Optimal. Leaf size=856 \[ \text{result too large to display} \]
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Rubi [A] time = 1.25523, antiderivative size = 856, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5281, 3297, 3303, 3298, 3301} \[ \frac{\cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac{\cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac{\sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac{\sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac{3 \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) d}{16 a^2 b}-\frac{3 \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) d}{16 a^2 b}+\frac{\sinh (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sinh (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt{b} x+\sqrt{-a}\right )}+\frac{3 \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) d}{16 a^2 b}-\frac{3 \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) d}{16 a^2 b}-\frac{3 \cosh (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{3 \cosh (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{b} x+\sqrt{-a}\right )}-\frac{\cosh (c+d x)}{16 (-a)^{3/2} \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )^2}+\frac{\cosh (c+d x)}{16 (-a)^{3/2} \sqrt{b} \left (\sqrt{b} x+\sqrt{-a}\right )^2}+\frac{3 \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{5/2} \sqrt{b}}-\frac{3 \cosh \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{3 \sinh \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{3 \sinh \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}} \]
Antiderivative was successfully verified.
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Rule 5281
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx &=\int \left (-\frac{b^{3/2} \cosh (c+d x)}{8 (-a)^{3/2} \left (\sqrt{-a} \sqrt{b}-b x\right )^3}-\frac{3 b \cosh (c+d x)}{16 a^2 \left (\sqrt{-a} \sqrt{b}-b x\right )^2}-\frac{b^{3/2} \cosh (c+d x)}{8 (-a)^{3/2} \left (\sqrt{-a} \sqrt{b}+b x\right )^3}-\frac{3 b \cosh (c+d x)}{16 a^2 \left (\sqrt{-a} \sqrt{b}+b x\right )^2}-\frac{3 b \cosh (c+d x)}{8 a^2 \left (-a b-b^2 x^2\right )}\right ) \, dx\\ &=-\frac{(3 b) \int \frac{\cosh (c+d x)}{\left (\sqrt{-a} \sqrt{b}-b x\right )^2} \, dx}{16 a^2}-\frac{(3 b) \int \frac{\cosh (c+d x)}{\left (\sqrt{-a} \sqrt{b}+b x\right )^2} \, dx}{16 a^2}-\frac{(3 b) \int \frac{\cosh (c+d x)}{-a b-b^2 x^2} \, dx}{8 a^2}-\frac{b^{3/2} \int \frac{\cosh (c+d x)}{\left (\sqrt{-a} \sqrt{b}-b x\right )^3} \, dx}{8 (-a)^{3/2}}-\frac{b^{3/2} \int \frac{\cosh (c+d x)}{\left (\sqrt{-a} \sqrt{b}+b x\right )^3} \, dx}{8 (-a)^{3/2}}\\ &=-\frac{\cosh (c+d x)}{16 (-a)^{3/2} \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )^2}-\frac{3 \cosh (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\cosh (c+d x)}{16 (-a)^{3/2} \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )^2}+\frac{3 \cosh (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{(3 b) \int \left (-\frac{\sqrt{-a} \cosh (c+d x)}{2 a b \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{\sqrt{-a} \cosh (c+d x)}{2 a b \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{8 a^2}+\frac{(3 d) \int \frac{\sinh (c+d x)}{\sqrt{-a} \sqrt{b}-b x} \, dx}{16 a^2}-\frac{(3 d) \int \frac{\sinh (c+d x)}{\sqrt{-a} \sqrt{b}+b x} \, dx}{16 a^2}+\frac{\left (\sqrt{b} d\right ) \int \frac{\sinh (c+d x)}{\left (\sqrt{-a} \sqrt{b}-b x\right )^2} \, dx}{16 (-a)^{3/2}}-\frac{\left (\sqrt{b} d\right ) \int \frac{\sinh (c+d x)}{\left (\sqrt{-a} \sqrt{b}+b x\right )^2} \, dx}{16 (-a)^{3/2}}\\ &=-\frac{\cosh (c+d x)}{16 (-a)^{3/2} \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )^2}-\frac{3 \cosh (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\cosh (c+d x)}{16 (-a)^{3/2} \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )^2}+\frac{3 \cosh (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{d \sinh (c+d x)}{16 (-a)^{3/2} b \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{d \sinh (c+d x)}{16 (-a)^{3/2} b \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{3 \int \frac{\cosh (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{16 (-a)^{5/2}}-\frac{3 \int \frac{\cosh (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{16 (-a)^{5/2}}-\frac{d^2 \int \frac{\cosh (c+d x)}{\sqrt{-a} \sqrt{b}-b x} \, dx}{16 (-a)^{3/2} \sqrt{b}}-\frac{d^2 \int \frac{\cosh (c+d x)}{\sqrt{-a} \sqrt{b}+b x} \, dx}{16 (-a)^{3/2} \sqrt{b}}-\frac{\left (3 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}+b x} \, dx}{16 a^2}-\frac{\left (3 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}-b x} \, dx}{16 a^2}-\frac{\left (3 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}+b x} \, dx}{16 a^2}+\frac{\left (3 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}-b x} \, dx}{16 a^2}\\ &=-\frac{\cosh (c+d x)}{16 (-a)^{3/2} \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )^2}-\frac{3 \cosh (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\cosh (c+d x)}{16 (-a)^{3/2} \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )^2}+\frac{3 \cosh (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{3 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^2 b}-\frac{3 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^2 b}+\frac{d \sinh (c+d x)}{16 (-a)^{3/2} b \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{d \sinh (c+d x)}{16 (-a)^{3/2} b \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{3 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^2 b}-\frac{3 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^2 b}-\frac{\left (3 \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}{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)^{3/2} \sqrt{b}}-\frac{\left (3 \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}{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)^{3/2} \sqrt{b}}-\frac{\left (3 \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}{16 (-a)^{5/2}}-\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)^{3/2} \sqrt{b}}+\frac{\left (3 \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}{16 (-a)^{5/2}}+\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)^{3/2} \sqrt{b}}\\ &=-\frac{\cosh (c+d x)}{16 (-a)^{3/2} \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )^2}-\frac{3 \cosh (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\cosh (c+d x)}{16 (-a)^{3/2} \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )^2}+\frac{3 \cosh (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{3 \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{5/2} \sqrt{b}}+\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)^{3/2} b^{3/2}}-\frac{3 \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 (-a)^{5/2} \sqrt{b}}-\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)^{3/2} b^{3/2}}-\frac{3 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^2 b}-\frac{3 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^2 b}+\frac{d \sinh (c+d x)}{16 (-a)^{3/2} b \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{d \sinh (c+d x)}{16 (-a)^{3/2} b \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{3 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^2 b}-\frac{3 \sinh \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{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)^{3/2} b^{3/2}}-\frac{3 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^2 b}-\frac{3 \sinh \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{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)^{3/2} b^{3/2}}\\ \end{align*}
Mathematica [C] time = 2.38417, size = 933, normalized size = 1.09 \[ \frac{\frac{6 b^{5/2} \cosh (c) \cosh (d x) x^3}{\left (b x^2+a\right )^2}+\frac{6 b^{5/2} \sinh (c) \sinh (d x) x^3}{\left (b x^2+a\right )^2}+\frac{2 a b^{3/2} d \cosh (d x) \sinh (c) x^2}{\left (b x^2+a\right )^2}+\frac{2 a b^{3/2} d \cosh (c) \sinh (d x) x^2}{\left (b x^2+a\right )^2}+\frac{10 a b^{3/2} \cosh (c) \cosh (d x) x}{\left (b x^2+a\right )^2}+\frac{10 a b^{3/2} \sinh (c) \sinh (d x) x}{\left (b x^2+a\right )^2}+\frac{2 a^2 \sqrt{b} d \cosh (d x) \sinh (c)}{\left (b x^2+a\right )^2}+\frac{\text{CosIntegral}\left (i d x-\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (i \left (3 b-a d^2\right ) \cosh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )-3 \sqrt{a} \sqrt{b} d \sinh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )}{\sqrt{a}}+\frac{i \text{CosIntegral}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\left (a d^2-3 b\right ) \cosh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )+3 i \sqrt{a} \sqrt{b} d \sinh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )}{\sqrt{a}}+\frac{2 a^2 \sqrt{b} d \cosh (c) \sinh (d x)}{\left (b x^2+a\right )^2}-3 i \sqrt{b} d \cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \cosh (c) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )-i \sqrt{a} d^2 \cosh (c) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )+\frac{3 i b \cosh (c) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )}{\sqrt{a}}+\sqrt{a} d^2 \cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sinh (c) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )-\frac{3 b \cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sinh (c) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )}{\sqrt{a}}-3 \sqrt{b} d \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sinh (c) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )+3 i \sqrt{b} d \cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \cosh (c) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )+i \sqrt{a} d^2 \cosh (c) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )-\frac{3 i b \cosh (c) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )}{\sqrt{a}}+\sqrt{a} d^2 \cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sinh (c) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )-\frac{3 b \cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sinh (c) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )}{\sqrt{a}}-3 \sqrt{b} d \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sinh (c) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )}{16 a^2 b^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.037, size = 1064, normalized size = 1.2 \begin{align*} \text{result too large to display} \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 = 2.33339, size = 4342, normalized size = 5.07 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}{{\left (b x^{2} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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