Integrand size = 16, antiderivative size = 186 \[ \int \left (a+b x^3\right )^2 \cosh (c+d x) \, dx=-\frac {12 a b \cosh (c+d x)}{d^4}-\frac {720 b^2 x \cosh (c+d x)}{d^6}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {720 b^2 \sinh (c+d x)}{d^7}+\frac {a^2 \sinh (c+d x)}{d}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {b^2 x^6 \sinh (c+d x)}{d} \]
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Time = 0.22 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5385, 2717, 3377, 2718} \[ \int \left (a+b x^3\right )^2 \cosh (c+d x) \, dx=\frac {a^2 \sinh (c+d x)}{d}-\frac {12 a b \cosh (c+d x)}{d^4}+\frac {12 a b x \sinh (c+d x)}{d^3}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {720 b^2 \sinh (c+d x)}{d^7}-\frac {720 b^2 x \cosh (c+d x)}{d^6}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {b^2 x^6 \sinh (c+d x)}{d} \]
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Rule 2717
Rule 2718
Rule 3377
Rule 5385
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 \cosh (c+d x)+2 a b x^3 \cosh (c+d x)+b^2 x^6 \cosh (c+d x)\right ) \, dx \\ & = a^2 \int \cosh (c+d x) \, dx+(2 a b) \int x^3 \cosh (c+d x) \, dx+b^2 \int x^6 \cosh (c+d x) \, dx \\ & = \frac {a^2 \sinh (c+d x)}{d}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d}-\frac {(6 a b) \int x^2 \sinh (c+d x) \, dx}{d}-\frac {\left (6 b^2\right ) \int x^5 \sinh (c+d x) \, dx}{d} \\ & = -\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {a^2 \sinh (c+d x)}{d}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d}+\frac {(12 a b) \int x \cosh (c+d x) \, dx}{d^2}+\frac {\left (30 b^2\right ) \int x^4 \cosh (c+d x) \, dx}{d^2} \\ & = -\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {a^2 \sinh (c+d x)}{d}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {b^2 x^6 \sinh (c+d x)}{d}-\frac {(12 a b) \int \sinh (c+d x) \, dx}{d^3}-\frac {\left (120 b^2\right ) \int x^3 \sinh (c+d x) \, dx}{d^3} \\ & = -\frac {12 a b \cosh (c+d x)}{d^4}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {a^2 \sinh (c+d x)}{d}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {b^2 x^6 \sinh (c+d x)}{d}+\frac {\left (360 b^2\right ) \int x^2 \cosh (c+d x) \, dx}{d^4} \\ & = -\frac {12 a b \cosh (c+d x)}{d^4}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {a^2 \sinh (c+d x)}{d}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {b^2 x^6 \sinh (c+d x)}{d}-\frac {\left (720 b^2\right ) \int x \sinh (c+d x) \, dx}{d^5} \\ & = -\frac {12 a b \cosh (c+d x)}{d^4}-\frac {720 b^2 x \cosh (c+d x)}{d^6}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {a^2 \sinh (c+d x)}{d}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {b^2 x^6 \sinh (c+d x)}{d}+\frac {\left (720 b^2\right ) \int \cosh (c+d x) \, dx}{d^6} \\ & = -\frac {12 a b \cosh (c+d x)}{d^4}-\frac {720 b^2 x \cosh (c+d x)}{d^6}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {720 b^2 \sinh (c+d x)}{d^7}+\frac {a^2 \sinh (c+d x)}{d}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {b^2 x^6 \sinh (c+d x)}{d} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.60 \[ \int \left (a+b x^3\right )^2 \cosh (c+d x) \, dx=\frac {-6 b d \left (a d^2 \left (2+d^2 x^2\right )+b x \left (120+20 d^2 x^2+d^4 x^4\right )\right ) \cosh (c+d x)+\left (a^2 d^6+2 a b d^4 x \left (6+d^2 x^2\right )+b^2 \left (720+360 d^2 x^2+30 d^4 x^4+d^6 x^6\right )\right ) \sinh (c+d x)}{d^7} \]
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Time = 0.24 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.85
method | result | size |
