+i x\right ) \cos \left (\frac {\pi }{2}+i x\right )^3}{b+a \sin \left (\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\cos \left (i x+\frac {\pi }{2}\right )^3 \sin \left (i x+\frac {\pi }{2}\right )}{b+a \sin \left (i x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3316 |
\(\displaystyle -\frac {\int \frac {\cosh (x) \left (a^2-a^2 \cosh ^2(x)\right )}{b+a \cosh (x)}d(a \cosh (x))}{a^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {a \cosh (x) \left (a^2-a^2 \cosh ^2(x)\right )}{b+a \cosh (x)}d(a \cosh (x))}{a^4}\) |
\(\Big \downarrow \) 522 |
\(\displaystyle -\frac {\int \left (-\cosh ^2(x) a^2+\left (1-\frac {b^2}{a^2}\right ) a^2+b \cosh (x) a+\frac {b^3-a^2 b}{b+a \cosh (x)}\right )d(a \cosh (x))}{a^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {1}{3} a^3 \cosh ^3(x)+a \left (a^2-b^2\right ) \cosh (x)-b \left (a^2-b^2\right ) \log (a \cosh (x)+b)+\frac {1}{2} a^2 b \cosh ^2(x)}{a^4}\) |
-((a*(a^2 - b^2)*Cosh[x] + (a^2*b*Cosh[x]^2)/2 - (a^3*Cosh[x]^3)/3 - b*(a^ 2 - b^2)*Log[b + a*Cosh[x]])/a^4)
3.1.61.3.1 Defintions of rubi rules used
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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(135\) vs. \(2(57)=114\).
Time = 10.41 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.23
method | result | size |
risch | \(-\frac {b x}{a^{2}}+\frac {b^{3} x}{a^{4}}+\frac {{\mathrm e}^{3 x}}{24 a}-\frac {b \,{\mathrm e}^{2 x}}{8 a^{2}}-\frac {3 \,{\mathrm e}^{x}}{8 a}+\frac {{\mathrm e}^{x} b^{2}}{2 a^{3}}-\frac {3 \,{\mathrm e}^{-x}}{8 a}+\frac {{\mathrm e}^{-x} b^{2}}{2 a^{3}}-\frac {b \,{\mathrm e}^{-2 x}}{8 a^{2}}+\frac {{\mathrm e}^{-3 x}}{24 a}+\frac {b \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{a^{2}}-\frac {b^{3} \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{a^{4}}\) | \(136\) |
default | \(-\frac {a^{2}-a b -2 b^{2}}{2 a^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {b \left (a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a^{4}}-\frac {a +b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{3 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {b \left (a^{3}-a^{2} b -a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b +a +b \right )}{a^{4} \left (a -b \right )}-\frac {-a^{2}+a b +2 b^{2}}{2 a^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {1}{3 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a +b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {b \left (a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a^{4}}\) | \(209\) |
-b*x/a^2+1/a^4*b^3*x+1/24/a*exp(3*x)-1/8/a^2*b*exp(2*x)-3/8/a*exp(x)+1/2/a ^3*exp(x)*b^2-3/8/a*exp(-x)+1/2/a^3*exp(-x)*b^2-1/8/a^2*b*exp(-2*x)+1/24/a *exp(-3*x)+1/a^2*b*ln(exp(2*x)+2*b/a*exp(x)+1)-1/a^4*b^3*ln(exp(2*x)+2*b/a *exp(x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (57) = 114\).
Time = 0.26 (sec) , antiderivative size = 490, normalized size of antiderivative = 8.03 \[ \int \frac {\sinh ^3(x)}{a+b \text {sech}(x)} \, dx=\frac {a^{3} \cosh \left (x\right )^{6} + a^{3} \sinh \left (x\right )^{6} - 3 \, a^{2} b \cosh \left (x\right )^{5} + 3 \, {\left (2 \, a^{3} \cosh \left (x\right ) - a^{2} b\right )} \sinh \left (x\right )^{5} - 24 \, {\left (a^{2} b - b^{3}\right )} x \cosh \left (x\right )^{3} - 3 \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} \cosh \left (x\right )^{4} + 3 \, {\left (5 \, a^{3} \cosh \left (x\right )^{2} - 5 \, a^{2} b \cosh \left (x\right ) - 3 \, a^{3} + 4 \, a b^{2}\right )} \sinh \left (x\right )^{4} - 3 \, a^{2} b \cosh \left (x\right ) + 2 \, {\left (10 \, a^{3} \cosh \left (x\right )^{3} - 15 \, a^{2} b \cosh \left (x\right )^{2} - 12 \, {\left (a^{2} b - b^{3}\right )} x - 6 \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + a^{3} - 3 \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} \cosh \left (x\right )^{2} + 3 \, {\left (5 \, a^{3} \cosh \left (x\right )^{4} - 10 \, a^{2} b \cosh \left (x\right )^{3} - 3 \, a^{3} + 4 \, a b^{2} - 24 \, {\left (a^{2} b - b^{3}\right )} x \cosh \left (x\right ) - 6 \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 24 \, {\left ({\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{3} + 3 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (a^{2} b - b^{3}\right )} \sinh \left (x\right )^{3}\right )} \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 3 \, {\left (2 \, a^{3} \cosh \left (x\right )^{5} - 5 \, a^{2} b \cosh \left (x\right )^{4} - 24 \, {\left (a^{2} b - b^{3}\right )} x \cosh \left (x\right )^{2} - 4 \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} \cosh \left (x\right )^{3} - a^{2} b - 2 \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{24 \, {\left (a^{4} \cosh \left (x\right )^{3} + 3 \, a^{4} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, a^{4} \cosh \left (x\right ) \sinh \left (x\right )^{2} + a^{4} \sinh \left (x\right )^{3}\right )}} \]
1/24*(a^3*cosh(x)^6 + a^3*sinh(x)^6 - 3*a^2*b*cosh(x)^5 + 3*(2*a^3*cosh(x) - a^2*b)*sinh(x)^5 - 24*(a^2*b - b^3)*x*cosh(x)^3 - 3*(3*a^3 - 4*a*b^2)*c osh(x)^4 + 3*(5*a^3*cosh(x)^2 - 5*a^2*b*cosh(x) - 3*a^3 + 4*a*b^2)*sinh(x) ^4 - 3*a^2*b*cosh(x) + 2*(10*a^3*cosh(x)^3 - 15*a^2*b*cosh(x)^2 - 12*(a^2* b - b^3)*x - 6*(3*a^3 - 4*a*b^2)*cosh(x))*sinh(x)^3 + a^3 - 3*(3*a^3 - 4*a *b^2)*cosh(x)^2 + 3*(5*a^3*cosh(x)^4 - 10*a^2*b*cosh(x)^3 - 3*a^3 + 4*a*b^ 2 - 24*(a^2*b - b^3)*x*cosh(x) - 6*(3*a^3 - 4*a*b^2)*cosh(x)^2)*sinh(x)^2 + 24*((a^2*b - b^3)*cosh(x)^3 + 3*(a^2*b - b^3)*cosh(x)^2*sinh(x) + 3*(a^2 *b - b^3)*cosh(x)*sinh(x)^2 + (a^2*b - b^3)*sinh(x)^3)*log(2*(a*cosh(x) + b)/(cosh(x) - sinh(x))) + 3*(2*a^3*cosh(x)^5 - 5*a^2*b*cosh(x)^4 - 24*(a^2 *b - b^3)*x*cosh(x)^2 - 4*(3*a^3 - 4*a*b^2)*cosh(x)^3 - a^2*b - 2*(3*a^3 - 4*a*b^2)*cosh(x))*sinh(x))/(a^4*cosh(x)^3 + 3*a^4*cosh(x)^2*sinh(x) + 3*a ^4*cosh(x)*sinh(x)^2 + a^4*sinh(x)^3)
\[ \int \frac {\sinh ^3(x)}{a+b \text {sech}(x)} \, dx=\int \frac {\sinh ^{3}{\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (57) = 114\).
Time = 0.20 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.10 \[ \int \frac {\sinh ^3(x)}{a+b \text {sech}(x)} \, dx=-\frac {{\left (3 \, a b e^{\left (-x\right )} - a^{2} + 3 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} e^{\left (-2 \, x\right )}\right )} e^{\left (3 \, x\right )}}{24 \, a^{3}} - \frac {3 \, a b e^{\left (-2 \, x\right )} - a^{2} e^{\left (-3 \, x\right )} + 3 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} e^{\left (-x\right )}}{24 \, a^{3}} + \frac {{\left (a^{2} b - b^{3}\right )} x}{a^{4}} + \frac {{\left (a^{2} b - b^{3}\right )} \log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a^{4}} \]
-1/24*(3*a*b*e^(-x) - a^2 + 3*(3*a^2 - 4*b^2)*e^(-2*x))*e^(3*x)/a^3 - 1/24 *(3*a*b*e^(-2*x) - a^2*e^(-3*x) + 3*(3*a^2 - 4*b^2)*e^(-x))/a^3 + (a^2*b - b^3)*x/a^4 + (a^2*b - b^3)*log(2*b*e^(-x) + a*e^(-2*x) + a)/a^4
Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.43 \[ \int \frac {\sinh ^3(x)}{a+b \text {sech}(x)} \, dx=\frac {a^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 3 \, a b {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 12 \, a^{2} {\left (e^{\left (-x\right )} + e^{x}\right )} + 12 \, b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}}{24 \, a^{3}} + \frac {{\left (a^{2} b - b^{3}\right )} \log \left ({\left | a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a^{4}} \]
1/24*(a^2*(e^(-x) + e^x)^3 - 3*a*b*(e^(-x) + e^x)^2 - 12*a^2*(e^(-x) + e^x ) + 12*b^2*(e^(-x) + e^x))/a^3 + (a^2*b - b^3)*log(abs(a*(e^(-x) + e^x) + 2*b))/a^4
Time = 2.22 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.02 \[ \int \frac {\sinh ^3(x)}{a+b \text {sech}(x)} \, dx=\frac {{\mathrm {e}}^{-3\,x}}{24\,a}+\frac {{\mathrm {e}}^{3\,x}}{24\,a}-\frac {x\,\left (a^2\,b-b^3\right )}{a^4}-\frac {{\mathrm {e}}^x\,\left (3\,a^2-4\,b^2\right )}{8\,a^3}-\frac {b\,{\mathrm {e}}^{-2\,x}}{8\,a^2}-\frac {b\,{\mathrm {e}}^{2\,x}}{8\,a^2}+\frac {\ln \left (a+2\,b\,{\mathrm {e}}^x+a\,{\mathrm {e}}^{2\,x}\right )\,\left (a^2\,b-b^3\right )}{a^4}-\frac {{\mathrm {e}}^{-x}\,\left (3\,a^2-4\,b^2\right )}{8\,a^3} \]