Integrand size = 13, antiderivative size = 132 \[ \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx=\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac {2 (a-b)^{3/2} b (a+b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^5}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 a^4}-\frac {(4 b-3 a \cosh (x)) \sinh ^3(x)}{12 a^2} \] Output:
1/8*(3*a^4-12*a^2*b^2+8*b^4)*x/a^5-2*(a-b)^(3/2)*b*(a+b)^(3/2)*arctan((a-b )^(1/2)*tanh(1/2*x)/(a+b)^(1/2))/a^5+1/8*(8*b*(a^2-b^2)-a*(3*a^2-4*b^2)*co sh(x))*sinh(x)/a^4-1/12*(4*b-3*a*cosh(x))*sinh(x)^3/a^2
Time = 0.52 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.66 \[ \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx=\frac {36 a^4 x-144 a^2 b^2 x+96 b^4 x+\frac {192 a^4 b \arctan \left (\frac {(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {384 a^2 b^3 \arctan \left (\frac {(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {192 b^5 \arctan \left (\frac {(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+24 a b \left (5 a^2-4 b^2\right ) \sinh (x)-24 a^2 \left (a^2-b^2\right ) \sinh (2 x)-8 a^3 b \sinh (3 x)+3 a^4 \sinh (4 x)}{96 a^5} \] Input:
Integrate[Sinh[x]^4/(a + b*Sech[x]),x]
Output:
(36*a^4*x - 144*a^2*b^2*x + 96*b^4*x + (192*a^4*b*ArcTan[((-a + b)*Tanh[x/ 2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (384*a^2*b^3*ArcTan[((-a + b)*Tanh [x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (192*b^5*ArcTan[((-a + b)*Tanh[ x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 24*a*b*(5*a^2 - 4*b^2)*Sinh[x] - 24*a^2*(a^2 - b^2)*Sinh[2*x] - 8*a^3*b*Sinh[3*x] + 3*a^4*Sinh[4*x])/(96*a ^5)
Time = 1.24 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.19, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.077, Rules used = {3042, 4360, 25, 25, 3042, 3344, 3042, 25, 3344, 3042, 3214, 3042, 3138, 218}
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.
| \(\displaystyle \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (-\frac {\pi }{2}+i x\right )^4}{a-b \csc \left (-\frac {\pi }{2}+i x\right )}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\sinh ^4(x) \cosh (x)}{-a \cosh (x)-b}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cosh (x) \sinh ^4(x)}{b+a \cosh (x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\sinh ^4(x) \cosh (x)}{a \cosh (x)+b}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (\frac {\pi }{2}+i x\right ) \cos \left (\frac {\pi }{2}+i x\right )^4}{b+a \sin \left (\frac {\pi }{2}+i x\right )}dx\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {\int \frac {\left (a b-\left (3 a^2-4 b^2\right ) \cosh (x)\right ) \sinh ^2(x)}{b+a \cosh (x)}dx}{4 a^2}-\frac {\sinh ^3(x) (4 b-3 a \cosh (x))}{12 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sinh ^3(x) (4 b-3 a \cosh (x))}{12 a^2}+\frac {\int -\frac {\cos \left (i x+\frac {\pi }{2}\right )^2 \left (a b+\left (4 b^2-3 a^2\right ) \sin \left (i x+\frac {\pi }{2}\right )\right )}{b+a \sin \left (i x+\frac {\pi }{2}\right )}dx}{4 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sinh ^3(x) (4 b-3 a \cosh (x))}{12 