Optimal. Leaf size=184 \[ \frac{2 a^2 \sinh (c+d x)}{d^3}-\frac{2 a^2 x \cosh (c+d x)}{d^2}+\frac{a^2 x^2 \sinh (c+d x)}{d}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}+\frac{12 a b x \sinh (c+d x)}{d^3}-\frac{12 a b \cosh (c+d x)}{d^4}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{12 b^2 x^2 \sinh (c+d x)}{d^3}-\frac{4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac{24 b^2 \sinh (c+d x)}{d^5}-\frac{24 b^2 x \cosh (c+d x)}{d^4}+\frac{b^2 x^4 \sinh (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.346497, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {6742, 3296, 2637, 2638} \[ \frac{2 a^2 \sinh (c+d x)}{d^3}-\frac{2 a^2 x \cosh (c+d x)}{d^2}+\frac{a^2 x^2 \sinh (c+d x)}{d}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}+\frac{12 a b x \sinh (c+d x)}{d^3}-\frac{12 a b \cosh (c+d x)}{d^4}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{12 b^2 x^2 \sinh (c+d x)}{d^3}-\frac{4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac{24 b^2 \sinh (c+d x)}{d^5}-\frac{24 b^2 x \cosh (c+d x)}{d^4}+\frac{b^2 x^4 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6742
Rule 3296
Rule 2637
Rule 2638
Rubi steps
\begin{align*} \int x^2 (a+b x)^2 \cosh (c+d x) \, dx &=\int \left (a^2 x^2 \cosh (c+d x)+2 a b x^3 \cosh (c+d x)+b^2 x^4 \cosh (c+d x)\right ) \, dx\\ &=a^2 \int x^2 \cosh (c+d x) \, dx+(2 a b) \int x^3 \cosh (c+d x) \, dx+b^2 \int x^4 \cosh (c+d x) \, dx\\ &=\frac{a^2 x^2 \sinh (c+d x)}{d}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{b^2 x^4 \sinh (c+d x)}{d}-\frac{\left (2 a^2\right ) \int x \sinh (c+d x) \, dx}{d}-\frac{(6 a b) \int x^2 \sinh (c+d x) \, dx}{d}-\frac{\left (4 b^2\right ) \int x^3 \sinh (c+d x) \, dx}{d}\\ &=-\frac{2 a^2 x \cosh (c+d x)}{d^2}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}-\frac{4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac{a^2 x^2 \sinh (c+d x)}{d}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{b^2 x^4 \sinh (c+d x)}{d}+\frac{\left (2 a^2\right ) \int \cosh (c+d x) \, dx}{d^2}+\frac{(12 a b) \int x \cosh (c+d x) \, dx}{d^2}+\frac{\left (12 b^2\right ) \int x^2 \cosh (c+d x) \, dx}{d^2}\\ &=-\frac{2 a^2 x \cosh (c+d x)}{d^2}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}-\frac{4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac{2 a^2 \sinh (c+d x)}{d^3}+\frac{12 a b x \sinh (c+d x)}{d^3}+\frac{12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac{a^2 x^2 \sinh (c+d x)}{d}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{b^2 x^4 \sinh (c+d x)}{d}-\frac{(12 a b) \int \sinh (c+d x) \, dx}{d^3}-\frac{\left (24 b^2\right ) \int x \sinh (c+d x) \, dx}{d^3}\\ &=-\frac{12 a b \cosh (c+d x)}{d^4}-\frac{24 b^2 x \cosh (c+d x)}{d^4}-\frac{2 a^2 x \cosh (c+d x)}{d^2}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}-\frac{4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac{2 a^2 \sinh (c+d x)}{d^3}+\frac{12 a b x \sinh (c+d x)}{d^3}+\frac{12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac{a^2 x^2 \sinh (c+d x)}{d}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{b^2 x^4 \sinh (c+d x)}{d}+\frac{\left (24 b^2\right ) \int \cosh (c+d x) \, dx}{d^4}\\ &=-\frac{12 a b \cosh (c+d x)}{d^4}-\frac{24 b^2 x \cosh (c+d x)}{d^4}-\frac{2 a^2 x \cosh (c+d x)}{d^2}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}-\frac{4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac{24 b^2 \sinh (c+d x)}{d^5}+\frac{2 a^2 \sinh (c+d x)}{d^3}+\frac{12 a b x \sinh (c+d x)}{d^3}+\frac{12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac{a^2 x^2 \sinh (c+d x)}{d}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{b^2 x^4 \sinh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.241922, size = 100, normalized size = 0.54 \[ \frac{\left (a^2 d^2 \left (d^2 x^2+2\right )+2 a b d^2 x \left (d^2 x^2+6\right )+b^2 \left (d^4 x^4+12 d^2 x^2+24\right )\right ) \sinh (c+d x)-2 d (a+2 b x) \left (a d^2 x+b \left (d^2 x^2+6\right )\right ) \cosh (c+d