Integrand size = 19, antiderivative size = 435 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\frac {\cosh (c+d x)}{2 a \left (a+b x^2\right )}+\frac {\cosh (c) \text {Chi}(d x)}{a^2}-\frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^2}-\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sinh (c) \text {Shi}(d x)}{a^2}-\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^2} \]
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Time = 0.67 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5401, 3384, 3379, 3382, 5397, 5388} \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^2} \, dx=-\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}+\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}+\frac {\cosh (c) \text {Chi}(d x)}{a^2}+\frac {\sinh (c) \text {Shi}(d x)}{a^2}-\frac {d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\cosh (c+d x)}{2 a \left (a+b x^2\right )} \]
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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^2 x}-\frac {b x \cosh (c+d x)}{a \left (a+b x^2\right )^2}-\frac {b x \cosh (c+d x)}{a^2 \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {\cosh (c+d x)}{x} \, dx}{a^2}-\frac {b \int \frac {x \cosh (c+d x)}{a+b x^2} \, dx}{a^2}-\frac {b \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{a} \\ & = \frac {\cosh (c+d x)}{2 a \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^2}-\frac {d \int \frac {\sinh (c+d x)}{a+b x^2} \, dx}{2 a}+\frac {\cosh (c) \int \frac {\cosh (d x)}{x} \, dx}{a^2}+\frac {\sinh (c) \int \frac {\sinh (d x)}{x} \, dx}{a^2} \\ & = \frac {\cosh (c+d x)}{2 a \left (a+b x^2\right )}+\frac {\cosh (c) \text {Chi}(d x)}{a^2}+\frac {\sinh (c) \text {Shi}(d x)}{a^2}+\frac {\sqrt {b} \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 a^2}-\frac {\sqrt {b} \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 a^2}-\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} \\ & = \frac {\cosh (c+d x)}{2 a \left (a+b x^2\right )}+\frac {\cosh (c) \text {Chi}(d x)}{a^2}+\frac {\sinh (c) \text {Shi}(d x)}{a^2}-\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 (-a)^{3/2}}-\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 (-a)^{3/2}}-\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^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^2}-\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^2}-\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^2} \\ & = \frac {\cosh (c+d x)}{2 a \left (a+b x^2\right )}+\frac {\cosh (c) \text {Chi}(d x)}{a^2}-\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^2}-\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^2}+\frac {\sinh (c) \text {Shi}(d x)}{a^2}+\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^2}-\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^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)^{3/2}}+\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)^{3/2}}-\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)^{3/2}}-\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)^{3/2}} \\ & = \frac {\cosh (c+d x)}{2 a \left (a+b x^2\right )}+\frac {\cosh (c) \text {Chi}(d x)}{a^2}-\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^2}-\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^2}-\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)^{3/2} \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)^{3/2} \sqrt {b}}+\frac {\sinh (c) \text {Shi}(d x)}{a^2}-\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)^{3/2} \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^2}-\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)^{3/2} \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^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.84 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.95 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\frac {\frac {4 a \cosh (c) \cosh (d x)}{a+b x^2}+\frac {i \sqrt {a} d e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )}{\sqrt {b}}-2 e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )+\frac {i \sqrt {a} d e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )}{\sqrt {b}}-2 e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )+\frac {4 a \sinh (c) \sinh (d x)}{a+b x^2}+8 (\cosh (c) \text {Chi}(d x)+\sinh (c) \text {Shi}(d x))}{8 a^2} \]
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Time = 0.28 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.26
method | result | size |
risch | \(\frac {{\mathrm e}^{-d x -c} d^{2}}{4 a \left (\left (d x +c \right )^{2} b -2 b \left (d x +c \right ) c +a \,d^{2}+c^{2} b \right )}+\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) d}{8 a \sqrt {-a b}}-\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) d}{8 a \sqrt {-a b}}+\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 a^{2}}+\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 a^{2}}-\frac {{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{2 a^{2}}+\frac {{\mathrm e}^{d x +c} d^{2}}{4 a \left (\left (d x +c \right )^{2} b -2 b \left (d x +c \right ) c +a \,d^{2}+c^{2} b \right )}+\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) d}{8 a \sqrt {-a b}}-\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) d}{8 a \sqrt {-a b}}+\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 a^{2}}+\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 a^{2}}-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2 a^{2}}\) | \(546\) |
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Leaf count of result is larger than twice the leaf count of optimal. 992 vs. \(2 (337) = 674\).
Time = 0.27 (sec) , antiderivative size = 992, normalized size of antiderivative = 2.28 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\int \frac {\cosh {\left (c + d x \right )}}{x \left (a + b x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2} x} \,d x } \]
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Exception generated. \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\text {Exception raised: AttributeError} \]
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Timed out. \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x\,{\left (b\,x^2+a\right )}^2} \,d x \]
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