Integrand size = 13, antiderivative size = 97 \[ \int \frac {\cosh ^4(x)}{a+b \sinh (x)} \, dx=-\frac {a \left (2 a^2+3 b^2\right ) x}{2 b^4}-\frac {2 \left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^4}+\frac {\cosh ^3(x)}{3 b}+\frac {\cosh (x) \left (2 \left (a^2+b^2\right )-a b \sinh (x)\right )}{2 b^3} \]
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Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2774, 2944, 2814, 2739, 632, 212} \[ \int \frac {\cosh ^4(x)}{a+b \sinh (x)} \, dx=-\frac {2 \left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^4}-\frac {a x \left (2 a^2+3 b^2\right )}{2 b^4}+\frac {\cosh (x) \left (2 \left (a^2+b^2\right )-a b \sinh (x)\right )}{2 b^3}+\frac {\cosh ^3(x)}{3 b} \]
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Rule 212
Rule 632
Rule 2739
Rule 2774
Rule 2814
Rule 2944
Rubi steps \begin{align*} \text {integral}& = \frac {\cosh ^3(x)}{3 b}+\frac {i \int \frac {\cosh ^2(x) (-i b+i a \sinh (x))}{a+b \sinh (x)} \, dx}{b} \\ & = \frac {\cosh ^3(x)}{3 b}+\frac {\cosh (x) \left (2 \left (a^2+b^2\right )-a b \sinh (x)\right )}{2 b^3}-\frac {i \int \frac {i b \left (a^2+2 b^2\right )-i a \left (2 a^2+3 b^2\right ) \sinh (x)}{a+b \sinh (x)} \, dx}{2 b^3} \\ & = -\frac {a \left (2 a^2+3 b^2\right ) x}{2 b^4}+\frac {\cosh ^3(x)}{3 b}+\frac {\cosh (x) \left (2 \left (a^2+b^2\right )-a b \sinh (x)\right )}{2 b^3}+\frac {\left (a^2+b^2\right )^2 \int \frac {1}{a+b \sinh (x)} \, dx}{b^4} \\ & = -\frac {a \left (2 a^2+3 b^2\right ) x}{2 b^4}+\frac {\cosh ^3(x)}{3 b}+\frac {\cosh (x) \left (2 \left (a^2+b^2\right )-a b \sinh (x)\right )}{2 b^3}+\frac {\left (2 \left (a^2+b^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b^4} \\ & = -\frac {a \left (2 a^2+3 b^2\right ) x}{2 b^4}+\frac {\cosh ^3(x)}{3 b}+\frac {\cosh (x) \left (2 \left (a^2+b^2\right )-a b \sinh (x)\right )}{2 b^3}-\frac {\left (4 \left (a^2+b^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{b^4} \\ & = -\frac {a \left (2 a^2+3 b^2\right ) x}{2 b^4}-\frac {2 \left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^4}+\frac {\cosh ^3(x)}{3 b}+\frac {\cosh (x) \left (2 \left (a^2+b^2\right )-a b \sinh (x)\right )}{2 b^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.76 (sec) , antiderivative size = 553, normalized size of antiderivative = 5.70 \[ \int \frac {\cosh ^4(x)}{a+b \sinh (x)} \, dx=\frac {\cosh ^3(x) \left (-12 \sqrt {a-i b} \sqrt {a+i b} \left (a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}{\sqrt {-\frac {b (-i+\sinh (x))}{a+i b}}}\right ) \sqrt {1+i \sinh (x)}+12 (a-i b)^2 (a+i b) \text {arctanh}\left (\frac {\sqrt {a-i b} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}{\sqrt {a+i b} \sqrt {-\frac {b (-i+\sinh (x))}{a+i b}}}\right ) \sqrt {1+i \sinh (x)}+\sqrt {a+i b} \sqrt {-\frac {b (-i+\sinh (x))}{a+i b}} \left ((3-3 i) \sqrt {2} \sqrt {b} \left (2 a^2-i a b+2 b^2\right ) \arcsin \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {a-i b} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}{\sqrt {b}}\right )+2 \sqrt {a-i b} \left (3 a^2+4 b^2\right ) \sqrt {1+i \sinh (x)} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}-3 a \sqrt {a-i b} b \sqrt {1+i \sinh (x)} \sinh (x) \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}+2 \sqrt {a-i b} b^2 \sqrt {1+i \sinh (x)} \sinh ^2(x) \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}\right )\right )}{6 (a-i b)^{3/2} (a+i b)^{3/2} b \sqrt {1+i \sinh (x)} \left (-\frac {b (-i+\sinh (x))}{a+i b}\right )^{3/2} \left (-\frac {b (i+\sinh (x))}{a-i b}\right )^{3/2}} \]
