Optimal. Leaf size=124 \[ \frac{2 a \sinh (c+d x)}{d^3}-\frac{2 a x \cosh (c+d x)}{d^2}+\frac{a x^2 \sinh (c+d x)}{d}+\frac{20 b x^3 \sinh (c+d x)}{d^3}-\frac{5 b x^4 \cosh (c+d x)}{d^2}-\frac{60 b x^2 \cosh (c+d x)}{d^4}+\frac{120 b x \sinh (c+d x)}{d^5}-\frac{120 b \cosh (c+d x)}{d^6}+\frac{b x^5 \sinh (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.228484, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {5287, 3296, 2637, 2638} \[ \frac{2 a \sinh (c+d x)}{d^3}-\frac{2 a x \cosh (c+d x)}{d^2}+\frac{a x^2 \sinh (c+d x)}{d}+\frac{20 b x^3 \sinh (c+d x)}{d^3}-\frac{5 b x^4 \cosh (c+d x)}{d^2}-\frac{60 b x^2 \cosh (c+d x)}{d^4}+\frac{120 b x \sinh (c+d x)}{d^5}-\frac{120 b \cosh (c+d x)}{d^6}+\frac{b x^5 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5287
Rule 3296
Rule 2637
Rule 2638
Rubi steps
\begin{align*} \int x^2 \left (a+b x^3\right ) \cosh (c+d x) \, dx &=\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*}
Mathematica [A] time = 0.133114, size = 84, normalized size = 0.68 \[ \frac{d \left (a d^2 \left (d^2 x^2+2\right )+b x \left (d^4 x^4+20 d^2 x^2+120\right )\right ) \sinh (c+d x)-\left (2 a d^4 x+5 b \left (d^4 x^4+12 d^2 x^2+24\right )\right ) \cosh (c+d x)}{d^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.007, size = 389, normalized size = 3.1 \begin{align*}{\frac{1}{{d}^{3}} \left ({\frac{b \left ( \left ( dx+c \right ) ^{5}\sinh \left ( dx+c \right ) -5\, \left ( dx+c \right ) ^{4}\cosh \left ( dx+c \right ) +20\, \left ( dx+c \right ) ^{3}\sinh \left ( dx+c \right ) -60\, \left ( dx+c \right ) ^{2}\cosh \left ( dx+c \right ) +120\, \left ( dx+c \right ) \sinh \left ( dx+c \right ) -120\,\cosh \left ( dx+c \right ) \right ) }{{d}^{3}}}-5\,{\frac{cb \left ( \left ( dx+c \right ) ^{4}\sinh \left ( dx+c \right ) -4\, \left ( dx+c \right ) ^{3}\cosh \left ( dx+c \right ) +12\, \left ( dx+c \right ) ^{2}\sinh \left ( dx+c \right ) -24\, \left ( dx+c \right ) \cosh \left ( dx+c \right ) +24\,\sinh \left ( dx+c \right ) \right ) }{{d}^{3}}}+10\,{\frac{b{c}^{2} \left ( \left ( dx+c \right ) ^{3}\sinh \left ( dx+c \right ) -3\, \left ( dx+c \right ) ^{2}\cosh \left ( dx+c \right ) +6\, \left ( dx+c \right ) \sinh \left ( dx+c \right ) -6\,\cosh \left ( dx+c \right ) \right ) }{{d}^{3}}}-10\,{\frac{b{c}^{3} \left ( \left ( dx+c \right ) ^{2}\sinh \left ( dx+c \right ) -2\, \left ( dx+c \right ) \cosh \left ( dx+c \right ) +2\,\sinh \left ( dx+c \right ) \right ) }{{d}^{3}}}+5\,{\frac{b{c}^{4} \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{{d}^{3}}}-{\frac{b{c}^{5}\sinh \left ( dx+c \right ) }{{d}^{3}}}+a \left ( \left ( dx+c \right ) ^{2}\sinh \left ( dx+c \right ) -2\, \left ( dx+c \right ) \cosh \left ( dx+c \right ) +2\,\sinh \left ( dx+c \right ) \right ) -2\,ac \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) +a{c}^{2}\sinh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.04248, size = 360, normalized size = 2.9 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.73341, size = 200, normalized size = 1.61 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 6.04663, size = 151, normalized size = 1.22 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18935, size = 211, normalized size = 1.7 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]