Integrand size = 13, antiderivative size = 138 \[ \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx=\frac {\left (a^2+b^2\right )^3 \log (a+b \sinh (x))}{b^7}-\frac {a \left (a^4+3 a^2 b^2+3 b^4\right ) \sinh (x)}{b^6}+\frac {\left (a^4+3 a^2 b^2+3 b^4\right ) \sinh ^2(x)}{2 b^5}-\frac {a \left (a^2+3 b^2\right ) \sinh ^3(x)}{3 b^4}+\frac {\left (a^2+3 b^2\right ) \sinh ^4(x)}{4 b^3}-\frac {a \sinh ^5(x)}{5 b^2}+\frac {\sinh ^6(x)}{6 b} \] Output:
(a^2+b^2)^3*ln(a+b*sinh(x))/b^7-a*(a^4+3*a^2*b^2+3*b^4)*sinh(x)/b^6+1/2*(a ^4+3*a^2*b^2+3*b^4)*sinh(x)^2/b^5-1/3*a*(a^2+3*b^2)*sinh(x)^3/b^4+1/4*(a^2 +3*b^2)*sinh(x)^4/b^3-1/5*a*sinh(x)^5/b^2+1/6*sinh(x)^6/b
Time = 0.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.88 \[ \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx=\frac {15 b^4 \left (a^2+b^2\right ) \cosh ^4(x)+10 b^6 \cosh ^6(x)+60 \left (a^2+b^2\right )^3 \log (a+b \sinh (x))-60 a b \left (a^4+3 a^2 b^2+3 b^4\right ) \sinh (x)+30 b^2 \left (a^2+b^2\right )^2 \sinh ^2(x)-20 a b^3 \left (a^2+3 b^2\right ) \sinh ^3(x)-12 a b^5 \sinh ^5(x)}{60 b^7} \] Input:
Integrate[Cosh[x]^7/(a + b*Sinh[x]),x]
Output:
(15*b^4*(a^2 + b^2)*Cosh[x]^4 + 10*b^6*Cosh[x]^6 + 60*(a^2 + b^2)^3*Log[a + b*Sinh[x]] - 60*a*b*(a^4 + 3*a^2*b^2 + 3*b^4)*Sinh[x] + 30*b^2*(a^2 + b^ 2)^2*Sinh[x]^2 - 20*a*b^3*(a^2 + 3*b^2)*Sinh[x]^3 - 12*a*b^5*Sinh[x]^5)/(6 0*b^7)
Time = 0.35 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 3147, 25, 476, 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.
| \(\displaystyle \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (i x)^7}{a-i b \sin (i x)}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle -\frac {\int -\frac {\left (\sinh ^2(x) b^2+b^2\right )^3}{a+b \sinh (x)}d(b \sinh (x))}{b^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (\sinh ^2(x) b^2+b^2\right )^3}{a+b \sinh (x)}d(b \sinh (x))}{b^7}\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \frac {\int \left (b^5 \sinh ^5(x)-a b^4 \sinh ^4(x)+b^3 \left (a^2+3 b^2\right ) \sinh ^3(x)-a b^2 \left (a^2+3 b^2\right ) \sinh ^2(x)+b \left (a^4+3 b^2 a^2+3 b^4\right ) \sinh (x)-a \left (a^4+3 b^2 a^2+3 b^4\right )+\frac {\left (a^2+b^2\right )^3}{a+b \sinh (x)}\right )d(b \sinh (x))}{b^7}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\left (a^2+b^2\right )^3 \log (a+b \sinh (x))-\frac {1}{4} b^4 \left (a^2+3 b^2\right ) \sinh ^4(x)+\frac {1}{3} a b^3 \left (a^2+3 b^2\right ) \sinh ^3(x)-\frac {1}{2} b^2 \left (a^4+3 a^2 b^2+3 b^4\right ) \sinh ^2(x)+a b \left (a^4+3 a^2 b^2+3 b^4\right ) \sinh (x)+\frac {1}{5} a b^5 \sinh ^5(x)-\frac {1}{6} b^6 \sinh ^6(x)}{b^7}\) |
Input:
