Optimal. Leaf size=49 \[ \frac {2 d^2 \sinh (a+b x)}{b^3}-\frac {2 d (c+d x) \cosh (a+b x)}{b^2}+\frac {(c+d x)^2 \sinh (a+b x)}{b} \]
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Rubi [A] time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3296, 2637} \[ -\frac {2 d (c+d x) \cosh (a+b x)}{b^2}+\frac {2 d^2 \sinh (a+b x)}{b^3}+\frac {(c+d x)^2 \sinh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rubi steps
\begin {align*} \int (c+d x)^2 \cosh (a+b x) \, dx &=\frac {(c+d x)^2 \sinh (a+b x)}{b}-\frac {(2 d) \int (c+d x) \sinh (a+b x) \, dx}{b}\\ &=-\frac {2 d (c+d x) \cosh (a+b x)}{b^2}+\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {\left (2 d^2\right ) \int \cosh (a+b x) \, dx}{b^2}\\ &=-\frac {2 d (c+d x) \cosh (a+b x)}{b^2}+\frac {2 d^2 \sinh (a+b x)}{b^3}+\frac {(c+d x)^2 \sinh (a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 44, normalized size = 0.90 \[ \frac {\sinh (a+b x) \left (b^2 (c+d x)^2+2 d^2\right )-2 b d (c+d x) \cosh (a+b x)}{b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 64, normalized size = 1.31 \[ -\frac {2 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, d^{2}\right )} \sinh \left (b x + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 112, normalized size = 2.29 \[ \frac {{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, b d^{2} x - 2 \, b c d + 2 \, d^{2}\right )} e^{\left (b x + a\right )}}{2 \, b^{3}} - \frac {{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, b d^{2} x + 2 \, b c d + 2 \, d^{2}\right )} e^{\left (-b x - a\right )}}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 147, normalized size = 3.00 \[ \frac {\frac {d^{2} \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{2}}-\frac {2 d^{2} a \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{2}}+\frac {2 d c \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b}+\frac {d^{2} a^{2} \sinh \left (b x +a \right )}{b^{2}}-\frac {2 d a c \sinh \left (b x +a \right )}{b}+c^{2} \sinh \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 135, normalized size = 2.76 \[ \frac {c^{2} e^{\left (b x + a\right )}}{2 \, b} + \frac {{\left (b x e^{a} - e^{a}\right )} c d e^{\left (b x\right )}}{b^{2}} - \frac {c^{2} e^{\left (-b x - a\right )}}{2 \, b} - \frac {{\left (b x + 1\right )} c d e^{\left (-b x - a\right )}}{b^{2}} + \frac {{\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} d^{2} e^{\left (b x\right )}}{2 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, b x + 2\right )} d^{2} e^{\left (-b x - a\right )}}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 82, normalized size = 1.67 \[ \frac {\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2\,c^2+2\,d^2\right )}{b^3}+\frac {d^2\,x^2\,\mathrm {sinh}\left (a+b\,x\right )}{b}-\frac {2\,c\,d\,\mathrm {cosh}\left (a+b\,x\right )}{b^2}-\frac {2\,d^2\,x\,\mathrm {cosh}\left (a+b\,x\right )}{b^2}+\frac {2\,c\,d\,x\,\mathrm {sinh}\left (a+b\,x\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.56, size = 112, normalized size = 2.29 \[ \begin {cases} \frac {c^{2} \sinh {\left (a + b x \right )}}{b} + \frac {2 c d x \sinh {\left (a + b x \right )}}{b} + \frac {d^{2} x^{2} \sinh {\left (a + b x \right )}}{b} - \frac {2 c d \cosh {\left (a + b x \right )}}{b^{2}} - \frac {2 d^{2} x \cosh {\left (a + b x \right )}}{b^{2}} + \frac {2 d^{2} \sinh {\left (a + b x \right )}}{b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \cosh {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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