Integrand size = 21, antiderivative size = 265 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {b e \left (15 c^4 d^2+5 c^2 d e+e^2\right ) \left (1-c^2 x^2\right )}{5 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^2 \left (5 c^2 d+2 e\right ) \left (1-c^2 x^2\right )^2}{15 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 \left (1-c^2 x^2\right )^3}{25 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 (a+b \text {arccosh}(c x))}{x}+3 d^2 e x (a+b \text {arccosh}(c x))+d e^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {arccosh}(c x))+\frac {b c d^3 \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]
[Out]
Time = 0.29 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {276, 5958, 1624, 1813, 1634, 65, 211} \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {d^3 (a+b \text {arccosh}(c x))}{x}+3 d^2 e x (a+b \text {arccosh}(c x))+d e^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {arccosh}(c x))+\frac {b c d^3 \sqrt {c^2 x^2-1} \arctan \left (\sqrt {c^2 x^2-1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b e^2 \left (1-c^2 x^2\right )^2 \left (5 c^2 d+2 e\right )}{15 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^3 \left (1-c^2 x^2\right )^3}{25 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e \left (1-c^2 x^2\right ) \left (15 c^4 d^2+5 c^2 d e+e^2\right )}{5 c^5 \sqrt {c x-1} \sqrt {c x+1}} \]
[In]
[Out]
Rule 65
Rule 211
Rule 276
Rule 1624
Rule 1634
Rule 1813
Rule 5958
Rubi steps \begin{align*} \text {integral}& = -\frac {d^3 (a+b \text {arccosh}(c x))}{x}+3 d^2 e x (a+b \text {arccosh}(c x))+d e^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {arccosh}(c x))-(b c) \int \frac {-d^3+3 d^2 e x^2+d e^2 x^4+\frac {e^3 x^6}{5}}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {d^3 (a+b \text {arccosh}(c x))}{x}+3 d^2 e x (a+b \text {arccosh}(c x))+d e^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {arccosh}(c x))-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {-d^3+3 d^2 e x^2+d e^2 x^4+\frac {e^3 x^6}{5}}{x \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {d^3 (a+b \text {arccosh}(c x))}{x}+3 d^2 e x (a+b \text {arccosh}(c x))+d e^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {arccosh}(c x))-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {-d^3+3 d^2 e x+d e^2 x^2+\frac {e^3 x^3}{5}}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {d^3 (a+b \text {arccosh}(c x))}{x}+3 d^2 e x (a+b \text {arccosh}(c x))+d e^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {arccosh}(c x))-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {e \left (15 c^4 d^2+5 c^2 d e+e^2\right )}{5 c^4 \sqrt {-1+c^2 x}}-\frac {d^3}{x \sqrt {-1+c^2 x}}+\frac {e^2 \left (5 c^2 d+2 e\right ) \sqrt {-1+c^2 x}}{5 c^4}+\frac {e^3 \left (-1+c^2 x\right )^{3/2}}{5 c^4}\right ) \, dx,x,x^2\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b e \left (15 c^4 d^2+5 c^2 d e+e^2\right ) \left (1-c^2 x^2\right )}{5 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^2 \left (5 c^2 d+2 e\right ) \left (1-c^2 x^2\right )^2}{15 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 \left (1-c^2 x^2\right )^3}{25 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 (a+b \text {arccosh}(c x))}{x}+3 d^2 e x (a+b \text {arccosh}(c x))+d e^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {arccosh}(c x))+\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b e \left (15 c^4 d^2+5 c^2 d e+e^2\right ) \left (1-c^2 x^2\right )}{5 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^2 \left (5 c^2 d+2 e\right ) \left (1-c^2 x^2\right )^2}{15 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 \left (1-c^2 x^2\right )^3}{25 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 (a+b \text {arccosh}(c x))}{x}+3 d^2 e x (a+b \text {arccosh}(c x))+d e^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {arccosh}(c x))+\frac {\left (b d^3 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b e \left (15 c^4 d^2+5 c^2 d e+e^2\right ) \left (1-c^2 x^2\right )}{5 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^2 \left (5 c^2 d+2 e\right ) \left (1-c^2 x^2\right )^2}{15 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 \left (1-c^2 x^2\right )^3}{25 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 (a+b \text {arccosh}(c x))}{x}+3 d^2 e x (a+b \text {arccosh}(c x))+d e^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {arccosh}(c x))+\frac {b c d^3 \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {a d^3}{x}+3 a d^2 e x+a d e^2 x^3+\frac {1}{5} a e^3 x^5-\frac {b e \sqrt {-1+c x} \sqrt {1+c x} \left (8 e^2+2 c^2 e \left (25 d+2 e x^2\right )+c^4 \left (225 d^2+25 d e x^2+3 e^2 x^4\right )\right )}{75 c^5}+\frac {b \left (-5 d^3+15 d^2 e x^2+5 d e^2 x^4+e^3 x^6\right ) \text {arccosh}(c x)}{5 x}-b c d^3 \arctan \left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \]
