Integrand size = 18, antiderivative size = 151 \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x^3} \, dx=-\frac {\sqrt {a+a \cosh (c+d x)}}{2 x^2}+\frac {1}{8} d^2 \cosh \left (\frac {c}{2}\right ) \sqrt {a+a \cosh (c+d x)} \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right )+\frac {1}{8} d^2 \sqrt {a+a \cosh (c+d x)} \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )-\frac {d \sqrt {a+a \cosh (c+d x)} \tanh \left (\frac {c}{2}+\frac {d x}{2}\right )}{4 x} \]
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Time = 0.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3400, 3378, 3384, 3379, 3382} \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x^3} \, dx=\frac {1}{8} d^2 \cosh \left (\frac {c}{2}\right ) \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a}+\frac {1}{8} d^2 \sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a}-\frac {\sqrt {a \cosh (c+d x)+a}}{2 x^2}-\frac {d \tanh \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a}}{4 x} \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 3400
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {a+a \cosh (c+d x)} \csc \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \int \frac {\sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )}{x^3} \, dx \\ & = -\frac {\sqrt {a+a \cosh (c+d x)}}{2 x^2}+\frac {1}{4} \left (d \sqrt {a+a \cosh (c+d x)} \csc \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \int \frac {\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{x^2} \, dx \\ & = -\frac {\sqrt {a+a \cosh (c+d x)}}{2 x^2}-\frac {d \sqrt {a+a \cosh (c+d x)} \tanh \left (\frac {c}{2}+\frac {d x}{2}\right )}{4 x}+\frac {1}{8} \left (d^2 \sqrt {a+a \cosh (c+d x)} \csc \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \int \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{x} \, dx \\ & = -\frac {\sqrt {a+a \cosh (c+d x)}}{2 x^2}-\frac {d \sqrt {a+a \cosh (c+d x)} \tanh \left (\frac {c}{2}+\frac {d x}{2}\right )}{4 x}+\frac {1}{8} \left (d^2 \cosh \left (\frac {c}{2}\right ) \sqrt {a+a \cosh (c+d x)} \csc \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \int \frac {\cosh \left (\frac {d x}{2}\right )}{x} \, dx+\frac {1}{8} \left (d^2 \sqrt {a+a \cosh (c+d x)} \csc \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right ) \sinh \left (\frac {c}{2}\right )\right ) \int \frac {\sinh \left (\frac {d x}{2}\right )}{x} \, dx \\ & = -\frac {\sqrt {a+a \cosh (c+d x)}}{2 x^2}+\frac {1}{8} d^2 \cosh \left (\frac {c}{2}\right ) \sqrt {a+a \cosh (c+d x)} \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right )+\frac {1}{8} d^2 \sqrt {a+a \cosh (c+d x)} \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )-\frac {d \sqrt {a+a \cosh (c+d x)} \tanh \left (\frac {c}{2}+\frac {d x}{2}\right )}{4 x} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x^3} \, dx=\frac {\sqrt {a (1+\cosh (c+d x))} \left (-4+d^2 x^2 \cosh \left (\frac {c}{2}\right ) \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {1}{2} (c+d x)\right )+d^2 x^2 \text {sech}\left (\frac {1}{2} (c+d x)\right ) \sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )-2 d x \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{8 x^2} \]
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\[\int \frac {\sqrt {a +a \cosh \left (d x +c \right )}}{x^{3}}d x\]
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Exception generated. \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x^3} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x^3} \, dx=\int \frac {\sqrt {a \left (\cosh {\left (c + d x \right )} + 1\right )}}{x^{3}}\, dx \]
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\[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x^3} \, dx=\int { \frac {\sqrt {a \cosh \left (d x + c\right ) + a}}{x^{3}} \,d x } \]
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none
Time = 0.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x^3} \, dx=\frac {\sqrt {2} {\left (\sqrt {a} d^{2} x^{2} {\rm Ei}\left (\frac {1}{2} \, d x\right ) e^{\left (\frac {1}{2} \, c\right )} + \sqrt {a} d^{2} x^{2} {\rm Ei}\left (-\frac {1}{2} \, d x\right ) e^{\left (-\frac {1}{2} \, c\right )} - 2 \, \sqrt {a} d x e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + 2 \, \sqrt {a} d x e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} - 4 \, \sqrt {a} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - 4 \, \sqrt {a} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}\right )}}{16 \, x^{2}} \]
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Timed out. \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x^3} \, dx=\int \frac {\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}}{x^3} \,d x \]
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