parallelrisch | \(\frac {6 d \left (x \left (b \,x^{3}+a \right ) d^{4}+20 b \,d^{2} x^{2}+120 b \right ) x b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \left (-\left (b \,x^{3}+a \right )^{2} d^{6}+6 \left (-5 b^{2} x^{4}-2 a b x \right ) d^{4}-360 x^{2} d^{2} b^{2}-720 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+6 d \left (x^{2} \left (b \,x^{3}+a \right ) d^{4}+4 \left (5 b \,x^{3}+a \right ) d^{2}+120 b x \right ) b}{d^{7} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\) | \(159\) |
risch | \(\frac {\left (b^{2} x^{6} d^{6}-6 b^{2} x^{5} d^{5}+2 a b \,d^{6} x^{3}+30 b^{2} x^{4} d^{4}-6 a b \,d^{5} x^{2}+a^{2} d^{6}-120 b^{2} d^{3} x^{3}+12 a b \,d^{4} x +360 x^{2} d^{2} b^{2}-12 a b \,d^{3}-720 b^{2} d x +720 b^{2}\right ) {\mathrm e}^{d x +c}}{2 d^{7}}-\frac {\left (b^{2} x^{6} d^{6}+6 b^{2} x^{5} d^{5}+2 a b \,d^{6} x^{3}+30 b^{2} x^{4} d^{4}+6 a b \,d^{5} x^{2}+a^{2} d^{6}+120 b^{2} d^{3} x^{3}+12 a b \,d^{4} x +360 x^{2} d^{2} b^{2}+12 a b \,d^{3}+720 b^{2} d x +720 b^{2}\right ) {\mathrm e}^{-d x -c}}{2 d^{7}}\) | \(245\) |
meijerg | \(\frac {64 i b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \left (\frac {21}{8} d^{4} x^{4}+\frac {105}{2} x^{2} d^{2}+315\right ) \cosh \left (d x \right )}{28 \sqrt {\pi }}-\frac {i \left (\frac {7}{16} x^{6} d^{6}+\frac {105}{8} d^{4} x^{4}+\frac {315}{2} x^{2} d^{2}+315\right ) \sinh \left (d x \right )}{28 \sqrt {\pi }}\right )}{d^{7}}+\frac {64 b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {45}{4 \sqrt {\pi }}+\frac {\left (\frac {1}{16} x^{6} d^{6}+\frac {15}{8} d^{4} x^{4}+\frac {45}{2} x^{2} d^{2}+45\right ) \cosh \left (d x \right )}{4 \sqrt {\pi }}-\frac {x d \left (\frac {3}{8} d^{4} x^{4}+\frac {15}{2} x^{2} d^{2}+45\right ) \sinh \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{7}}+\frac {16 a b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (\frac {3 x^{2} d^{2}}{2}+3\right ) \cosh \left (d x \right )}{4 \sqrt {\pi }}+\frac {d x \left (\frac {x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{4}}-\frac {16 i b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \left (\frac {5 x^{2} d^{2}}{2}+15\right ) \cosh \left (d x \right )}{20 \sqrt {\pi }}-\frac {i \left (\frac {15 x^{2} d^{2}}{2}+15\right ) \sinh \left (d x \right )}{20 \sqrt {\pi }}\right )}{d^{4}}+\frac {a^{2} \cosh \left (c \right ) \sinh \left (d x \right )}{d}-\frac {a^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (d x \right )}{\sqrt {\pi }}\right )}{d}\) | \(319\) |
parts | \(\frac {b^{2} x^{6} \sinh \left (d x +c \right )}{d}+\frac {2 a b \,x^{3} \sinh \left (d x +c \right )}{d}+\frac {a^{2} \sinh \left (d x +c \right )}{d}-\frac {6 b \left (-\frac {b \,c^{5} \cosh \left (d x +c \right )}{d^{3}}+\frac {5 b \,c^{4} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{3}}-\frac {10 b \,c^{3} \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )}{d^{3}}+\frac {10 b \,c^{2} \left (\left (d x +c \right )^{3} \cosh \left (d x +c \right )-3 \left (d x +c \right )^{2} \sinh \left (d x +c \right )+6 \left (d x +c \right ) \cosh \left (d x +c \right )-6 \sinh \left (d x +c \right )\right )}{d^{3}}-\frac {5 b c \left (\left (d x +c \right )^{4} \cosh \left (d x +c \right )-4 \left (d x +c \right )^{3} \sinh \left (d x +c \right )+12 \left (d x +c \right )^{2} \cosh \left (d x +c \right )-24 \left (d x +c \right ) \sinh \left (d x +c \right )+24 \cosh \left (d x +c \right )\right )}{d^{3}}+\frac {b \left (\left (d x +c \right )^{5} \cosh \left (d x +c \right )-5 \left (d x +c \right )^{4} \sinh \left (d x +c \right )+20 \left (d x +c \right )^{3} \cosh \left (d x +c \right )-60 \left (d x +c \right )^{2} \sinh \left (d x +c \right )+120 \left (d x +c \right ) \cosh \left (d x +c \right )-120 \sinh \left (d x +c \right )\right )}{d^{3}}+a \,c^{2} \cosh \left (d x +c \right )-2 a c \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )+a \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )\right )}{d^{4}}\) | \(437\) |
