Integrand size = 15, antiderivative size = 79 \[ \int x \left (a+b x^2\right ) \cosh (c+d x) \, dx=-\frac {6 b \cosh (c+d x)}{d^4}-\frac {a \cosh (c+d x)}{d^2}-\frac {3 b x^2 \cosh (c+d x)}{d^2}+\frac {6 b x \sinh (c+d x)}{d^3}+\frac {a x \sinh (c+d x)}{d}+\frac {b x^3 \sinh (c+d x)}{d} \]
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Time = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5395, 3377, 2718} \[ \int x \left (a+b x^2\right ) \cosh (c+d x) \, dx=-\frac {a \cosh (c+d x)}{d^2}+\frac {a x \sinh (c+d x)}{d}-\frac {6 b \cosh (c+d x)}{d^4}+\frac {6 b x \sinh (c+d x)}{d^3}-\frac {3 b x^2 \cosh (c+d x)}{d^2}+\frac {b x^3 \sinh (c+d x)}{d} \]
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Rule 2718
Rule 3377
Rule 5395
Rubi steps \begin{align*} \text {integral}& = \int \left (a x \cosh (c+d x)+b x^3 \cosh (c+d x)\right ) \, dx \\ & = a \int x \cosh (c+d x) \, dx+b \int x^3 \cosh (c+d x) \, dx \\ & = \frac {a x \sinh (c+d x)}{d}+\frac {b x^3 \sinh (c+d x)}{d}-\frac {a \int \sinh (c+d x) \, dx}{d}-\frac {(3 b) \int x^2 \sinh (c+d x) \, dx}{d} \\ & = -\frac {a \cosh (c+d x)}{d^2}-\frac {3 b x^2 \cosh (c+d x)}{d^2}+\frac {a x \sinh (c+d x)}{d}+\frac {b x^3 \sinh (c+d x)}{d}+\frac {(6 b) \int x \cosh (c+d x) \, dx}{d^2} \\ & = -\frac {a \cosh (c+d x)}{d^2}-\frac {3 b x^2 \cosh (c+d x)}{d^2}+\frac {6 b x \sinh (c+d x)}{d^3}+\frac {a x \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 {a \cosh (c+d x)}{d^2}-\frac {3 b x^2 \cosh (c+d x)}{d^2}+\frac {6 b x \sinh (c+d x)}{d^3}+\frac {a x \sinh (c+d x)}{d}+\frac {b x^3 \sinh (c+d x)}{d} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.72 \[ \int x \left (a+b x^2\right ) \cosh (c+d x) \, dx=\frac {-\left (\left (a d^2+3 b \left (2+d^2 x^2\right )\right ) \cosh (c+d x)\right )+d x \left (a d^2+b \left (6+d^2 x^2\right )\right ) \sinh (c+d x)}{d^4} \]
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Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.09
method | result | size |
parallelrisch | \(\frac {3 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} x^{2} d^{2}-2 d x \left (\left (b \,x^{2}+a \right ) d^{2}+6 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (3 b \,x^{2}+2 a \right ) d^{2}+12 b}{d^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\) | \(86\) |
risch | \(\frac {\left (b \,d^{3} x^{3}+a \,d^{3} x -3 b \,d^{2} x^{2}-a \,d^{2}+6 d x b -6 b \right ) {\mathrm e}^{d x +c}}{2 d^{4}}-\frac {\left (b \,d^{3} x^{3}+a \,d^{3} x +3 b \,d^{2} x^{2}+a \,d^{2}+6 d x b +6 b \right ) {\mathrm e}^{-d x -c}}{2 d^{4}}\) | \(102\) |
parts | \(\frac {b \,x^{3} \sinh \left (d x +c \right )}{d}+\frac {a x \sinh \left (d x +c \right )}{d}-\frac {\frac {3 b \,c^{2} \cosh \left (d x +c \right )}{d^{2}}-\frac {6 b c \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{2}}+\frac {3 b \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )}{d^{2}}+a \cosh \left (d x +c \right )}{d^{2}}\) | \(127\) |
meijerg | \(\frac {8 b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (\frac {3 x^{2} d^{2}}{2}+3\right ) \cosh \left (d x \right )}{4 \sqrt {\pi }}+\frac {d x \left (\frac {x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{4}}-\frac {8 i b \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \left (\frac {5 x^{2} d^{2}}{2}+15\right ) \cosh \left (d x \right )}{20 \sqrt {\pi }}-\frac {i \left (\frac {15 x^{2} d^{2}}{2}+15\right ) \sinh \left (d x \right )}{20 \sqrt {\pi }}\right )}{d^{4}}-\frac {2 a \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {a \sinh \left (c \right ) \left (\cosh \left (d x \right ) x d -\sinh \left (d x \right )\right )}{d^{2}}\) | \(172\) |
