Optimal. Leaf size=139 \[ -\frac {6 a \cosh (c+d x)}{d^4}+\frac {6 a x \sinh (c+d x)}{d^3}-\frac {3 a x^2 \cosh (c+d x)}{d^2}+\frac {a x^3 \sinh (c+d x)}{d}-\frac {120 b \cosh (c+d x)}{d^6}+\frac {120 b x \sinh (c+d x)}{d^5}-\frac {60 b x^2 \cosh (c+d x)}{d^4}+\frac {20 b x^3 \sinh (c+d x)}{d^3}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {b x^5 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.26, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {5287, 3296, 2638} \[ -\frac {3 a x^2 \cosh (c+d x)}{d^2}+\frac {6 a x \sinh (c+d x)}{d^3}-\frac {6 a \cosh (c+d x)}{d^4}+\frac {a x^3 \sinh (c+d x)}{d}+\frac {20 b x^3 \sinh (c+d x)}{d^3}-\frac {5 b x^4 \cosh (c+d x)}{d^2}-\frac {60 b x^2 \cosh (c+d x)}{d^4}+\frac {120 b x \sinh (c+d x)}{d^5}-\frac {120 b \cosh (c+d x)}{d^6}+\frac {b x^5 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 5287
Rubi steps
\begin {align*} \int x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx &=\int \left (a x^3 \cosh (c+d x)+b x^5 \cosh (c+d x)\right ) \, dx\\ &=a \int x^3 \cosh (c+d x) \, dx+b \int x^5 \cosh (c+d x) \, dx\\ &=\frac {a x^3 \sinh (c+d x)}{d}+\frac {b x^5 \sinh (c+d x)}{d}-\frac {(3 a) \int x^2 \sinh (c+d x) \, dx}{d}-\frac {(5 b) \int x^4 \sinh (c+d x) \, dx}{d}\\ &=-\frac {3 a x^2 \cosh (c+d x)}{d^2}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {a x^3 \sinh (c+d x)}{d}+\frac {b x^5 \sinh (c+d x)}{d}+\frac {(6 a) \int x \cosh (c+d x) \, dx}{d^2}+\frac {(20 b) \int x^3 \cosh (c+d x) \, dx}{d^2}\\ &=-\frac {3 a x^2 \cosh (c+d x)}{d^2}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {6 a x \sinh (c+d x)}{d^3}+\frac {20 b x^3 \sinh (c+d x)}{d^3}+\frac {a x^3 \sinh (c+d x)}{d}+\frac {b x^5 \sinh (c+d x)}{d}-\frac {(6 a) \int \sinh (c+d x) \, dx}{d^3}-\frac {(60 b) \int x^2 \sinh (c+d x) \, dx}{d^3}\\ &=-\frac {6 a \cosh (c+d x)}{d^4}-\frac {60 b x^2 \cosh (c+d x)}{d^4}-\frac {3 a x^2 \cosh (c+d x)}{d^2}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {6 a x \sinh (c+d x)}{d^3}+\frac {20 b x^3 \sinh (c+d x)}{d^3}+\frac {a x^3 \sinh (c+d x)}{d}+\frac {b x^5 \sinh (c+d x)}{d}+\frac {(120 b) \int x \cosh (c+d x) \, dx}{d^4}\\ &=-\frac {6 a \cosh (c+d x)}{d^4}-\frac {60 b x^2 \cosh (c+d x)}{d^4}-\frac {3 a x^2 \cosh (c+d x)}{d^2}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {120 b x \sinh (c+d x)}{d^5}+\frac {6 a x \sinh (c+d x)}{d^3}+\frac {20 b x^3 \sinh (c+d x)}{d^3}+\frac {a x^3 \sinh (c+d x)}{d}+\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 {6 a \cosh (c+d x)}{d^4}-\frac {60 b x^2 \cosh (c+d x)}{d^4}-\frac {3 a x^2 \cosh (c+d x)}{d^2}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {120 b x \sinh (c+d x)}{d^5}+\frac {6 a x \sinh (c+d x)}{d^3}+\frac {20 b x^3 \sinh (c+d x)}{d^3}+\frac {a x^3 \sinh (c+d x)}{d}+\frac {b x^5 \sinh (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 92, normalized size = 0.66 \[ \frac {d x \left (a d^2 \left (d^2 x^2+6\right )+b \left (d^4 x^4+20 d^2 x^2+120\right )\right ) \sinh (c+d x)-\left (3 a d^2 \left (d^2 x^2+2\right )+5 b \left (d^4 x^4+12 d^2 x^2+24\right )\right ) \cosh (c+d x)}{d^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 95, normalized size = 0.68 \[ -\frac {{\left (5 \, b d^{4} x^{4} + 6 \, a d^{2} + 3 \, {\left (a d^{4} + 20 \, b d^{2}\right )} x^{2} + 120 \, b\right )} \cosh \left (d x + c\right ) - {\left (b d^{5} x^{5} + {\left (a d^{5} + 20 \, b d^{3}\right )} x^{3} + 6 \, {\left (a d^{3} + 20 \, b d\right )} x\right )} \sinh \left (d x + c\right )}{d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 174, normalized size = 1.25 \[ \frac {{\left (b d^{5} x^{5} + a d^{5} x^{3} - 5 \, b