derivativedivides | \(\frac {\frac {b^{2} c^{6} \sinh \left (d x +c \right )}{d^{6}}+\frac {b^{2} \left (\left (d x +c \right )^{6} \sinh \left (d x +c \right )-6 \left (d x +c \right )^{5} \cosh \left (d x +c \right )+30 \left (d x +c \right )^{4} \sinh \left (d x +c \right )-120 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+360 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-720 \left (d x +c \right ) \cosh \left (d x +c \right )+720 \sinh \left (d x +c \right )\right )}{d^{6}}+a^{2} \sinh \left (d x +c \right )-\frac {6 b^{2} c \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^{6}}+\frac {2 b a \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 {6 b^{2} c^{5} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{6}}+\frac {15 b^{2} c^{4} \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^{6}}-\frac {20 b^{2} c^{3} \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^{6}}-\frac {2 b \,c^{3} a \sinh \left (d x +c \right )}{d^{3}}+\frac {15 b^{2} c^{2} \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^{6}}+\frac {6 b \,c^{2} a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{3}}-\frac {6 b c 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}}}{d}\) | \(592\) |
default | \(\frac {\frac {b^{2} c^{6} \sinh \left (d x +c \right )}{d^{6}}+\frac {b^{2} \left (\left (d x +c \right )^{6} \sinh \left (d x +c \right )-6 \left (d x +c \right )^{5} \cosh \left (d x +c \right )+30 \left (d x +c \right )^{4} \sinh \left (d x +c \right )-120 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+360 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-720 \left (d x +c \right ) \cosh \left (d x +c \right )+720 \sinh \left (d x +c \right )\right )}{d^{6}}+a^{2} \sinh \left (d x +c \right )-\frac {6 b^{2} c \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^{6}}+\frac {2 b a \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 {6 b^{2} c^{5} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{6}}+\frac {15 b^{2} c^{4} \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^{6}}-\frac {20 b^{2} c^{3} \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^{6}}-\frac {2 b \,c^{3} a \sinh \left (d x +c \right )}{d^{3}}+\frac {15 b^{2} c^{2} \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^{6}}+\frac {6 b \,c^{2} a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{3}}-\frac {6 b c 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}}}{d}\) | \(592\) |
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Time = 0.24 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.70 \[ \int \left (a+b x^3\right )^2 \cosh (c+d x) \, dx=-\frac {6 \, {\left (b^{2} d^{5} x^{5} + a b d^{5} x^{2} + 20 \, b^{2} d^{3} x^{3} + 2 \, a b d^{3} + 120 \, b^{2} d x\right )} \cosh \left (d x + c\right ) - {\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{3} + 30 \, b^{2} d^{4} x^{4} + a^{2} d^{6} + 12 \, a b d^{4} x + 360 \, b^{2} d^{2} x^{2} + 720 \, b^{2}\right )} \sinh \left (d x + c\right )}{d^{7}} \]