Int[Cosh[x]^7/(a + b*Sinh[x]),x]
Output:
-((-((a^2 + b^2)^3*Log[a + b*Sinh[x]]) + a*b*(a^4 + 3*a^2*b^2 + 3*b^4)*Sin h[x] - (b^2*(a^4 + 3*a^2*b^2 + 3*b^4)*Sinh[x]^2)/2 + (a*b^3*(a^2 + 3*b^2)* Sinh[x]^3)/3 - (b^4*(a^2 + 3*b^2)*Sinh[x]^4)/4 + (a*b^5*Sinh[x]^5)/5 - (b^ 6*Sinh[x]^6)/6)/b^7)
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 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*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Time = 122.62 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.06
| method | result | size |
| derivativedivides | \(-\frac {-\frac {\sinh \left (x \right )^{6} b^{5}}{6}+\frac {a \sinh \left (x \right )^{5} b^{4}}{5}-\frac {b \left (a^{2} b^{2}+3 b^{4}\right ) \sinh \left (x \right )^{4}}{4}+\frac {a \left (a^{2} b^{2}+3 b^{4}\right ) \sinh \left (x \right )^{3}}{3}-\frac {\left (a^{4}+3 a^{2} b^{2}+3 b^{4}\right ) \sinh \left (x \right )^{2} b}{2}+a \left (a^{4}+3 a^{2} b^{2}+3 b^{4}\right ) \sinh \left (x \right )}{b^{6}}+\frac {\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \sinh \left (x \right )\right )}{b^{7}}\) | \(146\) |
| default | \(-\frac {-\frac {\sinh \left (x \right )^{6} b^{5}}{6}+\frac {a \sinh \left (x \right )^{5} b^{4}}{5}-\frac {b \left (a^{2} b^{2}+3 b^{4}\right ) \sinh \left (x \right )^{4}}{4}+\frac {a \left (a^{2} b^{2}+3 b^{4}\right ) \sinh \left (x \right )^{3}}{3}-\frac {\left (a^{4}+3 a^{2} b^{2}+3 b^{4}\right ) \sinh \left (x \right )^{2} b}{2}+a \left (a^{4}+3 a^{2} b^{2}+3 b^{4}\right ) \sinh \left (x \right )}{b^{6}}+\frac {\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \sinh \left (x \right )\right )}{b^{7}}\) | \(146\) |
| risch | \(-\frac {3 a^{4} x}{b^{5}}-\frac {x}{b}-\frac {3 x \,a^{2}}{b^{3}}-\frac {19 a \,{\mathrm e}^{x}}{16 b^{2}}+\frac {{\mathrm e}^{4 x}}{32 b}+\frac {29 \,{\mathrm e}^{2 x}}{128 b}+\frac {29 \,{\mathrm e}^{-2 x}}{128 b}+\frac {{\mathrm e}^{-4 x}}{32 b}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{b}+\frac {{\mathrm e}^{-6 x}}{384 b}+\frac {{\mathrm e}^{6 x}}{384 b}+\frac {a^{5} {\mathrm e}^{-x}}{2 b^{6}}+\frac {11 a^{3} {\mathrm e}^{-x}}{8 b^{4}}+\frac {19 a \,{\mathrm e}^{-x}}{16 b^{2}}+\frac {{\mathrm e}^{-2 x} a^{4}}{8 b^{5}}+\frac {5 \,{\mathrm e}^{-2 x} a^{2}}{16 b^{3}}+\frac {a^{3} {\mathrm e}^{-3 x}}{24 b^{4}}+\frac {3 a \,{\mathrm e}^{-3 x}}{32 b^{2}}+\frac {{\mathrm e}^{-4 x} a^{2}}{64 b^{3}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right ) a^{6}}{b^{7}}+\frac {3 \ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right ) a^{4}}{b^{5}}+\frac {3 \ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right ) a^{2}}{b^{3}}-\frac {x \,a^{6}}{b^{7}}+\frac {{\mathrm e}^{4 x} a^{2}}{64 b^{3}}-\frac {a^{3} {\mathrm e}^{3 x}}{24 b^{4}}-\frac {3 a \,{\mathrm e}^{3 x}}{32 b^{2}}+\frac {{\mathrm e}^{2 x} a^{4}}{8 b^{5}}+\frac {5 \,{\mathrm e}^{2 x} a^{2}}{16 b^{3}}-\frac {a^{5} {\mathrm e}^{x}}{2 b^{6}}-\frac {11 a^{3} {\mathrm e}^{x}}{8 b^{4}}-\frac {a \,{\mathrm e}^{5 x}}{160 b^{2}}+\frac {a \,{\mathrm e}^{-5 x}}{160 b^{2}}\) | \(374\) |
Input:
int(cosh(x)^7/(a+b*sinh(x)),x,method=_RETURNVERBOSE)
Output:
-1/b^6*(-1/6*sinh(x)^6*b^5+1/5*a*sinh(x)^5*b^4-1/4*b*(a^2*b^2+3*b^4)*sinh( x)^4+1/3*a*(a^2*b^2+3*b^4)*sinh(x)^3-1/2*(a^4+3*a^2*b^2+3*b^4)*sinh(x)^2*b +a*(a^4+3*a^2*b^2+3*b^4)*sinh(x))+(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/b^7*ln(a+b *sinh(x))
Leaf count of result is larger than twice the leaf count of optimal. 2105 vs. \(2 (128) = 256\).
Time = 0.12 (sec) , antiderivative size = 2105, normalized size of antiderivative = 15.25 \[ \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx=\text {Too large to display} \] Input:
integrate(cosh(x)^7/(a+b*sinh(x)),x, algorithm="fricas")
Output:
1/1920*(5*b^6*cosh(x)^12 + 5*b^6*sinh(x)^12 - 12*a*b^5*cosh(x)^11 + 12*(5* b^6*cosh(x) - a*b^5)*sinh(x)^11 + 30*(a^2*b^4 + 2*b^6)*cosh(x)^10 + 6*(55* b^6*cosh(x)^2 - 22*a*b^5*cosh(x) + 5*a^2*b^4 + 10*b^6)*sinh(x)^10 - 20*(4* a^3*b^3 + 9*a*b^5)*cosh(x)^9 + 20*(55*b^6*cosh(x)^3 - 33*a*b^5*cosh(x)^2 - 4*a^3*b^3 - 9*a*b^5 + 15*(a^2*b^4 + 2*b^6)*cosh(x))*sinh(x)^9 + 15*(16*a^ 4*b^2 + 40*a^2*b^4 + 29*b^6)*cosh(x)^8 + 15*(165*b^6*cosh(x)^4 - 132*a*b^5 *cosh(x)^3 + 16*a^4*b^2 + 40*a^2*b^4 + 29*b^6 + 90*(a^2*b^4 + 2*b^6)*cosh( x)^2 - 12*(4*a^3*b^3 + 9*a*b^5)*cosh(x))*sinh(x)^8 - 1920*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x*cosh(x)^6 - 120*(8*a^5*b + 22*a^3*b^3 + 19*a*b^5)*co sh(x)^7 + 120*(33*b^6*cosh(x)^5 - 33*a*b^5*cosh(x)^4 - 8*a^5*b - 22*a^3*b^ 3 - 19*a*b^5 + 30*(a^2*b^4 + 2*b^6)*cosh(x)^3 - 6*(4*a^3*b^3 + 9*a*b^5)*co sh(x)^2 + (16*a^4*b^2 + 40*a^2*b^4 + 29*b^6)*cosh(x))*sinh(x)^7 + 12*a*b^5 *cosh(x) + 12*(385*b^6*cosh(x)^6 - 462*a*b^5*cosh(x)^5 + 525*(a^2*b^4 + 2* b^6)*cosh(x)^4 - 140*(4*a^3*b^3 + 9*a*b^5)*cosh(x)^3 + 35*(16*a^4*b^2 + 40 *a^2*b^4 + 29*b^6)*cosh(x)^2 - 160*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x - 70*(8*a^5*b + 22*a^3*b^3 + 19*a*b^5)*cosh(x))*sinh(x)^6 + 5*b^6 + 120*(8* a^5*b + 22*a^3*b^3 + 19*a*b^5)*cosh(x)^5 + 24*(165*b^6*cosh(x)^7 - 231*a*b ^5*cosh(x)^6 + 40*a^5*b + 110*a^3*b^3 + 95*a*b^5 + 315*(a^2*b^4 + 2*b^6)*c osh(x)^5 - 105*(4*a^3*b^3 + 9*a*b^5)*cosh(x)^4 + 35*(16*a^4*b^2 + 40*a^2*b ^4 + 29*b^6)*cosh(x)^3 - 480*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x*cosh...
