Optimal. Leaf size=154 \[ -\frac{3 a x^2 \cosh (c+d x)}{d^2}+\frac{6 a x \sinh (c+d x)}{d^3}-\frac{6 a \cosh (c+d x)}{d^4}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{30 b x^4 \sinh (c+d x)}{d^3}+\frac{360 b x^2 \sinh (c+d x)}{d^5}-\frac{6 b x^5 \cosh (c+d x)}{d^2}-\frac{120 b x^3 \cosh (c+d x)}{d^4}+\frac{720 b \sinh (c+d x)}{d^7}-\frac{720 b x \cosh (c+d x)}{d^6}+\frac{b x^6 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.297123, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {5287, 3296, 2638, 2637} \[ -\frac{3 a x^2 \cosh (c+d x)}{d^2}+\frac{6 a x \sinh (c+d x)}{d^3}-\frac{6 a \cosh (c+d x)}{d^4}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{30 b x^4 \sinh (c+d x)}{d^3}+\frac{360 b x^2 \sinh (c+d x)}{d^5}-\frac{6 b x^5 \cosh (c+d x)}{d^2}-\frac{120 b x^3 \cosh (c+d x)}{d^4}+\frac{720 b \sinh (c+d x)}{d^7}-\frac{720 b x \cosh (c+d x)}{d^6}+\frac{b x^6 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 5287
Rule 3296
Rule 2638
Rule 2637
Rubi steps
\begin{align*} \int x^3 \left (a+b x^3\right ) \cosh (c+d x) \, dx &=\int \left (a x^3 \cosh (c+d x)+b x^6 \cosh (c+d x)\right ) \, dx\\ &=a \int x^3 \cosh (c+d x) \, dx+b \int x^6 \cosh (c+d x) \, dx\\ &=\frac{a x^3 \sinh (c+d x)}{d}+\frac{b x^6 \sinh (c+d x)}{d}-\frac{(3 a) \int x^2 \sinh (c+d x) \, dx}{d}-\frac{(6 b) \int x^5 \sinh (c+d x) \, dx}{d}\\ &=-\frac{3 a x^2 \cosh (c+d x)}{d^2}-\frac{6 b x^5 \cosh (c+d x)}{d^2}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{b x^6 \sinh (c+d x)}{d}+\frac{(6 a) \int x \cosh (c+d x) \, dx}{d^2}+\frac{(30 b) \int x^4 \cosh (c+d x) \, dx}{d^2}\\ &=-\frac{3 a x^2 \cosh (c+d x)}{d^2}-\frac{6 b x^5 \cosh (c+d x)}{d^2}+\frac{6 a x \sinh (c+d x)}{d^3}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{30 b x^4 \sinh (c+d x)}{d^3}+\frac{b x^6 \sinh (c+d x)}{d}-\frac{(6 a) \int \sinh (c+d x) \, dx}{d^3}-\frac{(120 b) \int x^3 \sinh (c+d x) \, dx}{d^3}\\ &=-\frac{6 a \cosh (c+d x)}{d^4}-\frac{3 a x^2 \cosh (c+d x)}{d^2}-\frac{120 b x^3 \cosh (c+d x)}{d^4}-\frac{6 b x^5 \cosh (c+d x)}{d^2}+\frac{6 a x \sinh (c+d x)}{d^3}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{30 b x^4 \sinh (c+d x)}{d^3}+\frac{b x^6 \sinh (c+d x)}{d}+\frac{(360 b) \int x^2 \cosh (c+d x) \, dx}{d^4}\\ &=-\frac{6 a \cosh (c+d x)}{d^4}-\frac{3 a x^2 \cosh (c+d x)}{d^2}-\frac{120 b x^3 \cosh (c+d x)}{d^4}-\frac{6 b x^5 \cosh (c+d x)}{d^2}+\frac{6 a x \sinh (c+d x)}{d^3}+\frac{360 b x^2 \sinh (c+d x)}{d^5}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{30 b x^4 \sinh (c+d x)}{d^3}+\frac{b x^6 \sinh (c+d x)}{d}-\frac{(720 b) \int x \sinh (c+d x) \, dx}{d^5}\\ &=-\frac{6 a \cosh (c+d x)}{d^4}-\frac{720 b x \cosh (c+d x)}{d^6}-\frac{3 a x^2 \cosh (c+d x)}{d^2}-\frac{120 b x^3 \cosh (c+d x)}{d^4}-\frac{6 b x^5 \cosh (c+d x)}{d^2}+\frac{6 a x \sinh (c+d x)}{d^3}+\frac{360 b x^2 \sinh (c+d x)}{d^5}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{30 b x^4 \sinh (c+d x)}{d^3}+\frac{b x^6 \sinh (c+d x)}{d}+\frac{(720 b) \int \cosh (c+d x) \, dx}{d^6}\\ &=-\frac{6 a \cosh (c+d x)}{d^4}-\frac{720 b x \cosh (c+d x)}{d^6}-\frac{3 a x^2 \cosh (c+d x)}{d^2}-\frac{120 b x^3 \cosh (c+d x)}{d^4}-\frac{6 