Integrand size = 12, antiderivative size = 184 \[ \int \frac {1}{(a+b \cosh (c+d x))^4} \, dx=\frac {a \left (2 a^2+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac {b \sinh (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^3}-\frac {5 a b \sinh (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cosh (c+d x))^2}-\frac {b \left (11 a^2+4 b^2\right ) \sinh (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cosh (c+d x))} \]
[Out]
Time = 0.18 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2743, 2833, 12, 2738, 211} \[ \int \frac {1}{(a+b \cosh (c+d x))^4} \, dx=\frac {a \left (2 a^2+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac {b \left (11 a^2+4 b^2\right ) \sinh (c+d x)}{6 d \left (a^2-b^2\right )^3 (a+b \cosh (c+d x))}-\frac {5 a b \sinh (c+d x)}{6 d \left (a^2-b^2\right )^2 (a+b \cosh (c+d x))^2}-\frac {b \sinh (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cosh (c+d x))^3} \]
[In]
[Out]
Rule 12
Rule 211
Rule 2738
Rule 2743
Rule 2833
Rubi steps \begin{align*} \text {integral}& = -\frac {b \sinh (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^3}-\frac {\int \frac {-3 a+2 b \cosh (c+d x)}{(a+b \cosh (c+d x))^3} \, dx}{3 \left (a^2-b^2\right )} \\ & = -\frac {b \sinh (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^3}-\frac {5 a b \sinh (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cosh (c+d x))^2}+\frac {\int \frac {2 \left (3 a^2+2 b^2\right )-5 a b \cosh (c+d x)}{(a+b \cosh (c+d x))^2} \, dx}{6 \left (a^2-b^2\right )^2} \\ & = -\frac {b \sinh (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^3}-\frac {5 a b \sinh (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cosh (c+d x))^2}-\frac {b \left (11 a^2+4 b^2\right ) \sinh (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cosh (c+d x))}-\frac {\int -\frac {3 a \left (2 a^2+3 b^2\right )}{a+b \cosh (c+d x)} \, dx}{6 \left (a^2-b^2\right )^3} \\ & = -\frac {b \sinh (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^3}-\frac {5 a b \sinh (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cosh (c+d x))^2}-\frac {b \left (11 a^2+4 b^2\right ) \sinh (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cosh (c+d x))}+\frac {\left (a \left (2 a^2+3 b^2\right )\right ) \int \frac {1}{a+b \cosh (c+d x)} \, dx}{2 \left (a^2-b^2\right )^3} \\ & = -\frac {b \sinh (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^3}-\frac {5 a b \sinh (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cosh (c+d x))^2}-\frac {b \left (11 a^2+4 b^2\right ) \sinh (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cosh (c+d x))}-\frac {\left (i a \left (2 a^2+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{\left (a^2-b^2\right )^3 d} \\ & = \frac {a \left (2 a^2+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac {b \sinh (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^3}-\frac {5 a b \sinh (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cosh (c+d x))^2}-\frac {b \left (11 a^2+4 b^2\right ) \sinh (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cosh (c+d x))} \\ \end{align*}
Time = 0.77 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(a+b \cosh (c+d x))^4} \, dx=\frac {\frac {6 a \left (2 a^2+3 b^2\right ) \arctan \left (\frac {(a-b) \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{7/2}}-\frac {b \left (36 a^4+a^2 b^2+8 b^4+6 a b \left (9 a^2+b^2\right ) \cosh (c+d x)+\left (11 a^2 b^2+4 b^4\right ) \cosh (2 (c+d x))\right ) \sinh (c+d x)}{2 (a-b)^3 (a+b)^3 (a+b \cosh (c+d x))^3}}{6 d} \]
[In]
[Out]
Time = 0.51 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.54
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (6 a^{2}+3 a b +2 b^{2}\right ) b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (9 a^{2}+b^{2}\right ) b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (6 a^{2}-3 a b +2 b^{2}\right ) b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}+\frac {a \left (2 a^{2}+3 b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{d}\) | \(284\) |
default | \(\frac {-\frac {2 \left (-\frac {\left (6 a^{2}+3 a b +2 b^{2}\right ) b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (9 a^{2}+b^{2}\right ) b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (6 a^{2}-3 a b +2 b^{2}\right ) b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}+\frac {a \left (2 a^{2}+3 b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{d}\) | \(284\) |
risch | \(\frac {6 a^{3} b^{2} {\mathrm e}^{5 d x +5 c}+9 a \,b^{4} {\mathrm e}^{5 d x +5 c}+30 a^{4} b \,{\mathrm e}^{4 d x +4 c}+45 a^{2} b^{3} {\mathrm e}^{4 d x +4 c}+44 a^{5} {\mathrm e}^{3 d x +3 c}+82 a^{3} b^{2} {\mathrm e}^{3 d x +3 c}+24 a \,b^{4} {\mathrm e}^{3 d x +3 c}+102 a^{4} b \,{\mathrm e}^{2 d x +2 c}+36 a^{2} b^{3} {\mathrm e}^{2 d x +2 c}+12 b^{5} {\mathrm e}^{2 d x +2 c}+60 a^{3} b^{2} {\mathrm e}^{d x +c}+15 \,{\mathrm e}^{d x +c} a \,b^{4}+11 a^{2} b^{3}+4 b^{5}}{3 d \left (a^{2}-b^{2}\right )^{3} \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}+b \right )^{3}}+\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}+\frac {3 a \ln \left ({\mathrm e}^{d x +c}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) b^{2}}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}-\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}-\frac {3 a \ln \left ({\mathrm e}^{d x +c}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) b^{2}}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}\) | \(543\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 2798 vs. \(2 (169) = 338\).
Time = 0.32 (sec) , antiderivative size = 5705, normalized size of antiderivative = 31.01 \[ \int \frac {1}{(a+b \cosh (c+d x))^4} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(a+b \cosh (c+d x))^4} \, dx=\text {Timed out} \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{(a+b \cosh (c+d x))^4} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.79 \[ \int \frac {1}{(a+b \cosh (c+d x))^4} \, dx=\frac {\frac {3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \arctan \left (\frac {b e^{\left (d x + c\right )} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {6 \, a^{3} b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 9 \, a b^{4} e^{\left (5 \, d x + 5 \, c\right )} + 30 \, a^{4} b e^{\left (4 \, d x + 4 \, c\right )} + 45 \, a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 44 \, a^{5} e^{\left (3 \, d x + 3 \, c\right )} + 82 \, a^{3} b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a b^{4} e^{\left (3 \, d x + 3 \, c\right )} + 102 \, a^{4} b e^{\left (2 \, d x + 2 \, c\right )} + 36 \, a^{2} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b^{5} e^{\left (2 \, d x + 2 \, c\right )} + 60 \, a^{3} b^{2} e^{\left (d x + c\right )} + 15 \, a b^{4} e^{\left (d x + c\right )} + 11 \, a^{2} b^{3} + 4 \, b^{5}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} + b\right )}^{3}}}{3 \, d} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(a+b \cosh (c+d x))^4} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (c+d\,x\right )\right )}^4} \,d x \]
[In]
[Out]