Integrand size = 12, antiderivative size = 79 \[ \int \frac {1}{(a+b \sinh (c+d x))^2} \, dx=-\frac {2 a \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {b \cosh (c+d x)}{\left (a^2+b^2\right ) d (a+b \sinh (c+d x))} \]
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Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2743, 12, 2739, 632, 210} \[ \int \frac {1}{(a+b \sinh (c+d x))^2} \, dx=-\frac {2 a \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{d \left (a^2+b^2\right )^{3/2}}-\frac {b \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))} \]
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Rule 12
Rule 210
Rule 632
Rule 2739
Rule 2743
Rubi steps \begin{align*} \text {integral}& = -\frac {b \cosh (c+d x)}{\left (a^2+b^2\right ) d (a+b \sinh (c+d x))}+\frac {\int \frac {a}{a+b \sinh (c+d x)} \, dx}{a^2+b^2} \\ & = -\frac {b \cosh (c+d x)}{\left (a^2+b^2\right ) d (a+b \sinh (c+d x))}+\frac {a \int \frac {1}{a+b \sinh (c+d x)} \, dx}{a^2+b^2} \\ & = -\frac {b \cosh (c+d x)}{\left (a^2+b^2\right ) d (a+b \sinh (c+d x))}-\frac {(2 i a) \text {Subst}\left (\int \frac {1}{a-2 i b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{\left (a^2+b^2\right ) d} \\ & = -\frac {b \cosh (c+d x)}{\left (a^2+b^2\right ) d (a+b \sinh (c+d x))}+\frac {(4 i a) \text {Subst}\left (\int \frac {1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 i b+2 a \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{\left (a^2+b^2\right ) d} \\ & = -\frac {2 a \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {b \cosh (c+d x)}{\left (a^2+b^2\right ) d (a+b \sinh (c+d x))} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(a+b \sinh (c+d x))^2} \, dx=-\frac {\frac {2 a \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+\frac {b \cosh (c+d x)}{\left (a^2+b^2\right ) (a+b \sinh (c+d x))}}{d} \]
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Time = 0.86 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.49
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{a \left (a^{2}+b^{2}\right )}-\frac {b}{a^{2}+b^{2}}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a}+\frac {2 a \,\operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}}{d}\) | \(118\) |
default | \(\frac {-\frac {2 \left (-\frac {b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{a \left (a^{2}+b^{2}\right )}-\frac {b}{a^{2}+b^{2}}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a}+\frac {2 a \,\operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}}{d}\) | \(118\) |
risch | \(\frac {2 a \,{\mathrm e}^{d x +c}-2 b}{d \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}+\frac {a \ln \left ({\mathrm e}^{d x +c}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d}-\frac {a \ln \left ({\mathrm e}^{d x +c}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d}\) | \(181\) |
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Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (76) = 152\).
Time = 0.28 (sec) , antiderivative size = 423, normalized size of antiderivative = 5.35 \[ \int \frac {1}{(a+b \sinh (c+d x))^2} \, dx=-\frac {2 \, a^{2} b + 2 \, b^{3} - {\left (a b \cosh \left (d x + c\right )^{2} + a b \sinh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) - a b + 2 \, {\left (a b \cosh \left (d x + c\right ) + a^{2}\right )} \sinh \left (d x + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) - b}\right ) - 2 \, {\left (a^{3} + a b^{2}\right )} \cosh \left (d x + c\right ) - 2 \, {\left (a^{3} + a b^{2}\right )} \sinh \left (d x + c\right )}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \cosh \left (d x + c\right )^{2} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \sinh \left (d x + c\right )^{2} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \cosh \left (d x + c\right ) - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d + 2 \, {\left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \cosh \left (d x + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )} \sinh \left (d x + c\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2082 vs. \(2 (68) = 136\).
Time = 90.37 (sec) , antiderivative size = 2082, normalized size of antiderivative = 26.35 \[ \int \frac {1}{(a+b \sinh (c+d x))^2} \, dx=\text {Too large to display} \]
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Time = 0.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.75 \[ \int \frac {1}{(a+b \sinh (c+d x))^2} \, dx=\frac {a \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}} d} - \frac {2 \, {\left (a e^{\left (-d x - c\right )} + b\right )}}{{\left (a^{2} b + b^{3} + 2 \, {\left (a^{3} + a b^{2}\right )} e^{\left (-d x - c\right )} - {\left (a^{2} b + b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} \]
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Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.51 \[ \int \frac {1}{(a+b \sinh (c+d x))^2} \, dx=\frac {\frac {a \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, {\left (a e^{\left (d x + c\right )} - b\right )}}{{\left (a^{2} + b^{2}\right )} {\left (b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} - b\right )}}}{d} \]
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Time = 1.63 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.53 \[ \int \frac {1}{(a+b \sinh (c+d x))^2} \, dx=\frac {a\,\ln \left (\frac {2\,a\,\left (b-a\,{\mathrm {e}}^{c+d\,x}\right )}{b\,{\left (a^2+b^2\right )}^{3/2}}-\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{b\,\left (a^2+b^2\right )}\right )}{d\,{\left (a^2+b^2\right )}^{3/2}}-\frac {a\,\ln \left (-\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{b\,\left (a^2+b^2\right )}-\frac {2\,a\,\left (b-a\,{\mathrm {e}}^{c+d\,x}\right )}{b\,{\left (a^2+b^2\right )}^{3/2}}\right )}{d\,{\left (a^2+b^2\right )}^{3/2}}-\frac {\frac {2\,b^2}{d\,\left (a^2\,b+b^3\right )}-\frac {2\,a\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2\,b+b^3\right )}}{2\,a\,{\mathrm {e}}^{c+d\,x}-b+b\,{\mathrm {e}}^{2\,c+2\,d\,x}} \]
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