a^2}-\frac {\int \frac {\cos \left (i x+\frac {\pi }{2}\right )^2 \left (a b-\left (3 a^2-4 b^2\right ) \sin \left (i x+\frac {\pi }{2}\right )\right )}{b+a \sin \left (i x+\frac {\pi }{2}\right )}dx}{4 a^2}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle -\frac {\frac {\int \frac {a b \left (5 a^2-4 b^2\right )-\left (3 a^4-12 b^2 a^2+8 b^4\right ) \cosh (x)}{b+a \cosh (x)}dx}{2 a^2}-\frac {\sinh (x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right )}{2 a^2}}{4 a^2}-\frac {\sinh ^3(x) (4 b-3 a \cosh (x))}{12 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sinh ^3(x) (4 b-3 a \cosh (x))}{12 a^2}-\frac {-\frac {\sinh (x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right )}{2 a^2}+\frac {\int \frac {a b \left (5 a^2-4 b^2\right )+\left (-3 a^4+12 b^2 a^2-8 b^4\right ) \sin \left (i x+\frac {\pi }{2}\right )}{b+a \sin \left (i x+\frac {\pi }{2}\right )}dx}{2 a^2}}{4 a^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {\frac {\frac {8 b \left (a^2-b^2\right )^2 \int \frac {1}{b+a \cosh (x)}dx}{a}-\frac {x \left (3 a^4-12 a^2 b^2+8 b^4\right )}{a}}{2 a^2}-\frac {\sinh (x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right )}{2 a^2}}{4 a^2}-\frac {\sinh ^3(x) (4 b-3 a \cosh (x))}{12 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sinh ^3(x) (4 b-3 a \cosh (x))}{12 a^2}-\frac {-\frac {\sinh (x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right )}{2 a^2}+\frac {-\frac {x \left (3 a^4-12 a^2 b^2+8 b^4\right )}{a}+\frac {8 b \left (a^2-b^2\right )^2 \int \frac {1}{b+a \sin \left (i x+\frac {\pi }{2}\right )}dx}{a}}{2 a^2}}{4 a^2}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle -\frac {\frac {\frac {16 b \left (a^2-b^2\right )^2 \int \frac {1}{(a-b) \tanh ^2\left (\frac {x}{2}\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{a}-\frac {x \left (3 a^4-12 a^2 b^2+8 b^4\right )}{a}}{2 a^2}-\frac {\sinh (x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right )}{2 a^2}}{4 a^2}-\frac {\sinh ^3(x) (4 b-3 a \cosh (x))}{12 a^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\sinh ^3(x) (4 b-3 a \cosh (x))}{12 a^2}-\frac {\frac {\frac {16 b \left (a^2-b^2\right )^2 \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}}-\frac {x \left (3 a^4-12 a^2 b^2+8 b^4\right )}{a}}{2 a^2}-\frac {\sinh (x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right )}{2 a^2}}{4 a^2}\) |
Input:
Int[Sinh[x]^4/(a + b*Sech[x]),x]
Output:
-1/12*((4*b - 3*a*Cosh[x])*Sinh[x]^3)/a^2 - ((-(((3*a^4 - 12*a^2*b^2 + 8*b ^4)*x)/a) + (16*b*(a^2 - b^2)^2*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b] ])/(a*Sqrt[a - b]*Sqrt[a + b]))/(2*a^2) - ((8*b*(a^2 - b^2) - a*(3*a^2 - 4 *b^2)*Cosh[x])*Sinh[x])/(2*a^2))/(4*a^2)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*(g* Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d* p + b*d*(m + p)*Sin[e + f*x])/(b^2*f*(m + p)*(m + p + 1))), x] + Simp[g^2*( (p - 1)/(b^2*(m + p)*(m + p + 1))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Si n[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1) - d*(a^ 2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1 , 0] && IntegerQ[2*m]