b x^5 \cosh (c+d x)}{d^2}+\frac{720 b \sinh (c+d x)}{d^7}+\frac{6 a x \sinh (c+d x)}{d^3}+\frac{360 b x^2 \sinh (c+d x)}{d^5}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{30 b x^4 \sinh (c+d x)}{d^3}+\frac{b x^6 \sinh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.160777, size = 100, normalized size = 0.65 \[ \frac{\left (a d^4 x \left (d^2 x^2+6\right )+b \left (d^6 x^6+30 d^4 x^4+360 d^2 x^2+720\right )\right ) \sinh (c+d x)-3 d \left (a d^2 \left (d^2 x^2+2\right )+2 b x \left (d^4 x^4+20 d^2 x^2+120\right )\right ) \cosh (c+d x)}{d^7} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 551, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04636, size = 362, normalized size = 2.35 \begin{align*} -\frac{1}{56} \, d{\left (\frac{7 \,{\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{7 \,{\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{4 \,{\left (d^{7} x^{7} e^{c} - 7 \, d^{6} x^{6} e^{c} + 42 \, d^{5} x^{5} e^{c} - 210 \, d^{4} x^{4} e^{c} + 840 \, d^{3} x^{3} e^{c} - 2520 \, d^{2} x^{2} e^{c} + 5040 \, d x e^{c} - 5040 \, e^{c}\right )} b e^{\left (d x\right )}}{d^{8}} + \frac{4 \,{\left (d^{7} x^{7} + 7 \, d^{6} x^{6} + 42 \, d^{5} x^{5} + 210 \, d^{4} x^{4} + 840 \, d^{3} x^{3} + 2520 \, d^{2} x^{2} + 5040 \, d x + 5040\right )} b e^{\left (-d x - c\right )}}{d^{8}}\right )} + \frac{1}{28} \,{\left (4 \, b x^{7} + 7 \, a x^{4}\right )} \cosh \left (d x + c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76403, size = 240, normalized size = 1.56 \begin{align*} -\frac{3 \,{\left (2 \, b d^{5} x^{5} + a d^{5} x^{2} + 40 \, b d^{3} x^{3} + 2 \, a d^{3} + 240 \, b d x\right )} \cosh \left (d x + c\right ) -{\left (b d^{6} x^{6} + a d^{6} x^{3} + 30 \, b d^{4} x^{4} + 6 \, a d^{4} x + 360 \, b d^{2} x^{2} + 720 \, b\right )} \sinh \left (d x + c\right )}{d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.67527, size = 185, normalized size = 1.2 \begin{align*} \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^{6} \sinh{\left (c + d x \right )}}{d} - \frac{6 b x^{5} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{30 b x^{4} \sinh{\left (c + d x \right )}}{d^{3}} - \frac{120 b x^{3} \cosh{\left (c + d x \right )}}{d^{4}} + \frac{360 b x^{2} \sinh{\left (c + d x \right )}}{d^{5}} - \frac{720 b x \cosh{\left (c + d x \right )}}{d^{6}} + \frac{720 b \sinh{\left (c + d x \right )}}{d^{7}} & \text{for}\: d \neq 0 \\\left (\frac{a x^{4}}{4} + \frac{b x^{7}}{7}\right ) \cosh{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14007, size = 259, normalized size = 1.68 \begin{align*} \frac{{\left (b d^{6} x^{6} - 6 \, b d^{5} x^{5} + a d^{6} x^{3} + 30 \, b d^{4} x^{4} - 3 \, a d^{5} x^{2} - 120 \, b d^{3} x^{3} + 6 \, a d^{4} x + 360 \, b d^{2} x^{2} - 6 \, a d^{3} - 720 \, b d x + 720 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{7}} - \frac{{\left (b d^{6} x^{6} + 6 \, b d^{5} x^{5} + a d^{6} x^{3} + 30 \, b d^{4} x^{4} + 3 \, a d^{5} x^{2} + 120 \, b d^{3} x^{3} + 6 \, a d^{4} x + 360 \, b d^{2} x^{2} + 6 \, a d^{3} + 720 \, b d x + 720 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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