derivativedivides | \(\frac {\frac {3 b \,c^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}-\frac {3 b c \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}+\frac {b \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{2}}+a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-\frac {b \,c^{3} \sinh \left (d x +c \right )}{d^{2}}-c a \sinh \left (d x +c \right )}{d^{2}}\) | \(183\) |
default | \(\frac {\frac {3 b \,c^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}-\frac {3 b c \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}+\frac {b \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{2}}+a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-\frac {b \,c^{3} \sinh \left (d x +c \right )}{d^{2}}-c a \sinh \left (d x +c \right )}{d^{2}}\) | \(183\) |
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Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.76 \[ \int x \left (a+b x^2\right ) \cosh (c+d x) \, dx=-\frac {{\left (3 \, b d^{2} x^{2} + a d^{2} + 6 \, b\right )} \cosh \left (d x + c\right ) - {\left (b d^{3} x^{3} + {\left (a d^{3} + 6 \, b d\right )} x\right )} \sinh \left (d x + c\right )}{d^{4}} \]
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Time = 0.25 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.25 \[ \int x \left (a+b x^2\right ) \cosh (c+d x) \, dx=\begin {cases} \frac {a x \sinh {\left (c + d x \right )}}{d} - \frac {a \cosh {\left (c + d x \right )}}{d^{2}} + \frac {b x^{3} \sinh {\left (c + d x \right )}}{d} - \frac {3 b x^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {6 b x \sinh {\left (c + d x \right )}}{d^{3}} - \frac {6 b \cosh {\left (c + d x \right )}}{d^{4}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{2}}{2} + \frac {b x^{4}}{4}\right ) \cosh {\left (c \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (79) = 158\).
Time = 0.19 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.70 \[ \int x \left (a+b x^2\right ) \cosh (c+d x) \, dx=\frac {{\left (b x^{2} + a\right )}^{2} \cosh \left (d x + c\right )}{4 \, b} - \frac {{\left (\frac {a^{2} e^{\left (d x + c\right )}}{d} + \frac {a^{2} e^{\left (-d x - c\right )}}{d} + \frac {2 \, {\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{3}} + \frac {2 \, {\left (d^{2} x^{2} + 2 \, d x + 2\right )} a b e^{\left (-d x - c\right )}}{d^{3}} + \frac {{\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{5}} + \frac {{\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} b^{2} e^{\left (-d x - c\right )}}{d^{5}}\right )} d}{8 \, b} \]
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Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.28 \[ \int x \left (a+b x^2\right ) \cosh (c+d x) \, dx=\frac {{\left (b d^{3} x^{3} + a d^{3} x - 3 \, b d^{2} x^{2} - a d^{2} + 6 \, b d x - 6 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{4}} - \frac {{\left (b d^{3} x^{3} + a d^{3} x + 3 \, b d^{2} x^{2} + a d^{2} + 6 \, b d x + 6 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{4}} \]
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Time = 1.68 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int x \left (a+b x^2\right ) \cosh (c+d x) \, dx=\frac {x\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a\,d^2+6\,b\right )}{d^3}-\frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (a\,d^2+6\,b\right )}{d^4}-\frac {3\,b\,x^2\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {b\,x^3\,\mathrm {sinh}\left (c+d\,x\right )}{d} \]
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