Timed out. \[ \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx=\text {Timed out} \] Input:
integrate(cosh(x)**7/(a+b*sinh(x)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (128) = 256\).
Time = 0.04 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.23 \[ \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx=-\frac {{\left (12 \, a b^{4} e^{\left (-x\right )} - 5 \, b^{5} - 30 \, {\left (a^{2} b^{3} + 2 \, b^{5}\right )} e^{\left (-2 \, x\right )} + 20 \, {\left (4 \, a^{3} b^{2} + 9 \, a b^{4}\right )} e^{\left (-3 \, x\right )} - 15 \, {\left (16 \, a^{4} b + 40 \, a^{2} b^{3} + 29 \, b^{5}\right )} e^{\left (-4 \, x\right )} + 120 \, {\left (8 \, a^{5} + 22 \, a^{3} b^{2} + 19 \, a b^{4}\right )} e^{\left (-5 \, x\right )}\right )} e^{\left (6 \, x\right )}}{1920 \, b^{6}} + \frac {12 \, a b^{4} e^{\left (-5 \, x\right )} + 5 \, b^{5} e^{\left (-6 \, x\right )} + 120 \, {\left (8 \, a^{5} + 22 \, a^{3} b^{2} + 19 \, a b^{4}\right )} e^{\left (-x\right )} + 15 \, {\left (16 \, a^{4} b + 40 \, a^{2} b^{3} + 29 \, b^{5}\right )} e^{\left (-2 \, x\right )} + 20 \, {\left (4 \, a^{3} b^{2} + 9 \, a b^{4}\right )} e^{\left (-3 \, x\right )} + 30 \, {\left (a^{2} b^{3} + 2 \, b^{5}\right )} e^{\left (-4 \, x\right )}}{1920 \, b^{6}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} x}{b^{7}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{b^{7}} \] Input:
integrate(cosh(x)^7/(a+b*sinh(x)),x, algorithm="maxima")
Output:
-1/1920*(12*a*b^4*e^(-x) - 5*b^5 - 30*(a^2*b^3 + 2*b^5)*e^(-2*x) + 20*(4*a ^3*b^2 + 9*a*b^4)*e^(-3*x) - 15*(16*a^4*b + 40*a^2*b^3 + 29*b^5)*e^(-4*x) + 120*(8*a^5 + 22*a^3*b^2 + 19*a*b^4)*e^(-5*x))*e^(6*x)/b^6 + 1/1920*(12*a *b^4*e^(-5*x) + 5*b^5*e^(-6*x) + 120*(8*a^5 + 22*a^3*b^2 + 19*a*b^4)*e^(-x ) + 15*(16*a^4*b + 40*a^2*b^3 + 29*b^5)*e^(-2*x) + 20*(4*a^3*b^2 + 9*a*b^4 )*e^(-3*x) + 30*(a^2*b^3 + 2*b^5)*e^(-4*x))/b^6 + (a^6 + 3*a^4*b^2 + 3*a^2 *b^4 + b^6)*x/b^7 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*log(-2*a*e^(-x) + b*e^(-2*x) - b)/b^7
Time = 0.13 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.84 \[ \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx=\frac {5 \, b^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{6} + 12 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{5} + 30 \, a^{2} b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 90 \, b^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 80 \, a^{3} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 240 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 240 \, a^{4} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 720 \, a^{2} b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 720 \, b^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 960 \, a^{5} {\left (e^{\left (-x\right )} - e^{x}\right )} + 2880 \, a^{3} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )} + 2880 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}}{1920 \, b^{6}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{b^{7}} \] Input:
integrate(cosh(x)^7/(a+b*sinh(x)),x, algorithm="giac")
Output:
1/1920*(5*b^5*(e^(-x) - e^x)^6 + 12*a*b^4*(e^(-x) - e^x)^5 + 30*a^2*b^3*(e ^(-x) - e^x)^4 + 90*b^5*(e^(-x) - e^x)^4 + 80*a^3*b^2*(e^(-x) - e^x)^3 + 2 40*a*b^4*(e^(-x) - e^x)^3 + 240*a^4*b*(e^(-x) - e^x)^2 + 720*a^2*b^3*(e^(- x) - e^x)^2 + 720*b^5*(e^(-x) - e^x)^2 + 960*a^5*(e^(-x) - e^x) + 2880*a^3 *b^2*(e^(-x) - e^x) + 2880*a*b^4*(e^(-x) - e^x))/b^6 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*log(abs(-b*(e^(-x) - e^x) + 2*a))/b^7
Time = 2.14 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.08 \[ \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx=\frac {{\mathrm {e}}^{-6\,x}}{384\,b}+\frac {{\mathrm {e}}^{6\,x}}{384\,b}+\frac {{\mathrm {e}}^{-x}\,\left (8\,a^5+22\,a^3\,b^2+19\,a\,b^4\right )}{16\,b^6}+\frac {{\mathrm {e}}^{-3\,x}\,\left (4\,a^3+9\,a\,b^2\right )}{96\,b^4}-\frac {{\mathrm {e}}^{3\,x}\,\left (4\,a^3+9\,a\,b^2\right )}{96\,b^4}+\frac {{\mathrm {e}}^{-4\,x}\,\left (a^2+2\,b^2\right )}{64\,b^3}+\frac {{\mathrm {e}}^{4\,x}\,\left (a^2+2\,b^2\right )}{64\,b^3}+\frac {a\,{\mathrm {e}}^{-5\,x}}{160\,b^2}-\frac {a\,{\mathrm {e}}^{5\,x}}{160\,b^2}-\frac {x\,{\left (a^2+b^2\right )}^3}{b^7}+\frac {{\mathrm {e}}^{-2\,x}\,\left (16\,a^4+40\,a^2\,b^2+29\,b^4\right )}{128\,b^5}+\frac {{\mathrm {e}}^{2\,x}\,\left (16\,a^4+40\,a^2\,b^2+29\,b^4\right )}{128\,b^5}-\frac {{\mathrm {e}}^x\,\left (8\,a^5+22\,a^3\,b^2+19\,a\,b^4\right )}{16\,b^6}+\frac {\ln \left (2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}\right )\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}{b^7} \] Input:
int(cosh(x)^7/(a + b*sinh(x)),x)
Output:
