Optimal. Leaf size=44 \[ \frac {1}{2} \text {Int}\left (\frac {\cosh \left (2 a+2 b x+2 c x^2\right )}{d+e x},x\right )-\frac {\log (d+e x)}{2 e} \]
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Rubi [A] time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sinh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sinh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx &=\int \left (-\frac {1}{2 (d+e x)}+\frac {\cosh \left (2 a+2 b x+2 c x^2\right )}{2 (d+e x)}\right ) \, dx\\ &=-\frac {\log (d+e x)}{2 e}+\frac {1}{2} \int \frac {\cosh \left (2 a+2 b x+2 c x^2\right )}{d+e x} \, dx\\ \end {align*}
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Mathematica [A] time = 15.98, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sinh \left (c x^{2} + b x + a\right )^{2}}{e x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (c x^{2} + b x + a\right )^{2}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{2}\left (c \,x^{2}+b x +a \right )}{e x +d}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\log \left (e x + d\right )}{2 \, e} + \frac {1}{4} \, \int \frac {e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a\right )}}{e x + d}\,{d x} + \frac {1}{4} \, \int \frac {e^{\left (-2 \, c x^{2} - 2 \, b x\right )}}{e x e^{\left (2 \, a\right )} + d e^{\left (2 \, a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {sinh}\left (c\,x^2+b\,x+a\right )}^2}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{2}{\left (a + b x + c x^{2} \right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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