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Time = 0.59 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.22 \[ \int \left (a+b x^3\right )^2 \cosh (c+d x) \, dx=\begin {cases} \frac {a^{2} \sinh {\left (c + d x \right )}}{d} + \frac {2 a b x^{3} \sinh {\left (c + d x \right )}}{d} - \frac {6 a b x^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {12 a b x \sinh {\left (c + d x \right )}}{d^{3}} - \frac {12 a b \cosh {\left (c + d x \right )}}{d^{4}} + \frac {b^{2} x^{6} \sinh {\left (c + d x \right )}}{d} - \frac {6 b^{2} x^{5} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {30 b^{2} x^{4} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {120 b^{2} x^{3} \cosh {\left (c + d x \right )}}{d^{4}} + \frac {360 b^{2} x^{2} \sinh {\left (c + d x \right )}}{d^{5}} - \frac {720 b^{2} x \cosh {\left (c + d x \right )}}{d^{6}} + \frac {720 b^{2} \sinh {\left (c + d x \right )}}{d^{7}} & \text {for}\: d \neq 0 \\\left (a^{2} x + \frac {a b x^{4}}{2} + \frac {b^{2} x^{7}}{7}\right ) \cosh {\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.31 \[ \int \left (a+b x^3\right )^2 \cosh (c+d x) \, dx=\frac {a^{2} e^{\left (d x + c\right )}}{2 \, d} - \frac {a^{2} e^{\left (-d x - c\right )}}{2 \, d} + \frac {{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{4}} - \frac {{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a b e^{\left (-d x - c\right )}}{d^{4}} + \frac {{\left (d^{6} x^{6} e^{c} - 6 \, d^{5} x^{5} e^{c} + 30 \, d^{4} x^{4} e^{c} - 120 \, d^{3} x^{3} e^{c} + 360 \, d^{2} x^{2} e^{c} - 720 \, d x e^{c} + 720 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{2 \, d^{7}} - \frac {{\left (d^{6} x^{6} + 6 \, d^{5} x^{5} + 30 \, d^{4} x^{4} + 120 \, d^{3} x^{3} + 360 \, d^{2} x^{2} + 720 \, d x + 720\right )} b^{2} e^{\left (-d x - c\right )}}{2 \, d^{7}} \]
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Time = 0.27 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.31 \[ \int \left (a+b x^3\right )^2 \cosh (c+d x) \, dx=\frac {{\left (b^{2} d^{6} x^{6} - 6 \, b^{2} d^{5} x^{5} + 2 \, a b d^{6} x^{3} + 30 \, b^{2} d^{4} x^{4} - 6 \, a b d^{5} x^{2} + a^{2} d^{6} - 120 \, b^{2} d^{3} x^{3} + 12 \, a b d^{4} x + 360 \, b^{2} d^{2} x^{2} - 12 \, a b d^{3} - 720 \, b^{2} d x + 720 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{7}} - \frac {{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} d^{5} x^{5} + 2 \, a b d^{6} x^{3} + 30 \, b^{2} d^{4} x^{4} + 6 \, a b d^{5} x^{2} + a^{2} d^{6} + 120 \, b^{2} d^{3} x^{3} + 12 \, a b d^{4} x + 360 \, b^{2} d^{2} x^{2} + 12 \, a b d^{3} + 720 \, b^{2} d x + 720 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{7}} \]
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Time = 0.24 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.98 \[ \int \left (a+b x^3\right )^2 \cosh (c+d x) \, dx=\frac {\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^6+720\,b^2\right )}{d^7}-\frac {6\,b^2\,x^5\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}-\frac {120\,b^2\,x^3\,\mathrm {cosh}\left (c+d\,x\right )}{d^4}+\frac {b^2\,x^6\,\mathrm {sinh}\left (c+d\,x\right )}{d}+\frac {30\,b^2\,x^4\,\mathrm {sinh}\left (c+d\,x\right )}{d^3}+\frac {360\,b^2\,x^2\,\mathrm {sinh}\left (c+d\,x\right )}{d^5}-\frac {12\,a\,b\,\mathrm {cosh}\left (c+d\,x\right )}{d^4}-\frac {720\,b^2\,x\,\mathrm {cosh}\left (c+d\,x\right )}{d^6}-\frac {6\,a\,b\,x^2\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {2\,a\,b\,x^3\,\mathrm {sinh}\left (c+d\,x\right )}{d}+\frac {12\,a\,b\,x\,\mathrm {sinh}\left (c+d\,x\right )}{d^3} \]
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