Optimal. Leaf size=134 \[ -\frac {a^2 \cosh (c+d x)}{d^2}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {4 a b \sinh (c+d x)}{d^3}-\frac {4 a b x \cosh (c+d x)}{d^2}+\frac {2 a b x^2 \sinh (c+d x)}{d}-\frac {6 b^2 \cosh (c+d x)}{d^4}+\frac {6 b^2 x \sinh (c+d x)}{d^3}-\frac {3 b^2 x^2 \cosh (c+d x)}{d^2}+\frac {b^2 x^3 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.21, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6742, 3296, 2638, 2637} \[ -\frac {a^2 \cosh (c+d x)}{d^2}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {4 a b \sinh (c+d x)}{d^3}-\frac {4 a b x \cosh (c+d x)}{d^2}+\frac {2 a b x^2 \sinh (c+d x)}{d}-\frac {3 b^2 x^2 \cosh (c+d x)}{d^2}+\frac {6 b^2 x \sinh (c+d x)}{d^3}-\frac {6 b^2 \cosh (c+d x)}{d^4}+\frac {b^2 x^3 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3296
Rule 6742
Rubi steps
\begin {align*} \int x (a+b x)^2 \cosh (c+d x) \, dx &=\int \left (a^2 x \cosh (c+d x)+2 a b x^2 \cosh (c+d x)+b^2 x^3 \cosh (c+d x)\right ) \, dx\\ &=a^2 \int x \cosh (c+d x) \, dx+(2 a b) \int x^2 \cosh (c+d x) \, dx+b^2 \int x^3 \cosh (c+d x) \, dx\\ &=\frac {a^2 x \sinh (c+d x)}{d}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {b^2 x^3 \sinh (c+d x)}{d}-\frac {a^2 \int \sinh (c+d x) \, dx}{d}-\frac {(4 a b) \int x \sinh (c+d x) \, dx}{d}-\frac {\left (3 b^2\right ) \int x^2 \sinh (c+d x) \, dx}{d}\\ &=-\frac {a^2 \cosh (c+d x)}{d^2}-\frac {4 a b x \cosh (c+d x)}{d^2}-\frac {3 b^2 x^2 \cosh (c+d x)}{d^2}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {b^2 x^3 \sinh (c+d x)}{d}+\frac {(4 a b) \int \cosh (c+d x) \, dx}{d^2}+\frac {\left (6 b^2\right ) \int x \cosh (c+d x) \, dx}{d^2}\\ &=-\frac {a^2 \cosh (c+d x)}{d^2}-\frac {4 a b x \cosh (c+d x)}{d^2}-\frac {3 b^2 x^2 \cosh (c+d x)}{d^2}+\frac {4 a b \sinh (c+d x)}{d^3}+\frac {6 b^2 x \sinh (c+d x)}{d^3}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {b^2 x^3 \sinh (c+d x)}{d}-\frac {\left (6 b^2\right ) \int \sinh (c+d x) \, dx}{d^3}\\ &=-\frac {6 b^2 \cosh (c+d x)}{d^4}-\frac {a^2 \cosh (c+d x)}{d^2}-\frac {4 a b x \cosh (c+d x)}{d^2}-\frac {3 b^2 x^2 \cosh (c+d x)}{d^2}+\frac {4 a b \sinh (c+d x)}{d^3}+\frac {6 b^2 x \sinh (c+d x)}{d^3}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {b^2 x^3 \sinh (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 87, normalized size = 0.65 \[ \frac {d \left (a^2 d^2 x+2 a b \left (d^2 x^2+2\right )+b^2 x \left (d^2 x^2+6\right )\right ) \sinh (c+d x)-\left (a^2 d^2+4 a b d^2 x+3 b^2 \left (d^2 x^2+2\right )\right ) \cosh (c+d x)}{d^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 95, normalized size = 0.71 \[ -\frac {{\left (3 \, b^{2} d^{2} x^{2} + 4 \, a b d^{2} x + a^{2} d^{2} + 6 \, b^{2}\right )} \cosh \left (d x + c\right ) - {\left (b^{2} d^{3} x^{3} + 2 \, a b d^{3} x^{2} + 4 \, a b d + {\left (a^{2} d^{3} + 6 \, b^{2} d\right )} x\right )} \sinh \left (d x + c\right )}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 171, normalized size = 1.28 \[ \frac {{\left (b^{2} d^{3} x^{3} + 2 \, a b d^{3} x^{2} + a^{2} d^{3} x - 3 \, b^{2} d^{2} x^{2} - 4 \, a b d^{2} x - a^{2} d^{2} + 6 \, b^{2} d x + 4 \, a b d - 6 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{4}} - \frac {{\left (b^{2} d^{3} x^{3} + 2 \, a b d^{3} x^{2} + a^{2} d^{3} x + 3 \, b^{2} d^{2} x^{2} + 4 \, a b d^{2} x + a^{2} d^{2} + 6 \, b^{2} d x + 4 \, a b d + 6 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 283, normalized size = 2.11 \[ \frac {\frac {b^{2} \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{2}}-\frac {3 b^{2} c \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}+\frac {2 b a \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d}+\frac {3 b^{2} c^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}-\frac {4 b c a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d}+a^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-\frac {b^{2} c^{3} \sinh \left (d x +c \right )}{d^{2}}+\frac {2 b \,c^{2} a \sinh \left (d x +c \right )}{d}-c \,a^{2} \sinh \left (d x +c \right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 275, normalized size = 2.05 \[ -\frac {1}{24} \, d {\left (\frac {6 \, {\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} a^{2} e^{\left (d x\right )}}{d^{3}} + \frac {6 \, {\left (d^{2} x^{2} + 2 \, d x + 2\right )} a^{2} e^{\left (-d x - c\right )}}{d^{3}} + \frac {8 \, {\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{4}} + \frac {8 \, {\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a b e^{\left (-d x - c\right )}}{d^{4}} + \frac {3 \, {\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{5}} + \frac {3 \, {\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} b^{2} e^{\left (-d x - c\right )}}{d^{5}}\right )} + \frac {1}{12} \, {\left (3 \, b^{2} x^{4} + 8 \, a b x^{3} + 6 \, a^{2} x^{2}\right )} \cosh \left (d x + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 125, normalized size = 0.93 \[ \frac {b^2\,x^3\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {3\,b^2\,x^2\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}-\frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (a^2\,d^2+6\,b^2\right )}{d^4}+\frac {4\,a\,b\,\mathrm {sinh}\left (c+d\,x\right )}{d^3}+\frac {x\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^2+6\,b^2\right )}{d^3}+\frac {2\,a\,b\,x^2\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {4\,a\,b\,x\,\mathrm {cosh}\left (c+d\,x\right )}{d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.15, size = 172, normalized size = 1.28 \[ \begin {cases} \frac {a^{2} x \sinh {\left (c + d x \right )}}{d} - \frac {a^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 a b x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {4 a b x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {4 a b \sinh {\left (c + d x \right )}}{d^{3}} + \frac {b^{2} x^{3} \sinh {\left (c + d x \right )}}{d} - \frac {3 b^{2} x^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {6 b^{2} x \sinh {\left (c + d x \right )}}{d^{3}} - \frac {6 b^{2} \cosh {\left (c + d x \right )}}{d^{4}} & \text {for}\: d \neq 0 \\\left (\frac {a^{2} x^{2}}{2} + \frac {2 a b x^{3}}{3} + \frac {b^{2} x^{4}}{4}\right ) \cosh {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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