Integrand size = 19, antiderivative size = 177 \[ \int x^4 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {8 b \left (49 c^2 d+30 e\right ) \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^7}-\frac {4 b \left (49 c^2 d+30 e\right ) x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^5}-\frac {b \left (49 c^2 d+30 e\right ) x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c^3}-\frac {b e x^6 \sqrt {-1+c x} \sqrt {1+c x}}{49 c}+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e x^7 (a+b \text {arccosh}(c x)) \]
[Out]
Time = 0.10 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5956, 471, 102, 12, 75} \[ \int x^4 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e x^7 (a+b \text {arccosh}(c x))-\frac {8 b \sqrt {c x-1} \sqrt {c x+1} \left (49 c^2 d+30 e\right )}{3675 c^7}-\frac {4 b x^2 \sqrt {c x-1} \sqrt {c x+1} \left (49 c^2 d+30 e\right )}{3675 c^5}-\frac {b x^4 \sqrt {c x-1} \sqrt {c x+1} \left (49 c^2 d+30 e\right )}{1225 c^3}-\frac {b e x^6 \sqrt {c x-1} \sqrt {c x+1}}{49 c} \]
[In]
[Out]
Rule 12
Rule 75
Rule 102
Rule 471
Rule 5956
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} d x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e x^7 (a+b \text {arccosh}(c x))-\frac {1}{35} (b c) \int \frac {x^5 \left (7 d+5 e x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b e x^6 \sqrt {-1+c x} \sqrt {1+c x}}{49 c}+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e x^7 (a+b \text {arccosh}(c x))+\frac {1}{245} \left (b c \left (-49 d-\frac {30 e}{c^2}\right )\right ) \int \frac {x^5}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b \left (49 c^2 d+30 e\right ) x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c^3}-\frac {b e x^6 \sqrt {-1+c x} \sqrt {1+c x}}{49 c}+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e x^7 (a+b \text {arccosh}(c x))-\frac {\left (b \left (49 c^2 d+30 e\right )\right ) \int \frac {4 x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{1225 c^3} \\ & = -\frac {b \left (49 c^2 d+30 e\right ) x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c^3}-\frac {b e x^6 \sqrt {-1+c x} \sqrt {1+c x}}{49 c}+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e x^7 (a+b \text {arccosh}(c x))-\frac {\left (4 b \left (49 c^2 d+30 e\right )\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{1225 c^3} \\ & = -\frac {4 b \left (49 c^2 d+30 e\right ) x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^5}-\frac {b \left (49 c^2 d+30 e\right ) x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c^3}-\frac {b e x^6 \sqrt {-1+c x} \sqrt {1+c x}}{49 c}+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e x^7 (a+b \text {arccosh}(c x))-\frac {\left (4 b \left (49 c^2 d+30 e\right )\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3675 c^5} \\ & = -\frac {4 b \left (49 c^2 d+30 e\right ) x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^5}-\frac {b \left (49 c^2 d+30 e\right ) x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c^3}-\frac {b e x^6 \sqrt {-1+c x} \sqrt {1+c x}}{49 c}+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e x^7 (a+b \text {arccosh}(c x))-\frac {\left (8 b \left (49 c^2 d+30 e\right )\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3675 c^5} \\ & = -\frac {8 b \left (49 c^2 d+30 e\right ) \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^7}-\frac {4 b \left (49 c^2 d+30 e\right ) x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^5}-\frac {b \left (49 c^2 d+30 e\right ) x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c^3}-\frac {b e x^6 \sqrt {-1+c x} \sqrt {1+c x}}{49 c}+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e x^7 (a+b \text {arccosh}(c x)) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.69 \[ \int x^4 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{35} a x^5 \left (7 d+5 e x^2\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (240 e+8 c^2 \left (49 d+15 e x^2\right )+2 c^4 \left (98 d x^2+45 e x^4\right )+3 c^6 \left (49 d x^4+25 e x^6\right )\right )}{3675 c^7}+\frac {1}{35} b x^5 \left (7 d+5 e x^2\right ) \text {arccosh}(c x) \]
[In]
[Out]
Time = 0.58 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.69
method | result | size |
parts | \(a \left (\frac {1}{7} e \,x^{7}+\frac {1}{5} d \,x^{5}\right )+\frac {b \left (\frac {c^{5} \operatorname {arccosh}\left (c x \right ) e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5} d}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e \,x^{6}+147 c^{6} d \,x^{4}+90 c^{4} e \,x^{4}+196 c^{4} d \,x^{2}+120 c^{2} e \,x^{2}+392 c^{2} d +240 e \right )}{3675 c^{2}}\right )}{c^{5}}\) | \(123\) |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{5} d \,c^{7} x^{5}+\frac {1}{7} e \,c^{7} x^{7}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) d \,c^{7} x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) e \,c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e \,x^{6}+147 c^{6} d \,x^{4}+90 c^{4} e \,x^{4}+196 c^{4} d \,x^{2}+120 c^{2} e \,x^{2}+392 c^{2} d +240 e \right )}{3675}\right )}{c^{2}}}{c^{5}}\) | \(133\) |
default | \(\frac {\frac {a \left (\frac {1}{5} d \,c^{7} x^{5}+\frac {1}{7} e \,c^{7} x^{7}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) d \,c^{7} x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) e \,c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e \,x^{6}+147 c^{6} d \,x^{4}+90 c^{4} e \,x^{4}+196 c^{4} d \,x^{2}+120 c^{2} e \,x^{2}+392 c^{2} d +240 e \right )}{3675}\right )}{c^{2}}}{c^{5}}\) | \(133\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.79 \[ \int x^4 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {525 \, a c^{7} e x^{7} + 735 \, a c^{7} d x^{5} + 105 \, {\left (5 \, b c^{7} e x^{7} + 7 \, b c^{7} d x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (75 \, b c^{6} e x^{6} + 3 \, {\left (49 \, b c^{6} d + 30 \, b c^{4} e\right )} x^{4} + 392 \, b c^{2} d + 4 \, {\left (49 \, b c^{4} d + 30 \, b c^{2} e\right )} x^{2} + 240 \, b e\right )} \sqrt {c^{2} x^{2} - 1}}{3675 \, c^{7}} \]
[In]
[Out]
\[ \int x^4 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.01 \[ \int x^4 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{7} \, a e x^{7} + \frac {1}{5} \, a d x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b e \]
[In]
[Out]
Exception generated. \[ \int x^4 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Timed out. \[ \int x^4 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \]
[In]
[Out]