[In]
[Out]
Time = 0.67 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.03
method | result | size |
parts | \(a \left (\frac {e^{3} x^{5}}{5}+d \,e^{2} x^{3}+3 d^{2} e x -\frac {d^{3}}{x}\right )+b c \left (\frac {\operatorname {arccosh}\left (c x \right ) e^{3} x^{5}}{5 c}+\frac {\operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{3}}{c}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) d^{2} e x}{c}-\frac {\operatorname {arccosh}\left (c x \right ) d^{3}}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )+225 c^{4} d^{2} e \sqrt {c^{2} x^{2}-1}+25 c^{4} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{2}+3 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{4} x^{4}+50 c^{2} d \,e^{2} \sqrt {c^{2} x^{2}-1}+4 e^{3} c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+8 e^{3} \sqrt {c^{2} x^{2}-1}\right )}{75 c^{6} \sqrt {c^{2} x^{2}-1}}\right )\) | \(273\) |
derivativedivides | \(c \left (\frac {a \left (3 c^{5} x \,d^{2} e +c^{5} x^{3} d \,e^{2}+\frac {c^{5} x^{5} e^{3}}{5}-\frac {c^{5} d^{3}}{x}\right )}{c^{6}}+\frac {b \left (3 \,\operatorname {arccosh}\left (c x \right ) c^{5} d^{2} e x +\operatorname {arccosh}\left (c x \right ) c^{5} d \,e^{2} x^{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{5} x^{5}}{5}-\frac {\operatorname {arccosh}\left (c x \right ) c^{5} d^{3}}{x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )+225 c^{4} d^{2} e \sqrt {c^{2} x^{2}-1}+25 c^{4} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{2}+3 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{4} x^{4}+50 c^{2} d \,e^{2} \sqrt {c^{2} x^{2}-1}+4 e^{3} c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+8 e^{3} \sqrt {c^{2} x^{2}-1}\right )}{75 \sqrt {c^{2} x^{2}-1}}\right )}{c^{6}}\right )\) | \(289\) |
default | \(c \left (\frac {a \left (3 c^{5} x \,d^{2} e +c^{5} x^{3} d \,e^{2}+\frac {c^{5} x^{5} e^{3}}{5}-\frac {c^{5} d^{3}}{x}\right )}{c^{6}}+\frac {b \left (3 \,\operatorname {arccosh}\left (c x \right ) c^{5} d^{2} e x +\operatorname {arccosh}\left (c x \right ) c^{5} d \,e^{2} x^{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{5} x^{5}}{5}-\frac {\operatorname {arccosh}\left (c x \right ) c^{5} d^{3}}{x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )+225 c^{4} d^{2} e \sqrt {c^{2} x^{2}-1}+25 c^{4} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{2}+3 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{4} x^{4}+50 c^{2} d \,e^{2} \sqrt {c^{2} x^{2}-1}+4 e^{3} c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+8 e^{3} \sqrt {c^{2} x^{2}-1}\right )}{75 \sqrt {c^{2} x^{2}-1}}\right )}{c^{6}}\right )\) | \(289\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.23 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {15 \, a c^{5} e^{3} x^{6} + 75 \, a c^{5} d e^{2} x^{4} + 150 \, b c^{6} d^{3} x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 225 \, a c^{5} d^{2} e x^{2} - 75 \, a c^{5} d^{3} + 15 \, {\left (5 \, b c^{5} d^{3} - 15 \, b c^{5} d^{2} e - 5 \, b c^{5} d e^{2} - b c^{5} e^{3}\right )} x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 15 \, {\left (b c^{5} e^{3} x^{6} + 5 \, b c^{5} d e^{2} x^{4} + 15 \, b c^{5} d^{2} e x^{2} - 5 \, b c^{5} d^{3} + {\left (5 \, b c^{5} d^{3} - 15 \, b c^{5} d^{2} e - 5 \, b c^{5} d e^{2} - b c^{5} e^{3}\right )} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (3 \, b c^{4} e^{3} x^{5} + {\left (25 \, b c^{4} d e^{2} + 4 \, b c^{2} e^{3}\right )} x^{3} + {\left (225 \, b c^{4} d^{2} e + 50 \, b c^{2} d e^{2} + 8 \, b e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{75 \, c^{5} x} \]
[In]
[Out]
\[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}}{x^{2}}\, dx \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.84 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {1}{5} \, a e^{3} x^{5} + a d e^{2} x^{3} - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d e^{2} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b e^{3} + 3 \, a d^{2} e x + \frac {3 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{2} e}{c} - \frac {a d^{3}}{x} \]
[In]
[Out]
\[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3}{x^2} \,d x \]
[In]
[Out]