Integrand size = 19, antiderivative size = 730 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^3} \, dx=\frac {\cosh (c+d x)}{4 a \left (a+b x^2\right )^2}+\frac {\cosh (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac {\cosh (c) \text {Chi}(d x)}{a^3}-\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^3}+\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^2 b}-\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^3}+\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^2 b}+\frac {5 d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {5 d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {\sinh (c) \text {Shi}(d x)}{a^3}+\frac {5 d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{5/2} \sqrt {b}}+\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^3}-\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^2 b}+\frac {5 d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{5/2} \sqrt {b}}-\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^3}+\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^2 b} \]
[Out]
Time = 1.23 (sec) , antiderivative size = 730, normalized size of antiderivative = 1.00, number of steps used = 41, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5401, 3384, 3379, 3382, 5397, 5388, 3378} \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^3} \, 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^3}-\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^3}+\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^3}-\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^3}+\frac {\cosh (c) \text {Chi}(d x)}{a^3}+\frac {\sinh (c) \text {Shi}(d x)}{a^3}+\frac {d^2 \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^2 b}+\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^2 b}-\frac {d^2 \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^2 b}+\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^2 b}+\frac {\cosh (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {5 d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {5 d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {5 d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {5 d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {\cosh (c+d x)}{4 a \left (a+b x^2\right )^2} \]
[In]
[Out]
Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5388
Rule 5397
Rule 5401
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\cosh (c+d x)}{a^3 x}-\frac {b x \cosh (c+d x)}{a \left (a+b x^2\right )^3}-\frac {b x \cosh (c+d x)}{a^2 \left (a+b x^2\right )^2}-\frac {b x \cosh (c+d x)}{a^3 \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {\cosh (c+d x)}{x} \, dx}{a^3}-\frac {b \int \frac {x \cosh (c+d x)}{a+b x^2} \, dx}{a^3}-\frac {b \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{a^2}-\frac {b \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx}{a} \\ & = \frac {\cosh (c+d x)}{4 a \left (a+b x^2\right )^2}+\frac {\cosh (c+d x)}{2 a^2 \left (a+b x^2\right )}-\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^3}-\frac {d \int \frac {\sinh (c+d x)}{a+b x^2} \, dx}{2 a^2}-\frac {d \int \frac {\sinh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 a}+\frac {\cosh (c) \int \frac {\cosh (d x)}{x} \, dx}{a^3}+\frac {\sinh (c) \int \frac {\sinh (d x)}{x} \, dx}{a^3} \\ & = \frac {\cosh (c+d x)}{4 a \left (a+b x^2\right )^2}+\frac {\cosh (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac {\cosh (c) \text {Chi}(d x)}{a^3}+\frac {\sinh (c) \text {Shi}(d x)}{a^3}+\frac {\sqrt {b} \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 a^3}-\frac {\sqrt {b} \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 a^3}-\frac {d \int \left (\frac {\sqrt {-a} \sinh (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \sinh (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{2 a^2}-\frac {d \int \left (-\frac {b \sinh (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \sinh (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {b \sinh (c+d x)}{2 a \left (-a b-b^2 x^2\right )}\right ) \, dx}{4 a} \\ & = \frac {\cosh (c+d x)}{4 a \left (a+b x^2\right )^2}+\frac {\cosh (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac {\cosh (c) \text {Chi}(d x)}{a^3}+\frac {\sinh (c) \text {Shi}(d x)}{a^3}+\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 (-a)^{5/2}}+\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 (-a)^{5/2}}+\frac {(b d) \int \frac {\sinh (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 a^2}+\frac {(b d) \int \frac {\sinh (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 a^2}+\frac {(b d) \int \frac {\sinh (c+d x)}{-a b-b^2 x^2} \, dx}{8 a^2}-\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^3}+\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^3}-\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^3}-\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^3} \\ & = \frac {\cosh (c+d x)}{4 a \left (a+b x^2\right )^2}+\frac {\cosh (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac {\cosh (c) \text {Chi}(d x)}{a^3}-\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^3}-\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^3}+\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {\sinh (c) \text {Shi}(d x)}{a^3}+\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^3}-\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^3}+\frac {(b d) \int \left (-\frac {\sqrt {-a} \sinh (c+d x)}{2 a b \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {-a} \sinh (c+d x)}{2 a b \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{8 a^2}-\frac {d^2 \int \frac {\cosh (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a^2}+\frac {d^2 \int \frac {\cosh (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a^2}+\frac {\left (d \cosh \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}{4 (-a)^{5/2}}-\frac {\left (d \cosh \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}{4 (-a)^{5/2}}+\frac {\left (d \sinh \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}{4 (-a)^{5/2}}+\frac {\left (d \sinh \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}{4 (-a)^{5/2}} \\ & = \frac {\cosh (c+d x)}{4 a \left (a+b x^2\right )^2}+\frac {\cosh (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac {\cosh (c) \text {Chi}(d x)}{a^3}-\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^3}-\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^3}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2} \sqrt {b}}-\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2} \sqrt {b}}+\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {\sinh (c) \text {Shi}(d x)}{a^3}+\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2} \sqrt {b}}+\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^3}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{5/2} \sqrt {b}}-\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^3}+\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 (-a)^{5/2}}+\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 (-a)^{5/2}}+\frac {\left (d^2 \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}+b x} \, dx}{16 a^2}-\frac {\left (d^2 \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}-b x} \, dx}{16 a^2}+\frac {\left (d^2 \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}+b x} \, dx}{16 a^2}+\frac {\left (d^2 \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}-b x} \, dx}{16 a^2} \\ & = \frac {\cosh (c+d x)}{4 a \left (a+b x^2\right )^2}+\frac {\cosh (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac {\cosh (c) \text {Chi}(d x)}{a^3}-\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^3}+\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^2 b}-\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^3}+\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^2 b}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2} \sqrt {b}}-\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2} \sqrt {b}}+\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {\sinh (c) \text {Shi}(d x)}{a^3}+\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2} \sqrt {b}}+\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^3}-\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^2 b}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{5/2} \sqrt {b}}-\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^3}+\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^2 b}+\frac {\left (d \cosh \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}{16 (-a)^{5/2}}-\frac {\left (d \cosh \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}{16 (-a)^{5/2}}+\frac {\left (d \sinh \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}{16 (-a)^{5/2}}+\frac {\left (d \sinh \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}{16 (-a)^{5/2}} \\ & = \frac {\cosh (c+d x)}{4 a \left (a+b x^2\right )^2}+\frac {\cosh (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac {\cosh (c) \text {Chi}(d x)}{a^3}-\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^3}+\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^2 b}-\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^3}+\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^2 b}+\frac {5 d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {5 d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {\sinh (c) \text {Shi}(d x)}{a^3}+\frac {5 d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{5/2} \sqrt {b}}+\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^3}-\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^2 b}+\frac {5 d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{5/2} \sqrt {b}}-\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^3}+\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^2 b} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.60 (sec) , antiderivative size = 674, normalized size of antiderivative = 0.92 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^3} \, dx=\frac {\frac {8 a \left (3 a+2 b x^2\right ) \cosh (c+d x)}{\left (a+b x^2\right )^2}+32 \cosh (c) \text {Chi}(d x)+\frac {4 i \sqrt {a} d 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 )}{\sqrt {b}}-8 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 )+\frac {\sqrt {a} d e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\left (-i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )}{b}+\frac {4 i \sqrt {a} d 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 )}{\sqrt {b}}-8 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 )+\frac {\sqrt {a} d e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\left (-i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )}{b}-\frac {4 a d x \sinh (c+d x)}{a+b x^2}+32 \sinh (c) \text {Shi}(d x)}{32 a^3} \]
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Time = 0.38 (sec) , antiderivative size = 1090, normalized size of antiderivative = 1.49
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Leaf count of result is larger than twice the leaf count of optimal. 2076 vs. \(2 (575) = 1150\).
Time = 0.29 (sec) , antiderivative size = 2076, normalized size of antiderivative = 2.84 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^3} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3} x} \,d x } \]
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\[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^3} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3} x} \,d x } \]
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Timed out. \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^3} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x\,{\left (b\,x^2+a\right )}^3} \,d x \]
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