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. \(304\) vs. \(2(116)=232\).
Time = 86.60 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.31
| method | result | size |
| risch | \(\frac {3 x}{8 a}-\frac {3 x \,b^{2}}{2 a^{3}}+\frac {x \,b^{4}}{a^{5}}+\frac {{\mathrm e}^{4 x}}{64 a}-\frac {b \,{\mathrm e}^{3 x}}{24 a^{2}}-\frac {{\mathrm e}^{2 x}}{8 a}+\frac {{\mathrm e}^{2 x} b^{2}}{8 a^{3}}+\frac {5 b \,{\mathrm e}^{x}}{8 a^{2}}-\frac {b^{3} {\mathrm e}^{x}}{2 a^{4}}-\frac {5 b \,{\mathrm e}^{-x}}{8 a^{2}}+\frac {b^{3} {\mathrm e}^{-x}}{2 a^{4}}+\frac {{\mathrm e}^{-2 x}}{8 a}-\frac {{\mathrm e}^{-2 x} b^{2}}{8 a^{3}}+\frac {b \,{\mathrm e}^{-3 x}}{24 a^{2}}-\frac {{\mathrm e}^{-4 x}}{64 a}+\frac {\sqrt {-a^{2}+b^{2}}\, b \ln \left ({\mathrm e}^{x}-\frac {\sqrt {-a^{2}+b^{2}}-b}{a}\right )}{a^{3}}-\frac {\sqrt {-a^{2}+b^{2}}\, b^{3} \ln \left ({\mathrm e}^{x}-\frac {\sqrt {-a^{2}+b^{2}}-b}{a}\right )}{a^{5}}-\frac {\sqrt {-a^{2}+b^{2}}\, b \ln \left ({\mathrm e}^{x}+\frac {b +\sqrt {-a^{2}+b^{2}}}{a}\right )}{a^{3}}+\frac {\sqrt {-a^{2}+b^{2}}\, b^{3} \ln \left ({\mathrm e}^{x}+\frac {b +\sqrt {-a^{2}+b^{2}}}{a}\right )}{a^{5}}\) | \(305\) |
| default | \(-\frac {1}{4 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {-3 a -2 b}{6 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-a^{2}+4 a b +4 b^{2}}{8 a^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {\left (3 a^{4}-12 a^{2} b^{2}+8 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8 a^{5}}-\frac {3 a^{3}+8 a^{2} b -4 a \,b^{2}-8 b^{3}}{8 a^{4} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {1}{4 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {-3 a -2 b}{6 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a^{2}-4 a b -4 b^{2}}{8 a^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {\left (-3 a^{4}+12 a^{2} b^{2}-8 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 a^{5}}-\frac {3 a^{3}+8 a^{2} b -4 a \,b^{2}-8 b^{3}}{8 a^{4} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {2 b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5} \sqrt {\left (a -b \right ) \left (a +b \right )}}\) | \(310\) |
Input:
int(sinh(x)^4/(a+b*sech(x)),x,method=_RETURNVERBOSE)
Output:
3/8*x/a-3/2*x/a^3*b^2+x/a^5*b^4+1/64/a*exp(x)^4-1/24*b/a^2*exp(x)^3-1/8/a* exp(x)^2+1/8/a^3*exp(x)^2*b^2+5/8*b/a^2*exp(x)-1/2*b^3/a^4*exp(x)-5/8*b/a^ 2/exp(x)+1/2*b^3/a^4/exp(x)+1/8/a/exp(x)^2-1/8/a^3/exp(x)^2*b^2+1/24*b/a^2 /exp(x)^3-1/64/a/exp(x)^4+(-a^2+b^2)^(1/2)*b/a^3*ln(exp(x)-((-a^2+b^2)^(1/ 2)-b)/a)-(-a^2+b^2)^(1/2)*b^3/a^5*ln(exp(x)-((-a^2+b^2)^(1/2)-b)/a)-(-a^2+ b^2)^(1/2)*b/a^3*ln(exp(x)+(b+(-a^2+b^2)^(1/2))/a)+(-a^2+b^2)^(1/2)*b^3/a^ 5*ln(exp(x)+(b+(-a^2+b^2)^(1/2))/a)
Leaf count of result is larger than twice the leaf count of optimal. 866 vs. \(2 (115) = 230\).
Time = 0.12 (sec) , antiderivative size = 1812, normalized size of antiderivative = 13.73 \[ \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx=\text {Too large to display} \] Input:
integrate(sinh(x)^4/(a+b*sech(x)),x, algorithm="fricas")
Output:
[1/192*(3*a^4*cosh(x)^8 + 3*a^4*sinh(x)^8 - 8*a^3*b*cosh(x)^7 + 8*(3*a^4*c osh(x) - a^3*b)*sinh(x)^7 - 24*(a^4 - a^2*b^2)*cosh(x)^6 + 4*(21*a^4*cosh( x)^2 - 14*a^3*b*cosh(x) - 6*a^4 + 6*a^2*b^2)*sinh(x)^6 + 24*(3*a^4 - 12*a^ 2*b^2 + 8*b^4)*x*cosh(x)^4 + 24*(5*a^3*b - 4*a*b^3)*cosh(x)^5 + 24*(7*a^4* cosh(x)^3 - 7*a^3*b*cosh(x)^2 + 5*a^3*b - 4*a*b^3 - 6*(a^4 - a^2*b^2)*cosh (x))*sinh(x)^5 + 8*a^3*b*cosh(x) + 2*(105*a^4*cosh(x)^4 - 140*a^3*b*cosh(x )^3 - 180*(a^4 - a^2*b^2)*cosh(x)^2 + 12*(3*a^4 - 12*a^2*b^2 + 8*b^4)*x + 60*(5*a^3*b - 4*a*b^3)*cosh(x))*sinh(x)^4 - 3*a^4 - 24*(5*a^3*b - 4*a*b^3) *cosh(x)^3 + 8*(21*a^4*cosh(x)^5 - 35*a^3*b*cosh(x)^4 - 15*a^3*b + 12*a*b^ 3 - 60*(a^4 - a^2*b^2)*cosh(x)^3 + 12*(3*a^4 - 12*a^2*b^2 + 8*b^4)*x*cosh( x) + 30*(5*a^3*b - 4*a*b^3)*cosh(x)^2)*sinh(x)^3 + 24*(a^4 - a^2*b^2)*cosh (x)^2 + 12*(7*a^4*cosh(x)^6 - 14*a^3*b*cosh(x)^5 - 30*(a^4 - a^2*b^2)*cosh (x)^4 + 2*a^4 - 2*a^2*b^2 + 12*(3*a^4 - 12*a^2*b^2 + 8*b^4)*x*cosh(x)^2 + 20*(5*a^3*b - 4*a*b^3)*cosh(x)^3 - 6*(5*a^3*b - 4*a*b^3)*cosh(x))*sinh(x)^ 2 - 192*((a^2*b - b^3)*cosh(x)^4 + 4*(a^2*b - b^3)*cosh(x)^3*sinh(x) + 6*( a^2*b - b^3)*cosh(x)^2*sinh(x)^2 + 4*(a^2*b - b^3)*cosh(x)*sinh(x)^3 + (a^ 2*b - b^3)*sinh(x)^4)*sqrt(-a^2 + b^2)*log((a^2*cosh(x)^2 + a^2*sinh(x)^2 + 2*a*b*cosh(x) - a^2 + 2*b^2 + 2*(a^2*cosh(x) + a*b)*sinh(x) + 2*sqrt(-a^ 2 + b^2)*(a*cosh(x) + a*sinh(x) + b))/(a*cosh(x)^2 + a*sinh(x)^2 + 2*b*cos h(x) + 2*(a*cosh(x) + b)*sinh(x) + a)) + 8*(3*a^4*cosh(x)^7 - 7*a^3*b*c...
\[ \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx=\int \frac {\sinh ^{4}{\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \] Input:
integrate(sinh(x)**4/(a+b*sech(x)),x)
Output:
Integral(sinh(x)**4/(a + b*sech(x)), x)
Exception generated. \[ \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(sinh(x)^4/(a+b*sech(x)),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.12 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.49 \[ \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx=\frac {3 \, a^{3} e^{\left (4 \, x\right )} - 8 \, a^{2} b e^{\left (3 \, x\right )} - 24 \, a^{3} e^{\left (2 \, x\right )} + 24 \, a b^{2} e^{\left (2 \, x\right )} + 120 \, a^{2} b e^{x} - 96 \, b^{3} e^{x}}{192 \, a^{4}} + \frac {{\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} x}{8 \, a^{5}} + \frac {{\left (8 \, a^{3} b e^{x} - 3 \, a^{4} - 24 \, {\left (5 \, a^{3} b - 4 \, a b^{3}\right )} e^{\left (3 \, x\right )} + 24 \, {\left (a^{4} - a^{2} b^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-4 \, x\right )}}{192 \, a^{5}} - \frac {2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \arctan \left (\frac {a e^{x} + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a^{5}} \] Input:
integrate(sinh(x)^4/(a+b*sech(x)),x, algorithm="giac")
Output:
1/192*(3*a^3*e^(4*x) - 8*a^2*b*e^(3*x) - 24*a^3*e^(2*x) + 24*a*b^2*e^(2*x) + 120*a^2*b*e^x - 96*b^3*e^x)/a^4 + 1/8*(3*a^4 - 12*a^2*b^2 + 8*b^4)*x/a^ 5 + 1/192*(8*a^3*b*e^x - 3*a^4 - 24*(5*a^3*b - 4*a*b^3)*e^(3*x) + 24*(a^4 - a^2*b^2)*e^(2*x))*e^(-4*x)/a^5 - 2*(a^4*b - 2*a^2*b^3 + b^5)*arctan((a*e ^x + b)/sqrt(a^2 - b^2))/(sqrt(a^2 - b^2)*a^5)
