Integrand size = 13, antiderivative size = 138 \[ \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx=\frac {\left (a^2+b^2\right )^3 \log (a+b \sinh (x))}{b^7}-\frac {a \left (a^4+3 a^2 b^2+3 b^4\right ) \sinh (x)}{b^6}+\frac {\left (a^4+3 a^2 b^2+3 b^4\right ) \sinh ^2(x)}{2 b^5}-\frac {a \left (a^2+3 b^2\right ) \sinh ^3(x)}{3 b^4}+\frac {\left (a^2+3 b^2\right ) \sinh ^4(x)}{4 b^3}-\frac {a \sinh ^5(x)}{5 b^2}+\frac {\sinh ^6(x)}{6 b} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2747, 711} \[ \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx=\frac {\left (a^2+b^2\right )^3 \log (a+b \sinh (x))}{b^7}-\frac {a \left (a^2+3 b^2\right ) \sinh ^3(x)}{3 b^4}+\frac {\left (a^2+3 b^2\right ) \sinh ^4(x)}{4 b^3}-\frac {a \left (a^4+3 a^2 b^2+3 b^4\right ) \sinh (x)}{b^6}+\frac {\left (a^4+3 a^2 b^2+3 b^4\right ) \sinh ^2(x)}{2 b^5}-\frac {a \sinh ^5(x)}{5 b^2}+\frac {\sinh ^6(x)}{6 b} \]
[In]
[Out]
Rule 711
Rule 2747
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (-b^2-x^2\right )^3}{a+x} \, dx,x,b \sinh (x)\right )}{b^7} \\ & = -\frac {\text {Subst}\left (\int \left (a^5 \left (1+\frac {3 b^2 \left (a^2+b^2\right )}{a^4}\right )-\left (a^4+3 a^2 b^2+3 b^4\right ) x+a \left (a^2+3 b^2\right ) x^2-\left (a^2+3 b^2\right ) x^3+a x^4-x^5-\frac {\left (a^2+b^2\right )^3}{a+x}\right ) \, dx,x,b \sinh (x)\right )}{b^7} \\ & = \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} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.88 \[ \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx=\frac {15 b^4 \left (a^2+b^2\right ) \cosh ^4(x)+10 b^6 \cosh ^6(x)+60 \left (a^2+b^2\right )^3 \log (a+b \sinh (x))-60 a b \left (a^4+3 a^2 b^2+3 b^4\right ) \sinh (x)+30 b^2 \left (a^2+b^2\right )^2 \sinh ^2(x)-20 a b^3 \left (a^2+3 b^2\right ) \sinh ^3(x)-12 a b^5 \sinh ^5(x)}{60 b^7} \]
[In]
[Out]
Time = 128.98 (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 | \(-\frac {19 a \,{\mathrm e}^{x}}{16 b^{2}}-\frac {3 x \,a^{2}}{b^{3}}-\frac {x}{b}+\frac {29 \,{\mathrm e}^{-2 x}}{128 b}+\frac {{\mathrm e}^{-4 x}}{32 b}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{b}+\frac {{\mathrm e}^{4 x}}{32 b}+\frac {29 \,{\mathrm e}^{2 x}}{128 b}+\frac {{\mathrm e}^{-6 x}}{384 b}+\frac {{\mathrm e}^{6 x}}{384 b}+\frac {{\mathrm e}^{-2 x} a^{4}}{8 b^{5}}+\frac {5 \,{\mathrm e}^{-2 x} a^{2}}{16 b^{3}}+\frac {a^{3} {\mathrm e}^{-3 x}}{24 b^{4}}+\frac {3 a \,{\mathrm e}^{-3 x}}{32 b^{2}}+\frac {{\mathrm e}^{-4 x} a^{2}}{64 b^{3}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right ) a^{6}}{b^{7}}+\frac {3 \ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right ) a^{4}}{b^{5}}+\frac {3 \ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right ) a^{2}}{b^{3}}-\frac {x \,a^{6}}{b^{7}}-\frac {3 x \,a^{4}}{b^{5}}+\frac {{\mathrm e}^{4 x} a^{2}}{64 b^{3}}-\frac {a^{3} {\mathrm e}^{3 x}}{24 b^{4}}-\frac {3 a \,{\mathrm e}^{3 x}}{32 b^{2}}+\frac {{\mathrm e}^{2 x} a^{4}}{8 b^{5}}+\frac {5 \,{\mathrm e}^{2 x} a^{2}}{16 b^{3}}-\frac {a^{5} {\mathrm e}^{x}}{2 b^{6}}-\frac {11 a^{3} {\mathrm e}^{x}}{8 b^{4}}+\frac {a^{5} {\mathrm e}^{-x}}{2 b^{6}}+\frac {11 a^{3} {\mathrm e}^{-x}}{8 b^{4}}+\frac {19 a \,{\mathrm e}^{-x}}{16 b^{2}}+\frac {a \,{\mathrm e}^{-5 x}}{160 b^{2}}-\frac {a \,{\mathrm e}^{5 x}}{160 b^{2}}\) | \(374\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 2105 vs. \(2 (128) = 256\).
