Optimal. Leaf size=746 \[ \frac{d^2 \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d^2 \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a b^2}-\frac{d \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}-\frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{\cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{d^2 \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d^2 \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}+\frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{\sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{d \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}-\frac{d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a b^2}-\frac{d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2} \]
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Rubi [A] time = 1.10007, antiderivative size = 746, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {5291, 5281, 3297, 3303, 3298, 3301, 5288} \[ \frac{d^2 \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d^2 \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a b^2}-\frac{d \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}-\frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{\cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{d^2 \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d^2 \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}+\frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{\sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{d \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}-\frac{d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a b^2}-\frac{d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 5291
Rule 5281
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rule 5288
Rubi steps
\begin{align*} \int \frac{x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx &=-\frac{x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac{\int \frac{\cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 b}+\frac{d \int \frac{x \sinh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 b}\\ &=-\frac{x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}+\frac{\int \left (-\frac{b \cosh (c+d x)}{4 a \left (\sqrt{-a} \sqrt{b}-b x\right )^2}-\frac{b \cosh (c+d x)}{4 a \left (\sqrt{-a} \sqrt{b}+b x\right )^2}-\frac{b \cosh (c+d x)}{2 a \left (-a b-b^2 x^2\right )}\right ) \, dx}{4 b}+\frac{d^2 \int \frac{\cosh (c+d x)}{a+b x^2} \, dx}{8 b^2}\\ &=-\frac{x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{\int \frac{\cosh (c+d x)}{\left (\sqrt{-a} \sqrt{b}-b x\right )^2} \, dx}{16 a}-\frac{\int \frac{\cosh (c+d x)}{\left (\sqrt{-a} \sqrt{b}+b x\right )^2} \, dx}{16 a}-\frac{\int \frac{\cosh (c+d x)}{-a b-b^2 x^2} \, dx}{8 a}+\frac{d^2 \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}{8 b^2}\\ &=-\frac{\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{\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}+\frac{d \int \frac{\sinh (c+d x)}{\sqrt{-a} \sqrt{b}-b x} \, dx}{16 a b}-\frac{d \int \frac{\sinh (c+d x)}{\sqrt{-a} \sqrt{b}+b x} \, dx}{16 a b}-\frac{d^2 \int \frac{\cosh (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{16 \sqrt{-a} b^2}-\frac{d^2 \int \frac{\cosh (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{16 \sqrt{-a} b^2}\\ &=-\frac{\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}+\frac{\int \frac{\cosh (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{16 (-a)^{3/2} b}+\frac{\int \frac{\cosh (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{16 (-a)^{3/2} 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}+b x} \, dx}{16 a b}-\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} x} \, dx}{16 \sqrt{-a} b^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}-b x} \, dx}{16 a b}-\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} x} \, dx}{16 \sqrt{-a} b^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}+b x} \, dx}{16 a b}-\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} x} \, dx}{16 \sqrt{-a} b^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}-b x} \, dx}{16 a b}+\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} x} \, dx}{16 \sqrt{-a} b^2}\\ &=-\frac{\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}+\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 \sqrt{-a} b^{5/2}}-\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 \sqrt{-a} b^{5/2}}-\frac{d \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a b^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 )}{16 a b^2}-\frac{d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}+\frac{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 b^2}-\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 \sqrt{-a} b^{5/2}}-\frac{d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a b^2}-\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 \sqrt{-a} b^{5/2}}+\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}{16 (-a)^{3/2} b}+\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}{16 (-a)^{3/2} b}+\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}{16 (-a)^{3/2} b}-\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}{16 (-a)^{3/2} b}\\ &=-\frac{\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{\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{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 \sqrt{-a} b^{5/2}}+\frac{\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{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 \sqrt{-a} b^{5/2}}-\frac{d \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a b^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 )}{16 a b^2}-\frac{d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}+\frac{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 b^2}+\frac{\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{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 \sqrt{-a} b^{5/2}}-\frac{d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a b^2}+\frac{\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{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 \sqrt{-a} b^{5/2}}\\ \end{align*}
Mathematica [C] time = 2.50882, size = 932, normalized size = 1.25 \[ \frac{\frac{2 \sqrt{a} b^2 \cosh (c) \cosh (d x) x^3}{\left (b x^2+a\right )^2}+\frac{2 \sqrt{a} b^2 \sinh (c) \sinh (d x) x^3}{\left (b x^2+a\right )^2}-\frac{2 a^{3/2} b d \cosh (d x) \sinh (c) x^2}{\left (b x^2+a\right )^2}-\frac{2 a^{3/2} b d \cosh (c) \sinh (d x) x^2}{\left (b x^2+a\right )^2}-\frac{2 a^{3/2} b \cosh (c) \cosh (d x) x}{\left (b x^2+a\right )^2}-\frac{2 a^{3/2} b \sinh (c) \sinh (d x) x}{\left (b x^2+a\right )^2}-\frac{2 a^{5/2} d \cosh (d x) \sinh (c)}{\left (b x^2+a\right )^2}+\frac{i \text{CosIntegral}\left (i d x-\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\left (a d^2+b\right ) \cosh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )+i \sqrt{a} \sqrt{b} d \sinh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )}{\sqrt{b}}-\frac{i \text{CosIntegral}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\left (a d^2+b\right ) \cosh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )-i \sqrt{a} \sqrt{b} d \sinh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )}{\sqrt{b}}-\frac{2 a^{5/2} d \cosh (c) \sinh (d x)}{\left (b x^2+a\right )^2}-i \sqrt{a} 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 )+\frac{i 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 )}{\sqrt{b}}+i \sqrt{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 )-\frac{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 )}{\sqrt{b}}-\sqrt{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} 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 )+i \sqrt{a} 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 )-\frac{i 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 )}{\sqrt{b}}-i \sqrt{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 )-\frac{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 )}{\sqrt{b}}-\sqrt{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} 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^{3/2} b^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.108, size = 1064, normalized size = 1.4 \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 = 2.39734, size = 4169, normalized size = 5.59 \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{x^{2} \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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