Optimal. Leaf size=51 \[ \frac {a \sinh (c+d x)}{d}+\frac {2 b \sinh (c+d x)}{d^3}-\frac {2 b x \cosh (c+d x)}{d^2}+\frac {b x^2 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.07, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5277, 2637, 3296} \[ \frac {a \sinh (c+d x)}{d}+\frac {2 b \sinh (c+d x)}{d^3}-\frac {2 b x \cosh (c+d x)}{d^2}+\frac {b x^2 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 5277
Rubi steps
\begin {align*} \int \left (a+b x^2\right ) \cosh (c+d x) \, dx &=\int \left (a \cosh (c+d x)+b x^2 \cosh (c+d x)\right ) \, dx\\ &=a \int \cosh (c+d x) \, dx+b \int x^2 \cosh (c+d x) \, dx\\ &=\frac {a \sinh (c+d x)}{d}+\frac {b x^2 \sinh (c+d x)}{d}-\frac {(2 b) \int x \sinh (c+d x) \, dx}{d}\\ &=-\frac {2 b x \cosh (c+d x)}{d^2}+\frac {a \sinh (c+d x)}{d}+\frac {b x^2 \sinh (c+d x)}{d}+\frac {(2 b) \int \cosh (c+d x) \, dx}{d^2}\\ &=-\frac {2 b x \cosh (c+d x)}{d^2}+\frac {2 b \sinh (c+d x)}{d^3}+\frac {a \sinh (c+d x)}{d}+\frac {b x^2 \sinh (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 40, normalized size = 0.78 \[ \frac {\left (a d^2+b \left (d^2 x^2+2\right )\right ) \sinh (c+d x)-2 b d x \cosh (c+d x)}{d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 42, normalized size = 0.82 \[ -\frac {2 \, b d x \cosh \left (d x + c\right ) - {\left (b d^{2} x^{2} + a d^{2} + 2 \, b\right )} \sinh \left (d x + c\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 70, normalized size = 1.37 \[ \frac {{\left (b d^{2} x^{2} + a d^{2} - 2 \, b d x + 2 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{3}} - \frac {{\left (b d^{2} x^{2} + a d^{2} + 2 \, b d x + 2 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 97, normalized size = 1.90 \[ \frac {\frac {b \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}-\frac {2 b c \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}+\frac {b \,c^{2} \sinh \left (d x +c \right )}{d^{2}}+a \sinh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 86, normalized size = 1.69 \[ \frac {a e^{\left (d x + c\right )}}{2 \, d} - \frac {a e^{\left (-d x - c\right )}}{2 \, d} + \frac {{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} b e^{\left (d x\right )}}{2 \, d^{3}} - \frac {{\left (d^{2} x^{2} + 2 \, d x + 2\right )} b e^{\left (-d x - c\right )}}{2 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 47, normalized size = 0.92 \[ \frac {\mathrm {sinh}\left (c+d\,x\right )\,\left (a\,d^2+2\,b\right )}{d^3}-\frac {2\,b\,x\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {b\,x^2\,\mathrm {sinh}\left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 65, normalized size = 1.27 \[ \begin {cases} \frac {a \sinh {\left (c + d x \right )}}{d} + \frac {b x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {2 b x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 b \sinh {\left (c + d x \right )}}{d^{3}} & \text {for}\: d \neq 0 \\\left (a x + \frac {b x^{3}}{3}\right ) \cosh {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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