Time = 0.32 (sec) , antiderivative size = 2105, normalized size of antiderivative = 15.25 \[ \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (128) = 256\).
Time = 0.22 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.23 \[ \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx=-\frac {{\left (12 \, a b^{4} e^{\left (-x\right )} - 5 \, b^{5} - 30 \, {\left (a^{2} b^{3} + 2 \, b^{5}\right )} e^{\left (-2 \, x\right )} + 20 \, {\left (4 \, a^{3} b^{2} + 9 \, a b^{4}\right )} e^{\left (-3 \, x\right )} - 15 \, {\left (16 \, a^{4} b + 40 \, a^{2} b^{3} + 29 \, b^{5}\right )} e^{\left (-4 \, x\right )} + 120 \, {\left (8 \, a^{5} + 22 \, a^{3} b^{2} + 19 \, a b^{4}\right )} e^{\left (-5 \, x\right )}\right )} e^{\left (6 \, x\right )}}{1920 \, b^{6}} + \frac {12 \, a b^{4} e^{\left (-5 \, x\right )} + 5 \, b^{5} e^{\left (-6 \, x\right )} + 120 \, {\left (8 \, a^{5} + 22 \, a^{3} b^{2} + 19 \, a b^{4}\right )} e^{\left (-x\right )} + 15 \, {\left (16 \, a^{4} b + 40 \, a^{2} b^{3} + 29 \, b^{5}\right )} e^{\left (-2 \, x\right )} + 20 \, {\left (4 \, a^{3} b^{2} + 9 \, a b^{4}\right )} e^{\left (-3 \, x\right )} + 30 \, {\left (a^{2} b^{3} + 2 \, b^{5}\right )} e^{\left (-4 \, x\right )}}{1920 \, b^{6}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} x}{b^{7}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{b^{7}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.84 \[ \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx=\frac {5 \, b^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{6} + 12 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{5} + 30 \, a^{2} b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 90 \, b^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 80 \, a^{3} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 240 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 240 \, a^{4} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 720 \, a^{2} b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 720 \, b^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 960 \, a^{5} {\left (e^{\left (-x\right )} - e^{x}\right )} + 2880 \, a^{3} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )} + 2880 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}}{1920 \, b^{6}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{b^{7}} \]
[In]
[Out]
Time = 2.06 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.08 \[ \int \frac {\cosh ^7(x)}{a+b \sinh (x)} \, dx=\frac {{\mathrm {e}}^{-6\,x}}{384\,b}+\frac {{\mathrm {e}}^{6\,x}}{384\,b}+\frac {{\mathrm {e}}^{-x}\,\left (8\,a^5+22\,a^3\,b^2+19\,a\,b^4\right )}{16\,b^6}+\frac {{\mathrm {e}}^{-3\,x}\,\left (4\,a^3+9\,a\,b^2\right )}{96\,b^4}-\frac {{\mathrm {e}}^{3\,x}\,\left (4\,a^3+9\,a\,b^2\right )}{96\,b^4}+\frac {{\mathrm {e}}^{-4\,x}\,\left (a^2+2\,b^2\right )}{64\,b^3}+\frac {{\mathrm {e}}^{4\,x}\,\left (a^2+2\,b^2\right )}{64\,b^3}+\frac {a\,{\mathrm {e}}^{-5\,x}}{160\,b^2}-\frac {a\,{\mathrm {e}}^{5\,x}}{160\,b^2}-\frac {x\,{\left (a^2+b^2\right )}^3}{b^7}+\frac {{\mathrm {e}}^{-2\,x}\,\left (16\,a^4+40\,a^2\,b^2+29\,b^4\right )}{128\,b^5}+\frac {{\mathrm {e}}^{2\,x}\,\left (16\,a^4+40\,a^2\,b^2+29\,b^4\right )}{128\,b^5}-\frac {{\mathrm {e}}^x\,\left (8\,a^5+22\,a^3\,b^2+19\,a\,b^4\right )}{16\,b^6}+\frac {\ln \left (2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}\right )\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}{b^7} \]
[In]
[Out]