Integrand size = 17, antiderivative size = 124 \[ \int x^2 \left (a+b x^3\right ) \cosh (c+d x) \, dx=-\frac {120 b \cosh (c+d x)}{d^6}-\frac {2 a x \cosh (c+d x)}{d^2}-\frac {60 b x^2 \cosh (c+d x)}{d^4}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {2 a \sinh (c+d x)}{d^3}+\frac {120 b x \sinh (c+d x)}{d^5}+\frac {a x^2 \sinh (c+d x)}{d}+\frac {20 b x^3 \sinh (c+d x)}{d^3}+\frac {b x^5 \sinh (c+d x)}{d} \]
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Time = 0.16 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {5395, 3377, 2717, 2718} \[ \int x^2 \left (a+b x^3\right ) \cosh (c+d x) \, dx=\frac {2 a \sinh (c+d x)}{d^3}-\frac {2 a x \cosh (c+d x)}{d^2}+\frac {a x^2 \sinh (c+d x)}{d}-\frac {120 b \cosh (c+d x)}{d^6}+\frac {120 b x \sinh (c+d x)}{d^5}-\frac {60 b x^2 \cosh (c+d x)}{d^4}+\frac {20 b x^3 \sinh (c+d x)}{d^3}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {b x^5 \sinh (c+d x)}{d} \]
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Rule 2717
Rule 2718
Rule 3377
Rule 5395
Rubi steps \begin{align*} \text {integral}& = \int \left (a x^2 \cosh (c+d x)+b x^5 \cosh (c+d x)\right ) \, dx \\ & = a \int x^2 \cosh (c+d x) \, dx+b \int x^5 \cosh (c+d x) \, dx \\ & = \frac {a x^2 \sinh (c+d x)}{d}+\frac {b x^5 \sinh (c+d x)}{d}-\frac {(2 a) \int x \sinh (c+d x) \, dx}{d}-\frac {(5 b) \int x^4 \sinh (c+d x) \, dx}{d} \\ & = -\frac {2 a x \cosh (c+d x)}{d^2}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {a x^2 \sinh (c+d x)}{d}+\frac {b x^5 \sinh (c+d x)}{d}+\frac {(2 a) \int \cosh (c+d x) \, dx}{d^2}+\frac {(20 b) \int x^3 \cosh (c+d x) \, dx}{d^2} \\ & = -\frac {2 a x \cosh (c+d x)}{d^2}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {2 a \sinh (c+d x)}{d^3}+\frac {a x^2 \sinh (c+d x)}{d}+\frac {20 b x^3 \sinh (c+d x)}{d^3}+\frac {b x^5 \sinh (c+d x)}{d}-\frac {(60 b) \int x^2 \sinh (c+d x) \, dx}{d^3} \\ & = -\frac {2 a x \cosh (c+d x)}{d^2}-\frac {60 b x^2 \cosh (c+d x)}{d^4}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {2 a \sinh (c+d x)}{d^3}+\frac {a x^2 \sinh (c+d x)}{d}+\frac {20 b x^3 \sinh (c+d x)}{d^3}+\frac {b x^5 \sinh (c+d x)}{d}+\frac {(120 b) \int x \cosh (c+d x) \, dx}{d^4} \\ & = -\frac {2 a x \cosh (c+d x)}{d^2}-\frac {60 b x^2 \cosh (c+d x)}{d^4}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {2 a \sinh (c+d x)}{d^3}+\frac {120 b x \sinh (c+d x)}{d^5}+\frac {a x^2 \sinh (c+d x)}{d}+\frac {20 b x^3 \sinh (c+d x)}{d^3}+\frac {b x^5 \sinh (c+d x)}{d}-\frac {(120 b) \int \sinh (c+d x) \, dx}{d^5} \\ & = -\frac {120 b \cosh (c+d x)}{d^6}-\frac {2 a x \cosh (c+d x)}{d^2}-\frac {60 b x^2 \cosh (c+d x)}{d^4}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {2 a \sinh (c+d x)}{d^3}+\frac {120 b x \sinh (c+d x)}{d^5}+\frac {a x^2 \sinh (c+d x)}{d}+\frac {20 b x^3 \sinh (c+d x)}{d^3}+\frac {b x^5 \sinh (c+d x)}{d} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.68 \[ \int x^2 \left (a+b x^3\right ) \cosh (c+d x) \, dx=\frac {-\left (\left (2 a d^4 x+5 b \left (24+12 d^2 x^2+d^4 x^4\right )\right ) \cosh (c+d x)\right )+d \left (a d^2 \left (2+d^2 x^2\right )+b x \left (120+20 d^2 x^2+d^4 x^4\right )\right ) \sinh (c+d x)}{d^6} \]
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Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.04
method | result | size |
parallelrisch | \(\frac {\left (\left (5 b \,x^{4}+2 a x \right ) d^{4}+60 b \,d^{2} x^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 d \left (x^{2} \left (b \,x^{3}+a \right ) d^{4}+2 \left (10 b \,x^{3}+a \right ) d^{2}+120 b x \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (5 b \,x^{4}+2 a x \right ) d^{4}+60 b \,d^{2} x^{2}+240 b}{d^{6} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\) | \(129\) |