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Time = 8.01 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.67
method | result | size |
risch | \(-\frac {a^{3} x}{b^{4}}-\frac {3 a x}{2 b^{2}}+\frac {{\mathrm e}^{3 x}}{24 b}-\frac {a \,{\mathrm e}^{2 x}}{8 b^{2}}+\frac {{\mathrm e}^{x} a^{2}}{2 b^{3}}+\frac {5 \,{\mathrm e}^{x}}{8 b}+\frac {{\mathrm e}^{-x} a^{2}}{2 b^{3}}+\frac {5 \,{\mathrm e}^{-x}}{8 b}+\frac {a \,{\mathrm e}^{-2 x}}{8 b^{2}}+\frac {{\mathrm e}^{-3 x}}{24 b}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} \ln \left ({\mathrm e}^{x}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}{b^{4}}-\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} \ln \left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right )}{b^{4}}\) | \(162\) |
default | \(-\frac {2 \left (-a^{4}-2 a^{2} b^{2}-b^{4}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{4} \sqrt {a^{2}+b^{2}}}-\frac {1}{3 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a +b}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {2 a^{2}+a b +3 b^{2}}{2 b^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a \left (2 a^{2}+3 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b^{4}}+\frac {1}{3 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {b -a}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {-2 a^{2}+a b -3 b^{2}}{2 b^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a \left (2 a^{2}+3 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 b^{4}}\) | \(220\) |
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Leaf count of result is larger than twice the leaf count of optimal. 569 vs. \(2 (87) = 174\).
Time = 0.31 (sec) , antiderivative size = 569, normalized size of antiderivative = 5.87 \[ \int \frac {\cosh ^4(x)}{a+b \sinh (x)} \, dx=\frac {b^{3} \cosh \left (x\right )^{6} + b^{3} \sinh \left (x\right )^{6} - 3 \, a b^{2} \cosh \left (x\right )^{5} + 3 \, {\left (2 \, b^{3} \cosh \left (x\right ) - a b^{2}\right )} \sinh \left (x\right )^{5} - 12 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} x \cosh \left (x\right )^{3} + 3 \, {\left (4 \, a^{2} b + 5 \, b^{3}\right )} \cosh \left (x\right )^{4} + 3 \, {\left (5 \, b^{3} \cosh \left (x\right )^{2} - 5 \, a b^{2} \cosh \left (x\right ) + 4 \, a^{2} b + 5 \, b^{3}\right )} \sinh \left (x\right )^{4} + 3 \, a b^{2} \cosh \left (x\right ) + 2 \, {\left (10 \, b^{3} \cosh \left (x\right )^{3} - 15 \, a b^{2} \cosh \left (x\right )^{2} - 6 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} x + 6 \, {\left (4 \, a^{2} b + 5 \, b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + b^{3} + 3 \, {\left (4 \, a^{2} b + 5 \, b^{3}\right )} \cosh \left (x\right )^{2} + 3 \, {\left (5 \, b^{3} \cosh \left (x\right )^{4} - 10 \, a b^{2} \cosh \left (x\right )^{3} + 4 \, a^{2} b + 5 \, b^{3} - 12 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} x \cosh \left (x\right ) + 6 \, {\left (4 \, a^{2} b + 5 \, b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 24 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{3} + 3 \, {\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, {\left (a^{2} + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (a^{2} + b^{2}\right )} \sinh \left (x\right )^{3}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right ) + 3 \, {\left (2 \, b^{3} \cosh \left (x\right )^{5} - 5 \, a b^{2} \cosh \left (x\right )^{4} - 12 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} x \cosh \left (x\right )^{2} + 4 \, {\left (4 \, a^{2} b + 5 \, b^{3}\right )} \cosh \left (x\right )^{3} + a b^{2} + 2 \, {\left (4 \, a^{2} b + 5 \, b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{24 \, {\left (b^{4} \cosh \left (x\right )^{3} + 3 \, b^{4} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, b^{4} \cosh \left (x\right ) \sinh \left (x\right )^{2} + b^{4} \sinh \left (x\right )^{3}\right )}} \]
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Timed out. \[ \int \frac {\cosh ^4(x)}{a+b \sinh (x)} \, dx=\text {Timed out} \]
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Time = 0.32 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.75 \[ \int \frac {\cosh ^4(x)}{a+b \sinh (x)} \, dx=-\frac {{\left (3 \, a b e^{\left (-x\right )} - b^{2} - 3 \, {\left (4 \, a^{2} + 5 \, b^{2}\right )} e^{\left (-2 \, x\right )}\right )} e^{\left (3 \, x\right )}}{24 \, b^{3}} + \frac {3 \, a b e^{\left (-2 \, x\right )} + b^{2} e^{\left (-3 \, x\right )} + 3 \, {\left (4 \, a^{2} + 5 \, b^{2}\right )} e^{\left (-x\right )}}{24 \, b^{3}} - \frac {{\left (2 \, a^{3} + 3 \, a b^{2}\right )} x}{2 \, b^{4}} + \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.73 \[ \int \frac {\cosh ^4(x)}{a+b \sinh (x)} \, dx=\frac {b^{2} e^{\left (3 \, x\right )} - 3 \, a b e^{\left (2 \, x\right )} + 12 \, a^{2} e^{x} + 15 \, b^{2} e^{x}}{24 \, b^{3}} - \frac {{\left (2 \, a^{3} + 3 \, a b^{2}\right )} x}{2 \, b^{4}} + \frac {{\left (3 \, a b^{2} e^{x} + b^{3} + 3 \, {\left (4 \, a^{2} b + 5 \, b^{3}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-3 \, x\right )}}{24 \, b^{4}} + \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{4}} \]
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Time = 1.63 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.06 \[ \int \frac {\cosh ^4(x)}{a+b \sinh (x)} \, dx=\frac {{\mathrm {e}}^{-3\,x}}{24\,b}+\frac {{\mathrm {e}}^{3\,x}}{24\,b}-\frac {\ln \left (-\frac {2\,{\mathrm {e}}^x\,{\left (a^2+b^2\right )}^2}{b^5}-\frac {2\,\left (b-a\,{\mathrm {e}}^x\right )\,{\left (a^2+b^2\right )}^{3/2}}{b^5}\right )\,{\left (a^2+b^2\right )}^{3/2}}{b^4}+\frac {\ln \left (\frac {2\,\left (b-a\,{\mathrm {e}}^x\right )\,{\left (a^2+b^2\right )}^{3/2}}{b^5}-\frac {2\,{\mathrm {e}}^x\,{\left (a^2+b^2\right )}^2}{b^5}\right )\,{\left (a^2+b^2\right )}^{3/2}}{b^4}-\frac {x\,\left (2\,a^3+3\,a\,b^2\right )}{2\,b^4}+\frac {{\mathrm {e}}^x\,\left (4\,a^2+5\,b^2\right )}{8\,b^3}+\frac {a\,{\mathrm {e}}^{-2\,x}}{8\,b^2}-\frac {a\,{\mathrm {e}}^{2\,x}}{8\,b^2}+\frac {{\mathrm {e}}^{-x}\,\left (4\,a^2+5\,b^2\right )}{8\,b^3} \]
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