Optimal. Leaf size=500 \[ \frac {d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}-\frac {d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \sinh (c) \text {Chi}(d x)}{a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}-\frac {\cosh (c+d x)}{a^2 x}-\frac {3 \sqrt {b} \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}} \]
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Rubi [A] time = 1.25, antiderivative size = 500, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5293, 3297, 3303, 3298, 3301, 5281} \[ \frac {d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}-\frac {d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \sinh (c) \text {Chi}(d x)}{a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}-\frac {\cosh (c+d x)}{a^2 x}-\frac {3 \sqrt {b} \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5281
Rule 5293
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx &=\int \left (\frac {\cosh (c+d x)}{a^2 x^2}-\frac {b \cosh (c+d x)}{a \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^2 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\cosh (c+d x)}{x^2} \, dx}{a^2}-\frac {b \int \frac {\cosh (c+d x)}{a+b x^2} \, dx}{a^2}-\frac {b \int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{a}\\ &=-\frac {\cosh (c+d x)}{a^2 x}-\frac {b \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}{a^2}-\frac {b \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}{a}+\frac {d \int \frac {\sinh (c+d x)}{x} \, dx}{a^2}\\ &=-\frac {\cosh (c+d x)}{a^2 x}+\frac {b \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 (-a)^{5/2}}+\frac {b \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 (-a)^{5/2}}+\frac {b^2 \int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{4 a^2}+\frac {b^2 \int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{4 a^2}+\frac {b^2 \int \frac {\cosh (c+d x)}{-a b-b^2 x^2} \, dx}{2 a^2}+\frac {(d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx}{a^2}+\frac {(d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx}{a^2}\\ &=-\frac {\cosh (c+d x)}{a^2 x}+\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \text {Chi}(d x) \sinh (c)}{a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}+\frac {b^2 \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}{2 a^2}-\frac {(b d) \int \frac {\sinh (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{4 a^2}+\frac {(b d) \int \frac {\sinh (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{4 a^2}+\frac {\left (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)^{5/2}}+\frac {\left (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)^{5/2}}+\frac {\left (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)^{5/2}}-\frac {\left (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)^{5/2}}\\ &=-\frac {\cosh (c+d x)}{a^2 x}+\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{5/2}}+\frac {\sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{5/2}}+\frac {d \text {Chi}(d x) \sinh (c)}{a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}+\frac {\sqrt {b} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{5/2}}+\frac {\sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{5/2}}+\frac {b \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 (-a)^{5/2}}+\frac {b \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 (-a)^{5/2}}+\frac {\left (b 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}{4 a^2}+\frac {\left (b 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}{4 a^2}+\frac {\left (b 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}{4 a^2}-\frac {\left (b 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}{4 a^2}\\ &=-\frac {\cosh (c+d x)}{a^2 x}+\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{5/2}}+\frac {\sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{5/2}}+\frac {d \text {Chi}(d x) \sinh (c)}{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^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^2}+\frac {d \cosh (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^2}+\frac {\sqrt {b} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{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 )}{4 a^2}+\frac {\sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{5/2}}+\frac {\left (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}{4 (-a)^{5/2}}+\frac {\left (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}{4 (-a)^{5/2}}+\frac {\left (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}{4 (-a)^{5/2}}-\frac {\left (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}{4 (-a)^{5/2}}\\ &=-\frac {\cosh (c+d x)}{a^2 x}+\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 \sqrt {b} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{5/2}}+\frac {d \text {Chi}(d x) \sinh (c)}{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^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^2}+\frac {d \cosh (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^2}+\frac {3 \sqrt {b} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{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 )}{4 a^2}+\frac {3 \sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 1.09, size = 675, normalized size = 1.35 \[ \frac {i a^{3/2} d x \cosh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-i a^{3/2} d x \cosh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )+4 a^{3/2} d x \sinh (c) \text {Chi}(d x)+4 a^{3/2} d x \cosh (c) \text {Shi}(d x)-4 a^{3/2} \cosh (c+d x)+3 b^{3/2} x^3 \sinh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+3 b^{3/2} x^3 \sinh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )+4 \sqrt {a} b d x^3 \sinh (c) \text {Chi}(d x)+x \left (a+b x^2\right ) \text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\sqrt {a} d \sinh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )-3 i \sqrt {b} \cosh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )+x \left (a+b x^2\right ) \text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\sqrt {a} d \sinh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )+3 i \sqrt {b} \cosh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )+4 \sqrt {a} b d x^3 \cosh (c) \text {Shi}(d x)+i \sqrt {a} b d x^3 \cosh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-i \sqrt {a} b d x^3 \cosh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )+3 a \sqrt {b} x \sinh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+3 a \sqrt {b} x \sinh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )-6 \sqrt {a} b x^2 \cosh (c+d x)}{4 a^{5/2} x \left (a+b x^2\right )} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.64, size = 1310, normalized size = 2.62 \[ \text {result too large to display} \]
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 \[ \int \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 595, normalized size = 1.19 \[ -\frac {3 \,{\mathrm e}^{-d x -c} x \,d^{2} b}{4 a^{2} \left (b \,d^{2} x^{2}+a \,d^{2}\right )}-\frac {{\mathrm e}^{-d x -c} d^{2}}{2 a x \left (b \,d^{2} x^{2}+a \,d^{2}\right )}+\frac {d \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2 a^{2}}+\frac {d \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{8 a^{2}}+\frac {d \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{8 a^{2}}+\frac {3 \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) b}{8 a^{2} \sqrt {-a b}}-\frac {3 \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) b}{8 a^{2} \sqrt {-a b}}-\frac {3 \,{\mathrm e}^{d x +c} x \,d^{2} b}{4 a^{2} \left (b \,d^{2} x^{2}+a \,d^{2}\right )}-\frac {{\mathrm e}^{d x +c} d^{2}}{2 a x \left (b \,d^{2} x^{2}+a \,d^{2}\right )}-\frac {d \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2 a^{2}}-\frac {d \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{8 a^{2}}-\frac {d \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{8 a^{2}}+\frac {3 \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) b}{8 a^{2} \sqrt {-a b}}-\frac {3 \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) b}{8 a^{2} \sqrt {-a b}} \]
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 \[ \int \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2} x^{2}}\,{d x} \]
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 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x^2\,{\left (b\,x^2+a\right )}^2} \,d x \]
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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