Optimal. Leaf size=94 \[ \frac {2 a \sinh (c+d x)}{d^3}-\frac {2 a x \cosh (c+d x)}{d^2}+\frac {a x^2 \sinh (c+d x)}{d}-\frac {6 b \cosh (c+d x)}{d^4}+\frac {6 b x \sinh (c+d x)}{d^3}-\frac {3 b x^2 \cosh (c+d x)}{d^2}+\frac {b x^3 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.22, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6742, 3296, 2637, 2638} \[ \frac {2 a \sinh (c+d x)}{d^3}-\frac {2 a x \cosh (c+d x)}{d^2}+\frac {a x^2 \sinh (c+d x)}{d}-\frac {3 b x^2 \cosh (c+d x)}{d^2}+\frac {6 b x \sinh (c+d x)}{d^3}-\frac {6 b \cosh (c+d x)}{d^4}+\frac {b x^3 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3296
Rule 6742
Rubi steps
\begin {align*} \int x^2 (a+b x) \cosh (c+d x) \, dx &=\int \left (a x^2 \cosh (c+d x)+b x^3 \cosh (c+d x)\right ) \, dx\\ &=a \int x^2 \cosh (c+d x) \, dx+b \int x^3 \cosh (c+d x) \, dx\\ &=\frac {a x^2 \sinh (c+d x)}{d}+\frac {b x^3 \sinh (c+d x)}{d}-\frac {(2 a) \int x \sinh (c+d x) \, dx}{d}-\frac {(3 b) \int x^2 \sinh (c+d x) \, dx}{d}\\ &=-\frac {2 a x \cosh (c+d x)}{d^2}-\frac {3 b x^2 \cosh (c+d x)}{d^2}+\frac {a x^2 \sinh (c+d x)}{d}+\frac {b x^3 \sinh (c+d x)}{d}+\frac {(2 a) \int \cosh (c+d x) \, dx}{d^2}+\frac {(6 b) \int x \cosh (c+d x) \, dx}{d^2}\\ &=-\frac {2 a x \cosh (c+d x)}{d^2}-\frac {3 b x^2 \cosh (c+d x)}{d^2}+\frac {2 a \sinh (c+d x)}{d^3}+\frac {6 b x \sinh (c+d x)}{d^3}+\frac {a x^2 \sinh (c+d x)}{d}+\frac {b x^3 \sinh (c+d x)}{d}-\frac {(6 b) \int \sinh (c+d x) \, dx}{d^3}\\ &=-\frac {6 b \cosh (c+d x)}{d^4}-\frac {2 a x \cosh (c+d x)}{d^2}-\frac {3 b x^2 \cosh (c+d x)}{d^2}+\frac {2 a \sinh (c+d x)}{d^3}+\frac {6 b x \sinh (c+d x)}{d^3}+\frac {a x^2 \sinh (c+d x)}{d}+\frac {b x^3 \sinh (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 65, normalized size = 0.69 \[ \frac {d \left (a \left (d^2 x^2+2\right )+b x \left (d^2 x^2+6\right )\right ) \sinh (c+d x)-\left (2 a d^2 x+3 b \left (d^2 x^2+2\right )\right ) \cosh (c+d x)}{d^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 67, normalized size = 0.71 \[ -\frac {{\left (3 \, b d^{2} x^{2} + 2 \, a d^{2} x + 6 \, b\right )} \cosh \left (d x + c\right ) - {\left (b d^{3} x^{3} + a d^{3} x^{2} + 6 \, b d x + 2 \, a d\right )} \sinh \left (d x + c\right )}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 116, normalized size = 1.23 \[ \frac {{\left (b d^{3} x^{3} + a d^{3} x^{2} - 3 \, b d^{2} x^{2} - 2 \, a d^{2} x + 6 \, b d x + 2 \, a d - 6 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{4}} - \frac {{\left (b d^{3} x^{3} + a d^{3} x^{2} + 3 \, b d^{2} x^{2} + 2 \, a d^{2} x + 6 \, b d x + 2 \, a d + 6 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 224, normalized size = 2.38 \[ \frac {\frac {b \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d}-\frac {3 b c \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d}+\frac {3 b \,c^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d}-\frac {b \,c^{3} \sinh \left (d x +c \right )}{d}+a \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )-2 a c \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+a \,c^{2} \sinh \left (d x +c \right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 196, normalized size = 2.09 \[ -\frac {1}{24} \, d {\left (\frac {4 \, {\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a e^{\left (d x\right )}}{d^{4}} + \frac {4 \, {\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a e^{\left (-d x - c\right )}}{d^{4}} + \frac {3 \, {\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} b e^{\left (d x\right )}}{d^{5}} + \frac {3 \, {\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} b e^{\left (-d x - c\right )}}{d^{5}}\right )} + \frac {1}{12} \, {\left (3 \, b x^{4} + 4 \, a x^{3}\right )} \cosh \left (d x + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 92, normalized size = 0.98 \[ \frac {2\,a\,\mathrm {sinh}\left (c+d\,x\right )+6\,b\,x\,\mathrm {sinh}\left (c+d\,x\right )}{d^3}-\frac {2\,a\,x\,\mathrm {cosh}\left (c+d\,x\right )+3\,b\,x^2\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {a\,x^2\,\mathrm {sinh}\left (c+d\,x\right )+b\,x^3\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {6\,b\,\mathrm {cosh}\left (c+d\,x\right )}{d^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.94, size = 117, normalized size = 1.24 \[ \begin {cases} \frac {a x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {2 a x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 a \sinh {\left (c + d x \right )}}{d^{3}} + \frac {b x^{3} \sinh {\left (c + d x \right )}}{d} - \frac {3 b x^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {6 b x \sinh {\left (c + d x \right )}}{d^{3}} - \frac {6 b \cosh {\left (c + d x \right )}}{d^{4}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{3}}{3} + \frac {b x^{4}}{4}\right ) \cosh {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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