Optimal. Leaf size=184 \[ \frac {2 a^2 \sinh (c+d x)}{d^3}-\frac {2 a^2 x \cosh (c+d x)}{d^2}+\frac {a^2 x^2 \sinh (c+d x)}{d}-\frac {12 a b \cosh (c+d x)}{d^4}+\frac {12 a b x \sinh (c+d x)}{d^3}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {24 b^2 \sinh (c+d x)}{d^5}-\frac {24 b^2 x \cosh (c+d x)}{d^4}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {b^2 x^4 \sinh (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.35, antiderivative size = 184, normalized size of antiderivative = 1.00, 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 2637
Rule 2638
Rule 3296
Rule 6742
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 = 0.28, 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]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 127, normalized size = 0.69 \[ -\frac {2 \, {\left (2 \, b^{2} d^{3} x^{3} + 3 \, a b d^{3} x^{2} + 6 \, a b d + {\left (a^{2} d^{3} + 12 \, b^{2} d\right )} x\right )} \cosh \left (d x + c\right ) - {\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{3} + 12 \, a b d^{2} x + 2 \, a^{2} d^{2} + {\left (a^{2} d^{4} + 12 \, b^{2} d^{2}\right )} x^{2} + 24 \, b^{2}\right )} \sinh \left (d x + c\right )}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 236, normalized size = 1.28 \[ \frac {{\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{3} + a^{2} d^{4} x^{2} - 4 \, b^{2} d^{3} x^{3} - 6 \, a b d^{3} x^{2} - 2 \, a^{2} d^{3} x + 12 \, b^{2} d^{2} x^{2} + 12 \, a b d^{2} x + 2 \, a^{2} d^{2} - 24 \, b^{2} d x - 12 \, a b d + 24 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{5}} - \frac {{\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{3} + a^{2} d^{4} x^{2} + 4 \, b^{2} d^{3} x^{3} + 6 \, a b d^{3} x^{2} + 2 \, a^{2} d^{3} x + 12 \, b^{2} d^{2} x^{2} + 12 \, a b d^{2} x + 2 \, a^{2} d^{2} + 24 \, b^{2} d x + 12 \, a b d + 24 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 463, normalized size = 2.52 \[ \frac {\frac {b^{2} \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{2}}-\frac {4 b^{2} c \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{2}}+\frac {6 b^{2} c^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}-\frac {4 b^{2} c^{3} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}+\frac {2 b a \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d}-\frac {6 b c a \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d}+\frac {6 b a \,c^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d}+\frac {b^{2} c^{4} \sinh \left (d x +c \right )}{d^{2}}-\frac {2 b \,c^{3} a \sinh \left (d x +c \right )}{d}+a^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )-2 a^{2} c \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+a^{2} c^{2} \sinh \left (d x +c \right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 329, normalized size = 1.79 \[ -\frac {1}{60} \, d {\left (\frac {10 \, {\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a^{2} e^{\left (d x\right )}}{d^{4}} + \frac {10 \, {\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a^{2} e^{\left (-d x - c\right )}}{d^{4}} + \frac {15 \, {\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{5}} + \frac {15 \, {\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} a b e^{\left (-d x - c\right )}}{d^{5}} + \frac {6 \, {\left (d^{5} x^{5} e^{c} - 5 \, d^{4} x^{4} e^{c} + 20 \, d^{3} x^{3} e^{c} - 60 \, d^{2} x^{2} e^{c} + 120 \, d x e^{c} - 120 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{6}} + \frac {6 \, {\left (d^{5} x^{5} + 5 \, d^{4} x^{4} + 20 \, d^{3} x^{3} + 60 \, d^{2} x^{2} + 120 \, d x + 120\right )} b^{2} e^{\left (-d x - c\right )}}{d^{6}}\right )} + \frac {1}{30} \, {\left (6 \, b^{2} x^{5} + 15 \, a b x^{4} + 10 \, a^{2} x^{3}\right )} \cosh \left (d x + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.16, size = 168, normalized size = 0.91 \[ \frac {2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^2+12\,b^2\right )}{d^5}-\frac {4\,b^2\,x^3\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {b^2\,x^4\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {12\,a\,b\,\mathrm {cosh}\left (c+d\,x\right )}{d^4}-\frac {2\,x\,\mathrm {cosh}\left (c+d\,x\right )\,\left (a^2\,d^2+12\,b^2\right )}{d^4}+\frac {x^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^2+12\,b^2\right )}{d^3}-\frac {6\,a\,b\,x^2\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {2\,a\,b\,x^3\,\mathrm {sinh}\left (c+d\,x\right )}{d}+\frac {12\,a\,b\,x\,\mathrm {sinh}\left (c+d\,x\right )}{d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.24, size = 228, normalized size = 1.24 \[ \begin {cases} \frac {a^{2} x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {2 a^{2} x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 a^{2} \sinh {\left (c + d x \right )}}{d^{3}} + \frac {2 a b x^{3} \sinh {\left (c + d x \right )}}{d} - \frac {6 a b x^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {12 a b x \sinh {\left (c + d x \right )}}{d^{3}} - \frac {12 a b \cosh {\left (c + d x \right )}}{d^{4}} + \frac {b^{2} x^{4} \sinh {\left (c + d x \right )}}{d} - \frac {4 b^{2} x^{3} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {12 b^{2} x^{2} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {24 b^{2} x \cosh {\left (c + d x \right )}}{d^{4}} + \frac {24 b^{2} \sinh {\left (c + d x \right )}}{d^{5}} & \text {for}\: d \neq 0 \\\left (\frac {a^{2} x^{3}}{3} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{5}}{5}\right ) \cosh {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________