risch | \(\frac {\left (b \,x^{5} d^{5}-5 b \,x^{4} d^{4}+a \,d^{5} x^{2}+20 b \,d^{3} x^{3}-2 a \,d^{4} x -60 b \,d^{2} x^{2}+2 d^{3} a +120 d x b -120 b \right ) {\mathrm e}^{d x +c}}{2 d^{6}}-\frac {\left (b \,x^{5} d^{5}+5 b \,x^{4} d^{4}+a \,d^{5} x^{2}+20 b \,d^{3} x^{3}+2 a \,d^{4} x +60 b \,d^{2} x^{2}+2 d^{3} a +120 d x b +120 b \right ) {\mathrm e}^{-d x -c}}{2 d^{6}}\) | \(157\) |
meijerg | \(-\frac {32 b \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {15}{4 \sqrt {\pi }}+\frac {\left (\frac {15}{8} d^{4} x^{4}+\frac {45}{2} x^{2} d^{2}+45\right ) \cosh \left (d x \right )}{12 \sqrt {\pi }}-\frac {x d \left (\frac {3}{8} d^{4} x^{4}+\frac {15}{2} x^{2} d^{2}+45\right ) \sinh \left (d x \right )}{12 \sqrt {\pi }}\right )}{d^{6}}+\frac {32 i b \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {i x d \left (\frac {7}{8} d^{4} x^{4}+\frac {35}{2} x^{2} d^{2}+105\right ) \cosh \left (d x \right )}{28 \sqrt {\pi }}+\frac {i \left (\frac {35}{8} d^{4} x^{4}+\frac {105}{2} x^{2} d^{2}+105\right ) \sinh \left (d x \right )}{28 \sqrt {\pi }}\right )}{d^{6}}+\frac {4 i a \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {i \left (\frac {3 x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{3}}+\frac {4 a \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {x^{2} d^{2}}{2}+1\right ) \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{3}}\) | \(238\) |
parts | \(\frac {b \,x^{5} \sinh \left (d x +c \right )}{d}+\frac {a \,x^{2} \sinh \left (d x +c \right )}{d}-\frac {\frac {5 b \,c^{4} \cosh \left (d x +c \right )}{d^{4}}-\frac {20 b \,c^{3} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{4}}+\frac {30 b \,c^{2} \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )}{d^{4}}-\frac {20 b c \left (\left (d x +c \right )^{3} \cosh \left (d x +c \right )-3 \left (d x +c \right )^{2} \sinh \left (d x +c \right )+6 \left (d x +c \right ) \cosh \left (d x +c \right )-6 \sinh \left (d x +c \right )\right )}{d^{4}}+\frac {5 b \left (\left (d x +c \right )^{4} \cosh \left (d x +c \right )-4 \left (d x +c \right )^{3} \sinh \left (d x +c \right )+12 \left (d x +c \right )^{2} \cosh \left (d x +c \right )-24 \left (d x +c \right ) \sinh \left (d x +c \right )+24 \cosh \left (d x +c \right )\right )}{d^{4}}+\frac {2 a \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d}-\frac {2 a c \cosh \left (d x +c \right )}{d}}{d^{2}}\) | \(296\) |
derivativedivides | \(\frac {-\frac {b \,c^{5} \sinh \left (d x +c \right )}{d^{3}}+\frac {5 b \,c^{4} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{3}}-\frac {10 b \,c^{3} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{3}}+\frac {10 b \,c^{2} \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{3}}-\frac {5 b c \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{3}}+\frac {b \left (\left (d x +c \right )^{5} \sinh \left (d x +c \right )-5 \left (d x +c \right )^{4} \cosh \left (d x +c \right )+20 \left (d x +c \right )^{3} \sinh \left (d x +c \right )-60 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+120 \left (d x +c \right ) \sinh \left (d x +c \right )-120 \cosh \left (d x +c \right )\right )}{d^{3}}+a \,c^{2} \sinh \left (d x +c \right )-2 a c \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+a \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{3}}\) | \(389\) |
default | \(\frac {-\frac {b \,c^{5} \sinh \left (d x +c \right )}{d^{3}}+\frac {5 b \,c^{4} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{3}}-\frac {10 b \,c^{3} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{3}}+\frac {10 b \,c^{2} \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{3}}-\frac {5 b c \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{3}}+\frac {b \left (\left (d x +c \right )^{5} \sinh \left (d x +c \right )-5 \left (d x +c \right )^{4} \cosh \left (d x +c \right )+20 \left (d x +c \right )^{3} \sinh \left (d x +c \right )-60 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+120 \left (d x +c \right ) \sinh \left (d x +c \right )-120 \cosh \left (d x +c \right )\right )}{d^{3}}+a \,c^{2} \sinh \left (d x +c \right )-2 a c \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+a \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{3}}\) | \(389\) |
