Optimal. Leaf size=110 \[ a^2 \cosh (c) \text {Chi}(d x)+a^2 \sinh (c) \text {Shi}(d x)-\frac {2 a b \cosh (c+d x)}{d^2}+\frac {2 a b x \sinh (c+d x)}{d}-\frac {6 b^2 \cosh (c+d x)}{d^4}+\frac {6 b^2 x \sinh (c+d x)}{d^3}-\frac {3 b^2 x^2 \cosh (c+d x)}{d^2}+\frac {b^2 x^3 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.19, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5287, 3303, 3298, 3301, 3296, 2638} \[ a^2 \cosh (c) \text {Chi}(d x)+a^2 \sinh (c) \text {Shi}(d x)-\frac {2 a b \cosh (c+d x)}{d^2}+\frac {2 a b x \sinh (c+d x)}{d}-\frac {3 b^2 x^2 \cosh (c+d x)}{d^2}+\frac {6 b^2 x \sinh (c+d x)}{d^3}-\frac {6 b^2 \cosh (c+d x)}{d^4}+\frac {b^2 x^3 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3298
Rule 3301
Rule 3303
Rule 5287
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x} \, dx &=\int \left (\frac {a^2 \cosh (c+d x)}{x}+2 a b x \cosh (c+d x)+b^2 x^3 \cosh (c+d x)\right ) \, dx\\ &=a^2 \int \frac {\cosh (c+d x)}{x} \, dx+(2 a b) \int x \cosh (c+d x) \, dx+b^2 \int x^3 \cosh (c+d x) \, dx\\ &=\frac {2 a b x \sinh (c+d x)}{d}+\frac {b^2 x^3 \sinh (c+d x)}{d}-\frac {(2 a b) \int \sinh (c+d x) \, dx}{d}-\frac {\left (3 b^2\right ) \int x^2 \sinh (c+d x) \, dx}{d}+\left (a^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\left (a^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx\\ &=-\frac {2 a b \cosh (c+d x)}{d^2}-\frac {3 b^2 x^2 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text {Chi}(d x)+\frac {2 a b x \sinh (c+d x)}{d}+\frac {b^2 x^3 \sinh (c+d x)}{d}+a^2 \sinh (c) \text {Shi}(d x)+\frac {\left (6 b^2\right ) \int x \cosh (c+d x) \, dx}{d^2}\\ &=-\frac {2 a b \cosh (c+d x)}{d^2}-\frac {3 b^2 x^2 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text {Chi}(d x)+\frac {6 b^2 x \sinh (c+d x)}{d^3}+\frac {2 a b x \sinh (c+d x)}{d}+\frac {b^2 x^3 \sinh (c+d x)}{d}+a^2 \sinh (c) \text {Shi}(d x)-\frac {\left (6 b^2\right ) \int \sinh (c+d x) \, dx}{d^3}\\ &=-\frac {6 b^2 \cosh (c+d x)}{d^4}-\frac {2 a b \cosh (c+d x)}{d^2}-\frac {3 b^2 x^2 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text {Chi}(d x)+\frac {6 b^2 x \sinh (c+d x)}{d^3}+\frac {2 a b x \sinh (c+d x)}{d}+\frac {b^2 x^3 \sinh (c+d x)}{d}+a^2 \sinh (c) \text {Shi}(d x)\\ \end {align*}
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Mathematica [A] time = 0.39, size = 82, normalized size = 0.75 \[ a^2 \cosh (c) \text {Chi}(d x)+a^2 \sinh (c) \text {Shi}(d x)-\frac {b \left (2 a d^2+3 b \left (d^2 x^2+2\right )\right ) \cosh (c+d x)}{d^4}+\frac {b x \left (2 a d^2+b \left (d^2 x^2+6\right )\right ) \sinh (c+d x)}{d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 130, normalized size = 1.18 \[ -\frac {2 \, {\left (3 \, b^{2} d^{2} x^{2} + 2 \, a b d^{2} + 6 \, b^{2}\right )} \cosh \left (d x + c\right ) - {\left (a^{2} d^{4} {\rm Ei}\left (d x\right ) + a^{2} d^{4} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) - 2 \, {\left (b^{2} d^{3} x^{3} + 2 \, {\left (a b d^{3} + 3 \, b^{2} d\right )} x\right )} \sinh \left (d x + c\right ) - {\left (a^{2} d^{4} {\rm Ei}\left (d x\right ) - a^{2} d^{4} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{2 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 222, normalized