Optimal. Leaf size=213 \[ \frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 \sqrt{-a} \sqrt{b}}-\frac{\cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 \sqrt{-a} \sqrt{b}}-\frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 \sqrt{-a} \sqrt{b}}-\frac{\sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 \sqrt{-a} \sqrt{b}} \]
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Rubi [A] time = 0.258696, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5281, 3303, 3298, 3301} \[ \frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 \sqrt{-a} \sqrt{b}}-\frac{\cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 \sqrt{-a} \sqrt{b}}-\frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 \sqrt{-a} \sqrt{b}}-\frac{\sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 \sqrt{-a} \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 5281
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh (c+d x)}{a+b x^2} \, dx &=\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\\ &=-\frac{\int \frac{\cosh (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 \sqrt{-a}}-\frac{\int \frac{\cosh (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 \sqrt{-a}}\\ &=-\frac{\cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \int \frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 \sqrt{-a}}-\frac{\cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \int \frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 \sqrt{-a}}-\frac{\sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \int \frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 \sqrt{-a}}+\frac{\sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \int \frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 \sqrt{-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 \sqrt{-a} \sqrt{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 \sqrt{-a} \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 \sqrt{-a} \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 \sqrt{-a} \sqrt{b}}\\ \end{align*}
Mathematica [C] time = 0.200531, size = 180, normalized size = 0.85 \[ \frac{i \left (\cosh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (-\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )-\cosh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+i \left (\sinh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )+\sinh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )\right )\right )}{2 \sqrt{a} \sqrt{b}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.021, size = 212, normalized size = 1. \begin{align*}{\frac{1}{4}{{\rm e}^{{\frac{1}{b} \left ( d\sqrt{-ab}-cb \right ) }}}{\it Ei} \left ( 1,{\frac{1}{b} \left ( d\sqrt{-ab}+ \left ( dx+c \right ) b-cb \right ) } \right ){\frac{1}{\sqrt{-ab}}}}-{\frac{1}{4}{{\rm e}^{-{\frac{1}{b} \left ( d\sqrt{-ab}+cb \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{b} \left ( d\sqrt{-ab}- \left ( dx+c \right ) b+cb \right ) } \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{1}{4}{{\rm e}^{-{\frac{1}{b} \left ( d\sqrt{-ab}-cb \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{b} \left ( d\sqrt{-ab}+ \left ( dx+c \right ) b-cb \right ) } \right ){\frac{1}{\sqrt{-ab}}}}-{\frac{1}{4}{{\rm e}^{{\frac{1}{b} \left ( d\sqrt{-ab}+cb \right ) }}}{\it Ei} \left ( 1,{\frac{1}{b} \left ( d\sqrt{-ab}- \left ( dx+c \right ) b+cb \right ) } \right ){\frac{1}{\sqrt{-ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.15703, size = 624, normalized size = 2.93 \begin{align*} -\frac{{\left (\sqrt{-\frac{a d^{2}}{b}}{\rm Ei}\left (d x - \sqrt{-\frac{a d^{2}}{b}}\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 (\sqrt{-\frac{a d^{2}}{b}}{\rm Ei}\left (d x + \sqrt{-\frac{a d^{2}}{b}}\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 (\sqrt{-\frac{a d^{2}}{b}}{\rm Ei}\left (d x - \sqrt{-\frac{a d^{2}}{b}}\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 ) +{\left (\sqrt{-\frac{a d^{2}}{b}}{\rm Ei}\left (d x + \sqrt{-\frac{a d^{2}}{b}}\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 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh{\left (c + d x \right )}}{a + b x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (d x + c\right )}{b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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