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Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.70 \[ \int x^2 \left (a+b x^3\right ) \cosh (c+d x) \, dx=-\frac {{\left (5 \, b d^{4} x^{4} + 2 \, a d^{4} x + 60 \, b d^{2} x^{2} + 120 \, b\right )} \cosh \left (d x + c\right ) - {\left (b d^{5} x^{5} + a d^{5} x^{2} + 20 \, b d^{3} x^{3} + 2 \, a d^{3} + 120 \, b d x\right )} \sinh \left (d x + c\right )}{d^{6}} \]
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Time = 0.43 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.22 \[ \int x^2 \left (a+b x^3\right ) \cosh (c+d x) \, dx=\begin {cases} \frac {a x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {2 a x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 a \sinh {\left (c + d x \right )}}{d^{3}} + \frac {b x^{5} \sinh {\left (c + d x \right )}}{d} - \frac {5 b x^{4} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {20 b x^{3} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {60 b x^{2} \cosh {\left (c + d x \right )}}{d^{4}} + \frac {120 b x \sinh {\left (c + d x \right )}}{d^{5}} - \frac {120 b \cosh {\left (c + d x \right )}}{d^{6}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{3}}{3} + \frac {b x^{6}}{6}\right ) \cosh {\left (c \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (124) = 248\).
Time = 0.21 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.15 \[ \int x^2 \left (a+b x^3\right ) \cosh (c+d x) \, dx=\frac {{\left (b x^{3} + a\right )}^{2} \cosh \left (d x + c\right )}{6 \, b} - \frac {{\left (\frac {a^{2} e^{\left (d x + c\right )}}{d} + \frac {a^{2} e^{\left (-d x - c\right )}}{d} + \frac {2 \, {\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{4}} + \frac {2 \, {\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a b e^{\left (-d x - c\right )}}{d^{4}} + \frac {{\left (d^{6} x^{6} e^{c} - 6 \, d^{5} x^{5} e^{c} + 30 \, d^{4} x^{4} e^{c} - 120 \, d^{3} x^{3} e^{c} + 360 \, d^{2} x^{2} e^{c} - 720 \, d x e^{c} + 720 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{7}} + \frac {{\left (d^{6} x^{6} + 6 \, d^{5} x^{5} + 30 \, d^{4} x^{4} + 120 \, d^{3} x^{3} + 360 \, d^{2} x^{2} + 720 \, d x + 720\right )} b^{2} e^{\left (-d x - c\right )}}{d^{7}}\right )} d}{12 \, b} \]
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Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.26 \[ \int x^2 \left (a+b x^3\right ) \cosh (c+d x) \, dx=\frac {{\left (b d^{5} x^{5} - 5 \, b d^{4} x^{4} + a d^{5} x^{2} + 20 \, b d^{3} x^{3} - 2 \, a d^{4} x - 60 \, b d^{2} x^{2} + 2 \, a d^{3} + 120 \, b d x - 120 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{6}} - \frac {{\left (b d^{5} x^{5} + 5 \, b d^{4} x^{4} + a d^{5} x^{2} + 20 \, b d^{3} x^{3} + 2 \, a d^{4} x + 60 \, b d^{2} x^{2} + 2 \, a d^{3} + 120 \, b d x + 120 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{6}} \]
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Time = 1.66 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.98 \[ \int x^2 \left (a+b x^3\right ) \cosh (c+d x) \, dx=\frac {a\,x^2\,\mathrm {sinh}\left (c+d\,x\right )+b\,x^5\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {2\,a\,x\,\mathrm {cosh}\left (c+d\,x\right )+5\,b\,x^4\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {2\,a\,\mathrm {sinh}\left (c+d\,x\right )+20\,b\,x^3\,\mathrm {sinh}\left (c+d\,x\right )}{d^3}-\frac {120\,b\,\mathrm {cosh}\left (c+d\,x\right )}{d^6}+\frac {120\,b\,x\,\mathrm {sinh}\left (c+d\,x\right )}{d^5}-\frac {60\,b\,x^2\,\mathrm {cosh}\left (c+d\,x\right )}{d^4} \]
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