Integrand size = 17, antiderivative size = 122 \[ \int x \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {b \left (8 c^2 d+3 e\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {b e x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}-\frac {b \left (8 c^2 d+3 e\right ) \text {arccosh}(c x)}{32 c^4}+\frac {1}{2} d x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e x^4 (a+b \text {arccosh}(c x)) \]
[Out]
Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {5956, 471, 92, 54} \[ \int x \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{2} d x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e x^4 (a+b \text {arccosh}(c x))-\frac {b \text {arccosh}(c x) \left (8 c^2 d+3 e\right )}{32 c^4}-\frac {b x \sqrt {c x-1} \sqrt {c x+1} \left (8 c^2 d+3 e\right )}{32 c^3}-\frac {b e x^3 \sqrt {c x-1} \sqrt {c x+1}}{16 c} \]
[In]
[Out]
Rule 54
Rule 92
Rule 471
Rule 5956
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} d x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e x^4 (a+b \text {arccosh}(c x))-\frac {1}{8} (b c) \int \frac {x^2 \left (4 d+2 e x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b e x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}+\frac {1}{2} d x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e x^4 (a+b \text {arccosh}(c x))-\frac {1}{16} \left (b c \left (8 d+\frac {3 e}{c^2}\right )\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b \left (8 c^2 d+3 e\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {b e x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}+\frac {1}{2} d x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e x^4 (a+b \text {arccosh}(c x))-\frac {\left (b \left (8 c^2 d+3 e\right )\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 c^3} \\ & = -\frac {b \left (8 c^2 d+3 e\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {b e x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}-\frac {b \left (8 c^2 d+3 e\right ) \text {arccosh}(c x)}{32 c^4}+\frac {1}{2} d x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e x^4 (a+b \text {arccosh}(c x)) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.98 \[ \int x \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {c x \left (8 a c^3 x \left (2 d+e x^2\right )-b \sqrt {-1+c x} \sqrt {1+c x} \left (3 e+2 c^2 \left (4 d+e x^2\right )\right )\right )+8 b c^4 x^2 \left (2 d+e x^2\right ) \text {arccosh}(c x)-2 b \left (8 c^2 d+3 e\right ) \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )}{32 c^4} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(224\) vs. \(2(104)=208\).
Time = 0.34 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.84
method | result | size |
parts | \(\frac {a \left (e \,x^{2}+d \right )^{2}}{4 e}+\frac {b \left (\frac {c^{2} e \,\operatorname {arccosh}\left (c x \right ) x^{4}}{4}+\frac {\operatorname {arccosh}\left (c x \right ) c^{2} x^{2} d}{2}+\frac {c^{2} \operatorname {arccosh}\left (c x \right ) d^{2}}{4 e}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (8 c^{4} d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+8 d \,c^{3} e x \sqrt {c^{2} x^{2}-1}+2 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+8 d \,c^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+3 e^{2} c x \sqrt {c^{2} x^{2}-1}+3 e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{32 c^{2} e \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\) | \(225\) |
derivativedivides | \(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}{4 c^{2} e}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{4} d^{2}}{4 e}+\frac {\operatorname {arccosh}\left (c x \right ) c^{4} d \,x^{2}}{2}+\frac {e \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (8 c^{4} d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+8 d \,c^{3} e x \sqrt {c^{2} x^{2}-1}+2 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+8 d \,c^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+3 e^{2} c x \sqrt {c^{2} x^{2}-1}+3 e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{32 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}}{c^{2}}\) | \(236\) |
default | \(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}{4 c^{2} e}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{4} d^{2}}{4 e}+\frac {\operatorname {arccosh}\left (c x \right ) c^{4} d \,x^{2}}{2}+\frac {e \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (8 c^{4} d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+8 d \,c^{3} e x \sqrt {c^{2} x^{2}-1}+2 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+8 d \,c^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+3 e^{2} c x \sqrt {c^{2} x^{2}-1}+3 e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{32 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}}{c^{2}}\) | \(236\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.93 \[ \int x \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {8 \, a c^{4} e x^{4} + 16 \, a c^{4} d x^{2} + {\left (8 \, b c^{4} e x^{4} + 16 \, b c^{4} d x^{2} - 8 \, b c^{2} d - 3 \, b e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, b c^{3} e x^{3} + {\left (8 \, b c^{3} d + 3 \, b c e\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{32 \, c^{4}} \]
[In]
[Out]
\[ \int x \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int x \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.28 \[ \int x \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{4} \, a e x^{4} + \frac {1}{2} \, a d x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} b d + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b e \]
[In]
[Out]
Exception generated. \[ \int x \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Timed out. \[ \int x \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \]
[In]
[Out]