d^{4} x^{4} - 3 \, a d^{4} x^{2} + 20 \, b d^{3} x^{3} + 6 \, a d^{3} x - 60 \, b d^{2} x^{2} - 6 \, a d^{2} + 120 \, b d x - 120 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{6}} - \frac {{\left (b d^{5} x^{5} + a d^{5} x^{3} + 5 \, b d^{4} x^{4} + 3 \, a d^{4} x^{2} + 20 \, b d^{3} x^{3} + 6 \, a d^{3} x + 60 \, b d^{2} x^{2} + 6 \, a d^{2} + 120 \, b d x + 120 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 447, normalized size = 3.22 \[ \frac {\frac {b \left (\left (d x +c \right )^{5} \sinh \left (d x +c \right )-5 \left (d x +c \right )^{4} \cosh \left (d x +c \right )+20 \left (d x +c \right )^{3} \sinh \left (d x +c \right )-60 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+120 \left (d x +c \right ) \sinh \left (d x +c \right )-120 \cosh \left (d x +c \right )\right )}{d^{2}}-\frac {5 b c \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{2}}+\frac {10 b \,c^{2} \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{2}}-\frac {10 b \,c^{3} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}+\frac {5 b \,c^{4} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}-\frac {b \,c^{5} \sinh \left (d x +c \right )}{d^{2}}+a \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )-3 a c \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )+3 a \,c^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-a \,c^{3} \sinh \left (d x +c \right )}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 250, normalized size = 1.80 \[ -\frac {1}{24} \, d {\left (\frac {3 \, {\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} a e^{\left (d x\right )}}{d^{5}} + \frac {3 \, {\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} a e^{\left (-d x - c\right )}}{d^{5}} + \frac {2 \, {\left (d^{6} x^{6} e^{c} - 6 \, d^{5} x^{5} e^{c} + 30 \, d^{4} x^{4} e^{c} - 120 \, d^{3} x^{3} e^{c} + 360 \, d^{2} x^{2} e^{c} - 720 \, d x e^{c} + 720 \, e^{c}\right )} b e^{\left (d x\right )}}{d^{7}} + \frac {2 \, {\left (d^{6} x^{6} + 6 \, d^{5} x^{5} + 30 \, d^{4} x^{4} + 120 \, d^{3} x^{3} + 360 \, d^{2} x^{2} + 720 \, d x + 720\right )} b e^{\left (-d x - c\right )}}{d^{7}}\right )} + \frac {1}{12} \, {\left (2 \, b x^{6} + 3 \, a x^{4}\right )} \cosh \left (d x + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 116, normalized size = 0.83 \[ \frac {x^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a\,d^2+20\,b\right )}{d^3}-\frac {3\,x^2\,\mathrm {cosh}\left (c+d\,x\right )\,\left (a\,d^2+20\,b\right )}{d^4}-\frac {6\,\mathrm {cosh}\left (c+d\,x\right )\,\left (a\,d^2+20\,b\right )}{d^6}+\frac {6\,x\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a\,d^2+20\,b\right )}{d^5}-\frac {5\,b\,x^4\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {b\,x^5\,\mathrm {sinh}\left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.42, size = 168, normalized size = 1.21 \[ \begin {cases} \frac {a x^{3} \sinh {\left (c + d x \right )}}{d} - \frac {3 a x^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {6 a x \sinh {\left (c + d x \right )}}{d^{3}} - \frac {6 a \cosh {\left (c + d x \right )}}{d^{4}} + \frac {b x^{5} \sinh {\left (c + d x \right )}}{d} - \frac {5 b x^{4} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {20 b x^{3} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {60 b x^{2} \cosh {\left (c + d x \right )}}{d^{4}} + \frac {120 b x \sinh {\left (c + d x \right )}}{d^{5}} - \frac {120 b \cosh {\left (c + d x \right )}}{d^{6}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{4}}{4} + \frac {b x^{6}}{6}\right ) \cosh {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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