x)}{d^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.007, size = 463, normalized size = 2.5 \begin{align*}{\frac{1}{{d}^{3}} \left ({\frac{{b}^{2} \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}^{2}}}-4\,{\frac{c{b}^{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}^{2}}}+6\,{\frac{{c}^{2}{b}^{2} \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}^{2}}}-4\,{\frac{{b}^{2}{c}^{3} \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{{d}^{2}}}+2\,{\frac{ab \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}}-6\,{\frac{cba \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}}+6\,{\frac{ab{c}^{2} \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{d}}+{\frac{{b}^{2}{c}^{4}\sinh \left ( dx+c \right ) }{{d}^{2}}}-2\,{\frac{b{c}^{3}a\sinh \left ( dx+c \right ) }{d}}+{a}^{2} \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\,{a}^{2}c \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) +{a}^{2}{c}^{2}\sinh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.18353, size = 444, normalized size = 2.41 \begin{align*} -\frac{1}{60} \, d{\left (\frac{10 \,{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a^{2} e^{\left (d x\right )}}{d^{4}} + \frac{10 \,{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a^{2} e^{\left (-d x - c\right )}}{d^{4}} + \frac{15 \,{\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 b e^{\left (d x\right )}}{d^{5}} + \frac{15 \,{\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} a b e^{\left (-d x - c\right )}}{d^{5}} + \frac{6 \,{\left (d^{5} x^{5} e^{c} - 5 \, d^{4} x^{4} e^{c} + 20 \, d^{3} x^{3} e^{c} - 60 \, d^{2} x^{2} e^{c} + 120 \, d x e^{c} - 120 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{6}} + \frac{6 \,{\left (d^{5} x^{5} + 5 \, d^{4} x^{4} + 20 \, d^{3} x^{3} + 60 \, d^{2} x^{2} + 120 \, d x + 120\right )} b^{2} e^{\left (-d x - c\right )}}{d^{6}}\right )} + \frac{1}{30} \,{\left (6 \, b^{2} x^{5} + 15 \, a b x^{4} + 10 \, a^{2} x^{3}\right )} \cosh \left (d x + c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.01887, size = 273, normalized size = 1.48 \begin{align*} -\frac{2 \,{\left (2 \, b^{2} d^{3} x^{3} + 3 \, a b d^{3} x^{2} + 6 \, a b d +{\left (a^{2} d^{3} + 12 \, b^{2} d\right )} x\right )} \cosh \left (d x + c\right ) -{\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{3} + 12 \, a b d^{2} x + 2 \, a^{2} d^{2} +{\left (a^{2} d^{4} + 12 \, b^{2} d^{2}\right )} x^{2} + 24 \, b^{2}\right )} \sinh \left (d x + c\right )}{d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.53062, size = 228, normalized size = 1.24 \begin{align*} \begin{cases} \frac{a^{2} x^{2} \sinh{\left (c + d x \right )}}{d} - \frac{2 a^{2} x \cosh{\left (c + d x \right )}}{d^{2}} + \frac{2 a^{2} \sinh{\left (c + d x \right )}}{d^{3}} + \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^{4} \sinh{\left (c + d x \right )}}{d} - \frac{4 b^{2} x^{3} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{12 b^{2} x^{2} \sinh{\left (c + d x \right )}}{d^{3}} - \frac{24 b^{2} x \cosh{\left (c + d x \right )}}{d^{4}} + \frac{24 b^{2} \sinh{\left (c + d x \right )}}{d^{5}} & \text{for}\: d \neq 0 \\\left (\frac{a^{2} x^{3}}{3} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{5}}{5}\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 = 1.17721, size = 319, normalized size = 1.73 \begin{align*} \frac{{\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{3} + a^{2} d^{4} x^{2} - 4 \, b^{2} d^{3} x^{3} - 6 \, a b d^{3} x^{2} - 2 \, a^{2} d^{3} x + 12 \, b^{2} d^{2} x^{2} + 12 \, a b d^{2} x + 2 \, a^{2} d^{2} - 24 \, b^{2} d x - 12 \, a b d + 24 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{5}} - \frac{{\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{3} + a^{2} d^{4} x^{2} + 4 \, b^{2} d^{3} x^{3} + 6 \, a b d^{3} x^{2} + 2 \, a^{2} d^{3} x + 12 \, b^{2} d^{2} x^{2} + 12 \, a b d^{2} x + 2 \, a^{2} d^{2} + 24 \, b^{2} d x + 12 \, a b d + 24 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]