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 {\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 {\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 {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 {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 {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2} \]
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Rubi [A] time = 1.10, antiderivative size = 746, normalized size of antiderivative = 1.00, 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 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5281
Rule 5288
Rule 5291
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*}
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Mathematica [C] time = 2.67, 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 {Ci}\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 {Ci}\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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fricas [B] time = 0.48, size = 2047, normalized size = 2.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 1064, normalized size = 1.43 \[ \frac {d^{5} {\mathrm e}^{-d x -c} x^{2}}{16 b \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {d^{4} {\mathrm e}^{-d x -c} x^{3}}{16 a \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {d^{5} {\mathrm e}^{-d x -c} a}{16 b^{2} \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{4} {\mathrm e}^{-d x -c} x}{16 b \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 b a \sqrt {-a b}}-\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 b a \sqrt {-a b}}+\frac {d^{2} {\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 b^{2} \sqrt {-a b}}-\frac {d^{2} {\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 b^{2} \sqrt {-a b}}-\frac {d \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 b^{2} a}-\frac {d \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 b^{2} a}-\frac {d^{2} {\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 b^{2} \sqrt {-a b}}+\frac {d^{2} {\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 b^{2} \sqrt {-a b}}+\frac {d \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 b^{2} a}+\frac {d \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 b^{2} a}-\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 b a \sqrt {-a b}}+\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 b a \sqrt {-a b}}-\frac {d^{5} {\mathrm e}^{d x +c} x^{2}}{16 b \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {d^{4} {\mathrm e}^{d x +c} x^{3}}{16 a \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{5} {\mathrm e}^{d x +c} a}{16 b^{2} \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{4} {\mathrm e}^{d x +c} x}{16 b \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (d x^{2} e^{\left (2 \, c\right )} + 4 \, x e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} - {\left (d x^{2} - 4 \, x\right )} e^{\left (-d x\right )}}{2 \, {\left (b^{3} d^{2} x^{6} e^{c} + 3 \, a b^{2} d^{2} x^{4} e^{c} + 3 \, a^{2} b d^{2} x^{2} e^{c} + a^{3} d^{2} e^{c}\right )}} + \frac {1}{2} \, \int -\frac {2 \, {\left (3 \, a d x e^{c} - 10 \, b x^{2} e^{c} + 2 \, a e^{c}\right )} e^{\left (d x\right )}}{b^{4} d^{2} x^{8} + 4 \, a b^{3} d^{2} x^{6} + 6 \, a^{2} b^{2} d^{2} x^{4} + 4 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}}\,{d x} + \frac {1}{2} \, \int \frac {2 \, {\left (3 \, a d x + 10 \, b x^{2} - 2 \, a\right )} e^{\left (-d x\right )}}{b^{4} d^{2} x^{8} e^{c} + 4 \, a b^{3} d^{2} x^{6} e^{c} + 6 \, a^{2} b^{2} d^{2} x^{4} e^{c} + 4 \, a^{3} b d^{2} x^{2} e^{c} + a^{4} d^{2} e^{c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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