size = 2.02 \[ \frac {b^{2} d^{3} x^{3} e^{\left (d x + c\right )} - b^{2} d^{3} x^{3} e^{\left (-d x - c\right )} + a^{2} d^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{4} {\rm Ei}\left (d x\right ) e^{c} + 2 \, a b d^{3} x e^{\left (d x + c\right )} - 3 \, b^{2} d^{2} x^{2} e^{\left (d x + c\right )} - 2 \, a b d^{3} x e^{\left (-d x - c\right )} - 3 \, b^{2} d^{2} x^{2} e^{\left (-d x - c\right )} - 2 \, a b d^{2} e^{\left (d x + c\right )} + 6 \, b^{2} d x e^{\left (d x + c\right )} - 2 \, a b d^{2} e^{\left (-d x - c\right )} - 6 \, b^{2} d x e^{\left (-d x - c\right )} - 6 \, b^{2} e^{\left (d x + c\right )} - 6 \, b^{2} e^{\left (-d x - c\right )}}{2 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 226, normalized size = 2.05 \[ -\frac {3 b^{2} {\mathrm e}^{d x +c} x^{2}}{2 d^{2}}+\frac {3 b^{2} {\mathrm e}^{d x +c} x}{d^{3}}+\frac {b^{2} {\mathrm e}^{d x +c} x^{3}}{2 d}-\frac {3 b^{2} {\mathrm e}^{-d x -c} x}{d^{3}}-\frac {b^{2} {\mathrm e}^{-d x -c} x^{3}}{2 d}-\frac {3 b^{2} {\mathrm e}^{-d x -c} x^{2}}{2 d^{2}}-\frac {a b \,{\mathrm e}^{-d x -c}}{d^{2}}-\frac {a b \,{\mathrm e}^{d x +c}}{d^{2}}-\frac {a^{2} {\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}-\frac {a^{2} {\mathrm e}^{c} \Ei \left (1, -d x \right )}{2}-\frac {3 b^{2} {\mathrm e}^{d x +c}}{d^{4}}-\frac {3 b^{2} {\mathrm e}^{-d x -c}}{d^{4}}-\frac {a b \,{\mathrm e}^{-d x -c} x}{d}+\frac {a b \,{\mathrm e}^{d x +c} x}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 235, normalized size = 2.14 \[ -\frac {1}{8} \, {\left (4 \, a b {\left (\frac {{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} e^{\left (d x\right )}}{d^{3}} + \frac {{\left (d^{2} x^{2} + 2 \, d x + 2\right )} e^{\left (-d x - c\right )}}{d^{3}}\right )} + b^{2} {\left (\frac {{\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} e^{\left (d x\right )}}{d^{5}} + \frac {{\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} e^{\left (-d x - c\right )}}{d^{5}}\right )} + \frac {4 \, a^{2} \cosh \left (d x + c\right ) \log \left (x^{2}\right )}{d} - \frac {4 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} a^{2}}{d}\right )} d + \frac {1}{4} \, {\left (b^{2} x^{4} + 4 \, a b x^{2} + 2 \, a^{2} \log \left (x^{2}\right )\right )} \cosh \left (d x + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (b\,x^2+a\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.77, size = 121, normalized size = 1.10 \[ a^{2} \sinh {\relax (c )} \operatorname {Shi}{\left (d x \right )} + a^{2} \cosh {\relax (c )} \operatorname {Chi}\left (d x\right ) + 2 a b \left (\begin {cases} \frac {x \sinh {\left (c + d x \right )}}{d} - \frac {\cosh {\left (c + d x \right )}}{d^{2}} & \text {for}\: d \neq 0 \\\frac {x^{2} \cosh {\relax (c )}}{2} & \text {otherwise} \end {cases}\right ) + b^{2} \left (\begin {cases} \frac {x^{3} \sinh {\left (c + d x \right )}}{d} - \frac {3 x^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {6 x \sinh {\left (c + d x \right )}}{d^{3}} - \frac {6 \cosh {\left (c + d x \right )}}{d^{4}} & \text {for}\: d \neq 0 \\\frac {x^{4} \cosh {\relax (c )}}{4} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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