Integrand size = 15, antiderivative size = 74 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^2} \, dx=-\frac {2 (a A+b B) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {(A b-a B) \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))} \]
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Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2833, 12, 2739, 632, 212} \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^2} \, dx=-\frac {2 (a A+b B) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {\cosh (x) (A b-a B)}{\left (a^2+b^2\right ) (a+b \sinh (x))} \]
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Rule 12
Rule 212
Rule 632
Rule 2739
Rule 2833
Rubi steps \begin{align*} \text {integral}& = -\frac {(A b-a B) \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {\int \frac {-a A-b B}{a+b \sinh (x)} \, dx}{a^2+b^2} \\ & = -\frac {(A b-a B) \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {(a A+b B) \int \frac {1}{a+b \sinh (x)} \, dx}{a^2+b^2} \\ & = -\frac {(A b-a B) \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {(2 (a A+b B)) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^2+b^2} \\ & = -\frac {(A b-a B) \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {(4 (a A+b B)) \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 )}{a^2+b^2} \\ & = -\frac {2 (a A+b B) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {(A b-a B) \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.11 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {2 (a A+b B) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+\frac {(-A b+a B) \cosh (x)}{a+b \sinh (x)}}{a^2+b^2} \]
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Time = 0.56 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.53
method | result | size |
default | \(-\frac {2 \left (-\frac {b \left (A b -a B \right ) \tanh \left (\frac {x}{2}\right )}{a \left (a^{2}+b^{2}\right )}-\frac {A b -a B}{a^{2}+b^{2}}\right )}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}+\frac {2 \left (A a +B b \right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\) | \(113\) |
risch | \(\frac {2 \left (A b -a B \right ) \left ({\mathrm e}^{x} a -b \right )}{b \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right ) A a}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right ) B b}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right ) A a}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right ) B b}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\) | \(279\) |
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Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (69) = 138\).
Time = 0.31 (sec) , antiderivative size = 444, normalized size of antiderivative = 6.00 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^2} \, dx=-\frac {2 \, B a^{3} b - 2 \, A a^{2} b^{2} + 2 \, B a b^{3} - 2 \, A b^{4} - {\left (A a b^{2} + B b^{3} - {\left (A a b^{2} + B b^{3}\right )} \cosh \left (x\right )^{2} - {\left (A a b^{2} + B b^{3}\right )} \sinh \left (x\right )^{2} - 2 \, {\left (A a^{2} b + B a b^{2}\right )} \cosh \left (x\right ) - 2 \, {\left (A a^{2} b + B a b^{2} + {\left (A a b^{2} + B b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right ) - 2 \, {\left (B a^{4} - A a^{3} b + B a^{2} b^{2} - A a b^{3}\right )} \cosh \left (x\right ) - 2 \, {\left (B a^{4} - A a^{3} b + B a^{2} b^{2} - A a b^{3}\right )} \sinh \left (x\right )}{a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6} - {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \cosh \left (x\right )^{2} - {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \sinh \left (x\right )^{2} - 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5} + {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )} \]
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Timed out. \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (69) = 138\).
Time = 0.29 (sec) , antiderivative size = 229, normalized size of antiderivative = 3.09 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^2} \, dx=A {\left (\frac {a \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (a e^{\left (-x\right )} + b\right )}}{a^{2} b + b^{3} + 2 \, {\left (a^{3} + a b^{2}\right )} e^{\left (-x\right )} - {\left (a^{2} b + b^{3}\right )} e^{\left (-2 \, x\right )}}\right )} + B {\left (\frac {b \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, {\left (a^{2} e^{\left (-x\right )} + a b\right )}}{a^{2} b^{2} + b^{4} + 2 \, {\left (a^{3} b + a b^{3}\right )} e^{\left (-x\right )} - {\left (a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, x\right )}}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.61 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (A a + B b\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (B a^{2} e^{x} - A a b e^{x} - B a b + A b^{2}\right )}}{{\left (a^{2} b + b^{3}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} \]
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Time = 1.59 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.01 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^2} \, dx=\frac {\ln \left (\frac {2\,\left (b-a\,{\mathrm {e}}^x\right )\,\left (A\,a+B\,b\right )}{b\,{\left (a^2+b^2\right )}^{3/2}}-\frac {2\,{\mathrm {e}}^x\,\left (A\,a+B\,b\right )}{a^2\,b+b^3}\right )\,\left (A\,a+B\,b\right )}{{\left (a^2+b^2\right )}^{3/2}}-\frac {\ln \left (-\frac {2\,{\mathrm {e}}^x\,\left (A\,a+B\,b\right )}{a^2\,b+b^3}-\frac {2\,\left (b-a\,{\mathrm {e}}^x\right )\,\left (A\,a+B\,b\right )}{b\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (A\,a+B\,b\right )}{{\left (a^2+b^2\right )}^{3/2}}-\frac {\frac {2\,\left (A\,b^3-B\,a\,b^2\right )}{b\,\left (a^2\,b+b^3\right )}+\frac {2\,{\mathrm {e}}^x\,\left (B\,a^2\,b^2-A\,a\,b^3\right )}{b^2\,\left (a^2\,b+b^3\right )}}{2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}} \]
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