Optimal. Leaf size=79 \[ -\frac {a \cosh (c+d x)}{d^2}+\frac {a x \sinh (c+d x)}{d}-\frac {6 b \cosh (c+d x)}{d^4}+\frac {6 b x \sinh (c+d x)}{d^3}-\frac {3 b x^2 \cosh (c+d x)}{d^2}+\frac {b x^3 \sinh (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5287, 3296, 2638} \[ -\frac {a \cosh (c+d x)}{d^2}+\frac {a x \sinh (c+d x)}{d}-\frac {3 b x^2 \cosh (c+d x)}{d^2}+\frac {6 b x \sinh (c+d x)}{d^3}-\frac {6 b \cosh (c+d x)}{d^4}+\frac {b x^3 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2638
Rule 3296
Rule 5287
Rubi steps
\begin {align*} \int x \left (a+b x^2\right ) \cosh (c+d x) \, dx &=\int \left (a x \cosh (c+d x)+b x^3 \cosh (c+d x)\right ) \, dx\\ &=a \int x \cosh (c+d x) \, dx+b \int x^3 \cosh (c+d x) \, dx\\ &=\frac {a x \sinh (c+d x)}{d}+\frac {b x^3 \sinh (c+d x)}{d}-\frac {a \int \sinh (c+d x) \, dx}{d}-\frac {(3 b) \int x^2 \sinh (c+d x) \, dx}{d}\\ &=-\frac {a \cosh (c+d x)}{d^2}-\frac {3 b x^2 \cosh (c+d x)}{d^2}+\frac {a x \sinh (c+d x)}{d}+\frac {b x^3 \sinh (c+d x)}{d}+\frac {(6 b) \int x \cosh (c+d x) \, dx}{d^2}\\ &=-\frac {a \cosh (c+d x)}{d^2}-\frac {3 b x^2 \cosh (c+d x)}{d^2}+\frac {6 b x \sinh (c+d x)}{d^3}+\frac {a x \sinh (c+d x)}{d}+\frac {b x^3 \sinh (c+d x)}{d}-\frac {(6 b) \int \sinh (c+d x) \, dx}{d^3}\\ &=-\frac {6 b \cosh (c+d x)}{d^4}-\frac {a \cosh (c+d x)}{d^2}-\frac {3 b x^2 \cosh (c+d x)}{d^2}+\frac {6 b x \sinh (c+d x)}{d^3}+\frac {a x \sinh (c+d x)}{d}+\frac {b x^3 \sinh (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 57, normalized size = 0.72 \[ \frac {d x \left (a d^2+b \left (d^2 x^2+6\right )\right ) \sinh (c+d x)-\left (a d^2+3 b \left (d^2 x^2+2\right )\right ) \cosh (c+d x)}{d^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 60, normalized size = 0.76 \[ -\frac {{\left (3 \, b d^{2} x^{2} + a d^{2} + 6 \, b\right )} \cosh \left (d x + c\right ) - {\left (b d^{3} x^{3} + {\left (a d^{3} + 6 \, b d\right )} x\right )} \sinh \left (d x + c\right )}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.11, size = 101, normalized size = 1.28 \[ \frac {{\left (b d^{3} x^{3} + a d^{3} x - 3 \, b d^{2} x^{2} - a d^{2} + 6 \, b d x - 6 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{4}} - \frac {{\left (b d^{3} x^{3} + a d^{3} x + 3 \, b d^{2} x^{2} + a d^{2} + 6 \, b d x + 6 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 183, normalized size = 2.32 \[ \frac {\frac {b \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{2}}-\frac {3 b c \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}+\frac {3 b \,c^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}+a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-\frac {b \,c^{3} \sinh \left (d x +c \right )}{d^{2}}-a c \sinh \left (d x +c \right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.33, size = 213, normalized size = 2.70 \[ \frac {{\left (b x^{2} + a\right )}^{2} \cosh \left (d x + c\right )}{4 \, b} - \frac {{\left (\frac {a^{2} e^{\left (d x + c\right )}}{d} + \frac {a^{2} e^{\left (-d x - c\right )}}{d} + \frac {2 \, {\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{3}} + \frac {2 \, {\left (d^{2} x^{2} + 2 \, d x + 2\right )} a b e^{\left (-d x - c\right )}}{d^{3}} + \frac {{\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{5}} + \frac {{\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} b^{2} e^{\left (-d x - c\right )}}{d^{5}}\right )} d}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.98, size = 70, normalized size = 0.89 \[ \frac {x\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a\,d^2+6\,b\right )}{d^3}-\frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (a\,d^2+6\,b\right )}{d^4}-\frac {3\,b\,x^2\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {b\,x^3\,\mathrm {sinh}\left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.99, size = 99, normalized size = 1.25 \[ \begin {cases} \frac {a x \sinh {\left (c + d x \right )}}{d} - \frac {a \cosh {\left (c + d x \right )}}{d^{2}} + \frac {b x^{3} \sinh {\left (c + d x \right )}}{d} - \frac {3 b x^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {6 b x \sinh {\left (c + d x \right )}}{d^{3}} - \frac {6 b \cosh {\left (c + d x \right )}}{d^{4}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{2}}{2} + \frac {b x^{4}}{4}\right ) \cosh {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________