Optimal. Leaf size=109 \[ \frac {2 a \sinh (c+d x)}{d^3}-\frac {2 a x \cosh (c+d x)}{d^2}+\frac {a x^2 \sinh (c+d x)}{d}+\frac {24 b \sinh (c+d x)}{d^5}-\frac {24 b x \cosh (c+d x)}{d^4}+\frac {12 b x^2 \sinh (c+d x)}{d^3}-\frac {4 b x^3 \cosh (c+d x)}{d^2}+\frac {b x^4 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.19, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {5287, 3296, 2637} \[ \frac {2 a \sinh (c+d x)}{d^3}-\frac {2 a x \cosh (c+d x)}{d^2}+\frac {a x^2 \sinh (c+d x)}{d}+\frac {12 b x^2 \sinh (c+d x)}{d^3}-\frac {4 b x^3 \cosh (c+d x)}{d^2}+\frac {24 b \sinh (c+d x)}{d^5}-\frac {24 b x \cosh (c+d x)}{d^4}+\frac {b x^4 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 5287
Rubi steps
\begin {align*} \int x^2 \left (a+b x^2\right ) \cosh (c+d x) \, dx &=\int \left (a x^2 \cosh (c+d x)+b x^4 \cosh (c+d x)\right ) \, dx\\ &=a \int x^2 \cosh (c+d x) \, dx+b \int x^4 \cosh (c+d x) \, dx\\ &=\frac {a x^2 \sinh (c+d x)}{d}+\frac {b x^4 \sinh (c+d x)}{d}-\frac {(2 a) \int x \sinh (c+d x) \, dx}{d}-\frac {(4 b) \int x^3 \sinh (c+d x) \, dx}{d}\\ &=-\frac {2 a x \cosh (c+d x)}{d^2}-\frac {4 b x^3 \cosh (c+d x)}{d^2}+\frac {a x^2 \sinh (c+d x)}{d}+\frac {b x^4 \sinh (c+d x)}{d}+\frac {(2 a) \int \cosh (c+d x) \, dx}{d^2}+\frac {(12 b) \int x^2 \cosh (c+d x) \, dx}{d^2}\\ &=-\frac {2 a x \cosh (c+d x)}{d^2}-\frac {4 b x^3 \cosh (c+d x)}{d^2}+\frac {2 a \sinh (c+d x)}{d^3}+\frac {12 b x^2 \sinh (c+d x)}{d^3}+\frac {a x^2 \sinh (c+d x)}{d}+\frac {b x^4 \sinh (c+d x)}{d}-\frac {(24 b) \int x \sinh (c+d x) \, dx}{d^3}\\ &=-\frac {24 b x \cosh (c+d x)}{d^4}-\frac {2 a x \cosh (c+d x)}{d^2}-\frac {4 b x^3 \cosh (c+d x)}{d^2}+\frac {2 a \sinh (c+d x)}{d^3}+\frac {12 b x^2 \sinh (c+d x)}{d^3}+\frac {a x^2 \sinh (c+d x)}{d}+\frac {b x^4 \sinh (c+d x)}{d}+\frac {(24 b) \int \cosh (c+d x) \, dx}{d^4}\\ &=-\frac {24 b x \cosh (c+d x)}{d^4}-\frac {2 a x \cosh (c+d x)}{d^2}-\frac {4 b x^3 \cosh (c+d x)}{d^2}+\frac {24 b \sinh (c+d x)}{d^5}+\frac {2 a \sinh (c+d x)}{d^3}+\frac {12 b x^2 \sinh (c+d x)}{d^3}+\frac {a x^2 \sinh (c+d x)}{d}+\frac {b x^4 \sinh (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 74, normalized size = 0.68 \[ \frac {\left (a d^2 \left (d^2 x^2+2\right )+b \left (d^4 x^4+12 d^2 x^2+24\right )\right ) \sinh (c+d x)-2 d x \left (a d^2+2 b \left (d^2 x^2+6\right )\right ) \cosh (c+d x)}{d^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 78, normalized size = 0.72 \[ -\frac {2 \, {\left (2 \, b d^{3} x^{3} + {\left (a d^{3} + 12 \, b d\right )} x\right )} \cosh \left (d x + c\right ) - {\left (b d^{4} x^{4} + 2 \, a d^{2} + {\left (a d^{4} + 12 \, b d^{2}\right )} x^{2} + 24 \, b\right )} \sinh \left (d x + c\right )}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 138, normalized size = 1.27 \[ \frac {{\left (b d^{4} x^{4} + a d^{4} x^{2} - 4 \, b d^{3} x^{3} - 2 \, a d^{3} x + 12 \, b d^{2} x^{2} + 2 \, a d^{2} - 24 \, b d x + 24 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{5}} - \frac {{\left (b d^{4} x^{4} + a d^{4} x^{2} + 4 \, b d^{3} x^{3} + 2 \, a d^{3} x + 12 \, b d^{2} x^{2} + 2 \, a d^{2} + 24 \, b d x + 24 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 298, normalized size = 2.73 \[ \frac {\frac {b \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{2}}-\frac {4 b c \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{2}}+\frac {6 b \,c^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}-\frac {4 b \,c^{3} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}+\frac {b \,c^{4} \sinh \left (d x +c \right )}{d^{2}}+a \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )-2 a c \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+a \,c^{2} \sinh \left (d x +c \right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 214, normalized size = 1.96 \[ -\frac {1}{30} \, d {\left (\frac {5 \, {\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a e^{\left (d x\right )}}{d^{4}} + \frac {5 \, {\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a e^{\left (-d x - c\right )}}{d^{4}} + \frac {3 \, {\left (d^{5} x^{5} e^{c} - 5 \, d^{4} x^{4} e^{c} + 20 \, d^{3} x^{3} e^{c} - 60 \, d^{2} x^{2} e^{c} + 120 \, d x e^{c} - 120 \, e^{c}\right )} b e^{\left (d x\right )}}{d^{6}} + \frac {3 \, {\left (d^{5} x^{5} + 5 \, d^{4} x^{4} + 20 \, d^{3} x^{3} + 60 \, d^{2} x^{2} + 120 \, d x + 120\right )} b e^{\left (-d x - c\right )}}{d^{6}}\right )} + \frac {1}{15} \, {\left (3 \, b x^{5} + 5 \, a x^{3}\right )} \cosh \left (d x + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 93, normalized size = 0.85 \[ \frac {2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a\,d^2+12\,b\right )}{d^5}+\frac {x^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a\,d^2+12\,b\right )}{d^3}-\frac {2\,x\,\mathrm {cosh}\left (c+d\,x\right )\,\left (a\,d^2+12\,b\right )}{d^4}-\frac {4\,b\,x^3\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {b\,x^4\,\mathrm {sinh}\left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.96, size = 134, normalized size = 1.23 \[ \begin {cases} \frac {a x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {2 a x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 a \sinh {\left (c + d x \right )}}{d^{3}} + \frac {b x^{4} \sinh {\left (c + d x \right )}}{d} - \frac {4 b x^{3} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {12 b x^{2} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {24 b x \cosh {\left (c + d x \right )}}{d^{4}} + \frac {24 b \sinh {\left (c + d x \right )}}{d^{5}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{3}}{3} + \frac {b x^{5}}{5}\right ) \cosh {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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