Integrand size = 27, antiderivative size = 212 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {3 b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{16 c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x}}{16 c \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 c^2 d}+\frac {3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{16 b c^5 \sqrt {d-c^2 d x^2}} \]
[Out]
Time = 0.18 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5938, 5892, 30} \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 c^2 d}+\frac {3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{16 b c^5 \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^4 d}-\frac {b x^4 \sqrt {c x-1} \sqrt {c x+1}}{16 c \sqrt {d-c^2 d x^2}}-\frac {3 b x^2 \sqrt {c x-1} \sqrt {c x+1}}{16 c^3 \sqrt {d-c^2 d x^2}} \]
[In]
[Out]
Rule 30
Rule 5892
Rule 5938
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 c^2 d}+\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx}{4 c^2}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^3 \, dx}{4 c \sqrt {d-c^2 d x^2}} \\ & = -\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x}}{16 c \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 c^2 d}+\frac {3 \int \frac {a+b \text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}} \, dx}{8 c^4}-\frac {\left (3 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x \, dx}{8 c^3 \sqrt {d-c^2 d x^2}} \\ & = -\frac {3 b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{16 c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x}}{16 c \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 c^2 d}+\frac {3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{16 b c^5 \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 0.94 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.81 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {-\frac {16 a c x \left (3+2 c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{d}-\frac {48 a \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{\sqrt {d}}+\frac {b \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (-16 \cosh (2 \text {arccosh}(c x))-\cosh (4 \text {arccosh}(c x))+4 \text {arccosh}(c x) (6 \text {arccosh}(c x)+8 \sinh (2 \text {arccosh}(c x))+\sinh (4 \text {arccosh}(c x))))}{\sqrt {d-c^2 d x^2}}}{128 c^5} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(567\) vs. \(2(180)=360\).
Time = 0.94 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.68
method | result | size |
default | \(-\frac {a \,x^{3} \sqrt {-c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{4} \sqrt {c^{2} d}}+b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}}{16 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 d \,c^{5} \left (c^{2} x^{2}-1\right )}\right )\) | \(568\) |
parts | \(-\frac {a \,x^{3} \sqrt {-c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{4} \sqrt {c^{2} d}}+b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}}{16 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 d \,c^{5} \left (c^{2} x^{2}-1\right )}\right )\) | \(568\) |
[In]
[Out]
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
[In]
[Out]
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
[In]
[Out]