Optimal. Leaf size=64 \[ -\frac {a \cosh (c+d x)}{d^2}+\frac {a x \sinh (c+d x)}{d}+\frac {2 b \sinh (c+d x)}{d^3}-\frac {2 b x \cosh (c+d x)}{d^2}+\frac {b x^2 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.11, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6742, 3296, 2638, 2637} \[ -\frac {a \cosh (c+d x)}{d^2}+\frac {a x \sinh (c+d x)}{d}+\frac {2 b \sinh (c+d x)}{d^3}-\frac {2 b x \cosh (c+d x)}{d^2}+\frac {b x^2 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3296
Rule 6742
Rubi steps
\begin {align*} \int x (a+b x) \cosh (c+d x) \, dx &=\int \left (a x \cosh (c+d x)+b x^2 \cosh (c+d x)\right ) \, dx\\ &=a \int x \cosh (c+d x) \, dx+b \int x^2 \cosh (c+d x) \, dx\\ &=\frac {a x \sinh (c+d x)}{d}+\frac {b x^2 \sinh (c+d x)}{d}-\frac {a \int \sinh (c+d x) \, dx}{d}-\frac {(2 b) \int x \sinh (c+d x) \, dx}{d}\\ &=-\frac {a \cosh (c+d x)}{d^2}-\frac {2 b x \cosh (c+d x)}{d^2}+\frac {a x \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 {a \cosh (c+d x)}{d^2}-\frac {2 b x \cosh (c+d x)}{d^2}+\frac {2 b \sinh (c+d x)}{d^3}+\frac {a x \sinh (c+d x)}{d}+\frac {b x^2 \sinh (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 45, normalized size = 0.70 \[ \frac {\left (a d^2 x+b \left (d^2 x^2+2\right )\right ) \sinh (c+d x)-d (a+2 b x) \cosh (c+d x)}{d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 48, normalized size = 0.75 \[ -\frac {{\left (2 \, b d x + a d\right )} \cosh \left (d x + c\right ) - {\left (b d^{2} x^{2} + a d^{2} x + 2 \, b\right )} \sinh \left (d x + c\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 79, normalized size = 1.23 \[ \frac {{\left (b d^{2} x^{2} + a d^{2} x - 2 \, b d x - a d + 2 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{3}} - \frac {{\left (b d^{2} x^{2} + a d^{2} x + 2 \, b d x + a d + 2 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 122, normalized size = 1.91 \[ \frac {\frac {b \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d}-\frac {2 b c \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d}+a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+\frac {b \,c^{2} \sinh \left (d x +c \right )}{d}-a c \sinh \left (d x +c \right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 160, normalized size = 2.50 \[ -\frac {1}{12} \, d {\left (\frac {3 \, {\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} a e^{\left (d x\right )}}{d^{3}} + \frac {3 \, {\left (d^{2} x^{2} + 2 \, d x + 2\right )} a e^{\left (-d x - c\right )}}{d^{3}} + \frac {2 \, {\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} b e^{\left (d x\right )}}{d^{4}} + \frac {2 \, {\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} b e^{\left (-d x - c\right )}}{d^{4}}\right )} + \frac {1}{6} \, {\left (2 \, b x^{3} + 3 \, a x^{2}\right )} \cosh \left (d x + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 62, normalized size = 0.97 \[ \frac {b\,x^2\,\mathrm {sinh}\left (c+d\,x\right )+a\,x\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {a\,\mathrm {cosh}\left (c+d\,x\right )+2\,b\,x\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {2\,b\,\mathrm {sinh}\left (c+d\,x\right )}{d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 82, normalized size = 1.28 \[ \begin {cases} \frac {a x \sinh {\left (c + d x \right )}}{d} - \frac {a \cosh {\left (c + d x \right )}}{d^{2}} + \frac {b x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {2 b x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 b \sinh {\left (c + d x \right )}}{d^{3}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{2}}{2} + \frac {b x^{3}}{3}\right ) \cosh {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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