Integrand size = 19, antiderivative size = 197 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )} \, dx=\frac {\cosh (c) \text {Chi}(d x)}{a}-\frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a}+\frac {\sinh (c) \text {Shi}(d x)}{a}+\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a} \]
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Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {5401, 3384, 3379, 3382} \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )} \, dx=-\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a}+\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a}+\frac {\cosh (c) \text {Chi}(d x)}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5401
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\cosh (c+d x)}{a x}-\frac {b x \cosh (c+d x)}{a \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {\cosh (c+d x)}{x} \, dx}{a}-\frac {b \int \frac {x \cosh (c+d x)}{a+b x^2} \, dx}{a} \\ & = -\frac {b \int \left (-\frac {\cosh (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{a}+\frac {\cosh (c) \int \frac {\cosh (d x)}{x} \, dx}{a}+\frac {\sinh (c) \int \frac {\sinh (d x)}{x} \, dx}{a} \\ & = \frac {\cosh (c) \text {Chi}(d x)}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a}+\frac {\sqrt {b} \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 a}-\frac {\sqrt {b} \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 a} \\ & = \frac {\cosh (c) \text {Chi}(d x)}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a}-\frac {\left (\sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 a}+\frac {\left (\sqrt {b} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 a}-\frac {\left (\sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 a}-\frac {\left (\sqrt {b} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 a} \\ & = \frac {\cosh (c) \text {Chi}(d x)}{a}-\frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a}+\frac {\sinh (c) \text {Shi}(d x)}{a}+\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.88 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )} \, dx=-\frac {-4 \cosh (c) \text {Chi}(d x)+e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{2 c+\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+e^{2 c} \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )-4 \sinh (c) \text {Shi}(d x)}{4 a} \]
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Time = 0.26 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.15
method | result | size |
risch | \(-\frac {{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{2 a}-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2 a}+\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 a}+\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 a}+\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 a}+\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 a}\) | \(227\) |
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none
Time = 0.26 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.26 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )} \, dx=-\frac {{\left ({\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) - 2 \, {\left ({\rm Ei}\left (d x\right ) + {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + {\left ({\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left ({\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) - 2 \, {\left ({\rm Ei}\left (d x\right ) - {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) - {\left ({\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right )}{4 \, a} \]
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\[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )} \, dx=\int \frac {\cosh {\left (c + d x \right )}}{x \left (a + b x^{2}\right )}\, dx \]
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\[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )} x} \,d x } \]
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\[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )} x} \,d x } \]
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Timed out. \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x\,\left (b\,x^2+a\right )} \,d x \]
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