Time = 2.89 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.08 \[ \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx=\frac {{\mathrm {e}}^{4\,x}}{64\,a}-\frac {{\mathrm {e}}^{-4\,x}}{64\,a}+\frac {x\,\left (3\,a^4-12\,a^2\,b^2+8\,b^4\right )}{8\,a^5}-\frac {{\mathrm {e}}^{-x}\,\left (5\,a^2\,b-4\,b^3\right )}{8\,a^4}+\frac {{\mathrm {e}}^{-2\,x}\,\left (a^2-b^2\right )}{8\,a^3}-\frac {{\mathrm {e}}^{2\,x}\,\left (a^2-b^2\right )}{8\,a^3}+\frac {b\,{\mathrm {e}}^{-3\,x}}{24\,a^2}-\frac {b\,{\mathrm {e}}^{3\,x}}{24\,a^2}+\frac {{\mathrm {e}}^x\,\left (5\,a^2\,b-4\,b^3\right )}{8\,a^4}+\frac {b\,\ln \left (\frac {2\,{\mathrm {e}}^x\,\left (a^4\,b-2\,a^2\,b^3+b^5\right )}{a^6}-\frac {2\,b\,{\left (a+b\right )}^{3/2}\,\left (a+b\,{\mathrm {e}}^x\right )\,{\left (b-a\right )}^{3/2}}{a^6}\right )\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}{a^5}-\frac {b\,\ln \left (\frac {2\,{\mathrm {e}}^x\,\left (a^4\,b-2\,a^2\,b^3+b^5\right )}{a^6}+\frac {2\,b\,{\left (a+b\right )}^{3/2}\,\left (a+b\,{\mathrm {e}}^x\right )\,{\left (b-a\right )}^{3/2}}{a^6}\right )\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}{a^5} \] Input:
int(sinh(x)^4/(a + b/cosh(x)),x)
Output:
exp(4*x)/(64*a) - exp(-4*x)/(64*a) + (x*(3*a^4 + 8*b^4 - 12*a^2*b^2))/(8*a ^5) - (exp(-x)*(5*a^2*b - 4*b^3))/(8*a^4) + (exp(-2*x)*(a^2 - b^2))/(8*a^3 ) - (exp(2*x)*(a^2 - b^2))/(8*a^3) + (b*exp(-3*x))/(24*a^2) - (b*exp(3*x)) /(24*a^2) + (exp(x)*(5*a^2*b - 4*b^3))/(8*a^4) + (b*log((2*exp(x)*(a^4*b + b^5 - 2*a^2*b^3))/a^6 - (2*b*(a + b)^(3/2)*(a + b*exp(x))*(b - a)^(3/2))/ a^6)*(a + b)^(3/2)*(b - a)^(3/2))/a^5 - (b*log((2*exp(x)*(a^4*b + b^5 - 2* a^2*b^3))/a^6 + (2*b*(a + b)^(3/2)*(a + b*exp(x))*(b - a)^(3/2))/a^6)*(a + b)^(3/2)*(b - a)^(3/2))/a^5
Time = 0.21 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.95 \[ \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx=\frac {-384 e^{4 x} \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +b}{\sqrt {a^{2}-b^{2}}}\right ) a^{2} b +384 e^{4 x} \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +b}{\sqrt {a^{2}-b^{2}}}\right ) b^{3}+3 e^{8 x} a^{4}-8 e^{7 x} a^{3} b -24 e^{6 x} a^{4}+24 e^{6 x} a^{2} b^{2}+120 e^{5 x} a^{3} b -96 e^{5 x} a \,b^{3}+72 e^{4 x} a^{4} x -288 e^{4 x} a^{2} b^{2} x +192 e^{4 x} b^{4} x -120 e^{3 x} a^{3} b +96 e^{3 x} a \,b^{3}+24 e^{2 x} a^{4}-24 e^{2 x} a^{2} b^{2}+8 e^{x} a^{3} b -3 a^{4}}{192 e^{4 x} a^{5}} \] Input:
int(sinh(x)^4/(a+b*sech(x)),x)
Output:
( - 384*e**(4*x)*sqrt(a**2 - b**2)*atan((e**x*a + b)/sqrt(a**2 - b**2))*a* *2*b + 384*e**(4*x)*sqrt(a**2 - b**2)*atan((e**x*a + b)/sqrt(a**2 - b**2)) *b**3 + 3*e**(8*x)*a**4 - 8*e**(7*x)*a**3*b - 24*e**(6*x)*a**4 + 24*e**(6* x)*a**2*b**2 + 120*e**(5*x)*a**3*b - 96*e**(5*x)*a*b**3 + 72*e**(4*x)*a**4 *x - 288*e**(4*x)*a**2*b**2*x + 192*e**(4*x)*b**4*x - 120*e**(3*x)*a**3*b + 96*e**(3*x)*a*b**3 + 24*e**(2*x)*a**4 - 24*e**(2*x)*a**2*b**2 + 8*e**x*a **3*b - 3*a**4)/(192*e**(4*x)*a**5)