exp(-6*x)/(384*b) + exp(6*x)/(384*b) + (exp(-x)*(19*a*b^4 + 8*a^5 + 22*a^3 *b^2))/(16*b^6) + (exp(-3*x)*(9*a*b^2 + 4*a^3))/(96*b^4) - (exp(3*x)*(9*a* b^2 + 4*a^3))/(96*b^4) + (exp(-4*x)*(a^2 + 2*b^2))/(64*b^3) + (exp(4*x)*(a ^2 + 2*b^2))/(64*b^3) + (a*exp(-5*x))/(160*b^2) - (a*exp(5*x))/(160*b^2) - (x*(a^2 + b^2)^3)/b^7 + (exp(-2*x)*(16*a^4 + 29*b^4 + 40*a^2*b^2))/(128*b ^5) + (exp(2*x)*(16*a^4 + 29*b^4 + 40*a^2*b^2))/(128*b^5) - (exp(x)*(19*a* b^4 + 8*a^5 + 22*a^3*b^2))/(16*b^6) + (log(2*a*exp(x) - b + b*exp(2*x))*(a ^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))/b^7
Time = 0.20 (sec) , antiderivative size = 452, normalized size of antiderivative = 3.28 \[ \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx=\frac {5 b^{6}+5760 e^{6 x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a -b \right ) a^{4} b^{2}+5760 e^{6 x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a -b \right ) a^{2} b^{4}-5760 e^{6 x} a^{4} b^{2} x -5760 e^{6 x} a^{2} b^{4} x +60 e^{2 x} b^{6}+240 e^{4 x} a^{4} b^{2}+435 e^{4 x} b^{6}+600 e^{4 x} a^{2} b^{4}+80 e^{3 x} a^{3} b^{3}+180 e^{3 x} a \,b^{5}+12 e^{x} a \,b^{5}+30 e^{2 x} a^{2} b^{4}+5 e^{12 x} b^{6}+60 e^{10 x} b^{6}+435 e^{8 x} b^{6}-12 e^{11 x} a \,b^{5}+30 e^{10 x} a^{2} b^{4}-80 e^{9 x} a^{3} b^{3}-180 e^{9 x} a \,b^{5}+240 e^{8 x} a^{4} b^{2}+600 e^{8 x} a^{2} b^{4}-960 e^{7 x} a^{5} b -2640 e^{7 x} a^{3} b^{3}-2280 e^{7 x} a \,b^{5}+1920 e^{6 x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a -b \right ) a^{6}+1920 e^{6 x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a -b \right ) b^{6}-1920 e^{6 x} a^{6} x -1920 e^{6 x} b^{6} x +960 e^{5 x} a^{5} b +2640 e^{5 x} a^{3} b^{3}+2280 e^{5 x} a \,b^{5}}{1920 e^{6 x} b^{7}} \] Input:
int(cosh(x)^7/(a+b*sinh(x)),x)
Output:
(5*e**(12*x)*b**6 - 12*e**(11*x)*a*b**5 + 30*e**(10*x)*a**2*b**4 + 60*e**( 10*x)*b**6 - 80*e**(9*x)*a**3*b**3 - 180*e**(9*x)*a*b**5 + 240*e**(8*x)*a* *4*b**2 + 600*e**(8*x)*a**2*b**4 + 435*e**(8*x)*b**6 - 960*e**(7*x)*a**5*b - 2640*e**(7*x)*a**3*b**3 - 2280*e**(7*x)*a*b**5 + 1920*e**(6*x)*log(e**( 2*x)*b + 2*e**x*a - b)*a**6 + 5760*e**(6*x)*log(e**(2*x)*b + 2*e**x*a - b) *a**4*b**2 + 5760*e**(6*x)*log(e**(2*x)*b + 2*e**x*a - b)*a**2*b**4 + 1920 *e**(6*x)*log(e**(2*x)*b + 2*e**x*a - b)*b**6 - 1920*e**(6*x)*a**6*x - 576 0*e**(6*x)*a**4*b**2*x - 5760*e**(6*x)*a**2*b**4*x - 1920*e**(6*x)*b**6*x + 960*e**(5*x)*a**5*b + 2640*e**(5*x)*a**3*b**3 + 2280*e**(5*x)*a*b**5 + 2 40*e**(4*x)*a**4*b**2 + 600*e**(4*x)*a**2*b**4 + 435*e**(4*x)*b**6 + 80*e* *(3*x)*a**3*b**3 + 180*e**(3*x)*a*b**5 + 30*e**(2*x)*a**2*b**4 + 60*e**(2* x)*b**6 + 12*e**x*a*b**5 + 5*b**6)/(1920*e**(6*x)*b**7)