Integrand size = 19, antiderivative size = 249 \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )} \, dx=-\frac {\cosh (c+d x)}{a x}+\frac {\sqrt {b} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{3/2}}-\frac {\sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{3/2}}+\frac {d \text {Chi}(d x) \sinh (c)}{a}+\frac {d \cosh (c) \text {Shi}(d x)}{a}-\frac {\sqrt {b} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{3/2}}-\frac {\sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{3/2}} \]
[Out]
Time = 0.39 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5401, 3378, 3384, 3379, 3382, 5389} \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )} \, dx=\frac {\sqrt {b} \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{3/2}}-\frac {\sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{3/2}}-\frac {\sqrt {b} \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{3/2}}-\frac {\sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{3/2}}+\frac {d \sinh (c) \text {Chi}(d x)}{a}+\frac {d \cosh (c) \text {Shi}(d x)}{a}-\frac {\cosh (c+d x)}{a x} \]
[In]
[Out]
Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5389
Rule 5401
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\cosh (c+d x)}{a x^2}-\frac {b \cosh (c+d x)}{a \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {\cosh (c+d x)}{x^2} \, dx}{a}-\frac {b \int \frac {\cosh (c+d x)}{a+b x^2} \, dx}{a} \\ & = -\frac {\cosh (c+d x)}{a x}-\frac {b \int \left (\frac {\sqrt {-a} \cosh (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \cosh (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{a}+\frac {d \int \frac {\sinh (c+d x)}{x} \, dx}{a} \\ & = -\frac {\cosh (c+d x)}{a x}-\frac {b \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 (-a)^{3/2}}-\frac {b \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 (-a)^{3/2}}+\frac {(d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx}{a}+\frac {(d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx}{a} \\ & = -\frac {\cosh (c+d x)}{a x}+\frac {d \text {Chi}(d x) \sinh (c)}{a}+\frac {d \cosh (c) \text {Shi}(d x)}{a}-\frac {\left (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)^{3/2}}-\frac {\left (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)^{3/2}}-\frac {\left (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)^{3/2}}+\frac {\left (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)^{3/2}} \\ & = -\frac {\cosh (c+d x)}{a x}+\frac {\sqrt {b} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{3/2}}-\frac {\sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{3/2}}+\frac {d \text {Chi}(d x) \sinh (c)}{a}+\frac {d \cosh (c) \text {Shi}(d x)}{a}-\frac {\sqrt {b} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{3/2}}-\frac {\sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.69 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.98 \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )} \, dx=-\frac {\cosh (c) \cosh (d x)}{a x}+\frac {i \sqrt {b} e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )}{4 a^{3/2}}-\frac {i \sqrt {b} e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )}{4 a^{3/2}}-\frac {\sinh (c) \sinh (d x)}{a x}+\frac {d (\text {Chi}(d x) \sinh (c)+\cosh (c) \text {Shi}(d x))}{a} \]
[In]
[Out]
Time = 0.22 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.16
method | result | size |
risch | \(-\frac {{\mathrm e}^{-d x -c}}{2 a x}+\frac {b \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 a \sqrt {-a b}}-\frac {b \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 a \sqrt {-a b}}+\frac {d \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{2 a}-\frac {{\mathrm e}^{d x +c}}{2 a x}-\frac {b \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 a \sqrt {-a b}}+\frac {b \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 a \sqrt {-a b}}-\frac {d \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2 a}\) | \(288\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (193) = 386\).
Time = 0.27 (sec) , antiderivative size = 599, normalized size of antiderivative = 2.41 \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )} \, dx=-\frac {4 \, a d \cosh \left (d x + c\right ) - {\left ({\left (b x \cosh \left (d x + c\right )^{2} - b x \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (b x \cosh \left (d x + c\right )^{2} - b x \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\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 (a d^{2} x {\rm Ei}\left (d x\right ) - a d^{2} x {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + {\left ({\left (b x \cosh \left (d x + c\right )^{2} - b x \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (b x \cosh \left (d x + c\right )^{2} - b x \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (b x \cosh \left (d x + c\right )^{2} - b x \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (b x \cosh \left (d x + c\right )^{2} - b x \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\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 (a d^{2} x {\rm Ei}\left (d x\right ) + a d^{2} x {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) - {\left ({\left (b x \cosh \left (d x + c\right )^{2} - b x \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (b x \cosh \left (d x + c\right )^{2} - b x \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right )}{4 \, {\left (a^{2} d x \cosh \left (d x + c\right )^{2} - a^{2} d x \sinh \left (d x + c\right )^{2}\right )}} \]
[In]
[Out]
\[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )} \, dx=\int \frac {\cosh {\left (c + d x \right )}}{x^{2} \left (a + b x^{2}\right )}\, dx \]
[In]
[Out]
\[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )} x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )} x^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x^2\,\left (b\,x^2+a\right )} \,d x \]
[In]
[Out]