Integrand size = 16, antiderivative size = 88 \[ \int \sqrt {d x} (a+b \arccos (c x)) \, dx=-\frac {4 b \sqrt {d x} \sqrt {1-c^2 x^2}}{9 c}+\frac {2 (d x)^{3/2} (a+b \arccos (c x))}{3 d}+\frac {4 b \sqrt {d} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{9 c^{3/2}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4724, 327, 335, 227} \[ \int \sqrt {d x} (a+b \arccos (c x)) \, dx=\frac {2 (d x)^{3/2} (a+b \arccos (c x))}{3 d}+\frac {4 b \sqrt {d} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{9 c^{3/2}}-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {d x}}{9 c} \]
[In]
[Out]
Rule 227
Rule 327
Rule 335
Rule 4724
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d x)^{3/2} (a+b \arccos (c x))}{3 d}+\frac {(2 b c) \int \frac {(d x)^{3/2}}{\sqrt {1-c^2 x^2}} \, dx}{3 d} \\ & = -\frac {4 b \sqrt {d x} \sqrt {1-c^2 x^2}}{9 c}+\frac {2 (d x)^{3/2} (a+b \arccos (c x))}{3 d}+\frac {(2 b d) \int \frac {1}{\sqrt {d x} \sqrt {1-c^2 x^2}} \, dx}{9 c} \\ & = -\frac {4 b \sqrt {d x} \sqrt {1-c^2 x^2}}{9 c}+\frac {2 (d x)^{3/2} (a+b \arccos (c x))}{3 d}+\frac {(4 b) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {c^2 x^4}{d^2}}} \, dx,x,\sqrt {d x}\right )}{9 c} \\ & = -\frac {4 b \sqrt {d x} \sqrt {1-c^2 x^2}}{9 c}+\frac {2 (d x)^{3/2} (a+b \arccos (c x))}{3 d}+\frac {4 b \sqrt {d} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{9 c^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.28 \[ \int \sqrt {d x} (a+b \arccos (c x)) \, dx=\frac {2}{9} \sqrt {d x} \left (3 a x-\frac {2 b \sqrt {1-c^2 x^2}}{c}+3 b x \arccos (c x)-\frac {2 i b \sqrt {-\frac {1}{c}} \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {1}{c}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {1-c^2 x^2}}\right ) \]
[In]
[Out]
Time = 1.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.35
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (d x \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d x \right )^{\frac {3}{2}} \arccos \left (c x \right )}{3}+\frac {2 c \left (-\frac {d^{2} \sqrt {d x}\, \sqrt {-c^{2} x^{2}+1}}{3 c^{2}}+\frac {d^{2} \sqrt {-c x +1}\, \sqrt {c x +1}\, \operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{3 c^{2} \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{3 d}\right )}{d}\) | \(119\) |
default | \(\frac {\frac {2 \left (d x \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d x \right )^{\frac {3}{2}} \arccos \left (c x \right )}{3}+\frac {2 c \left (-\frac {d^{2} \sqrt {d x}\, \sqrt {-c^{2} x^{2}+1}}{3 c^{2}}+\frac {d^{2} \sqrt {-c x +1}\, \sqrt {c x +1}\, \operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{3 c^{2} \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{3 d}\right )}{d}\) | \(119\) |
parts | \(\frac {2 a \left (d x \right )^{\frac {3}{2}}}{3 d}+\frac {2 b \left (\frac {\left (d x \right )^{\frac {3}{2}} \arccos \left (c x \right )}{3}+\frac {2 c \left (-\frac {d^{2} \sqrt {d x}\, \sqrt {-c^{2} x^{2}+1}}{3 c^{2}}+\frac {d^{2} \sqrt {-c x +1}\, \sqrt {c x +1}\, \operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{3 c^{2} \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{3 d}\right )}{d}\) | \(121\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.78 \[ \int \sqrt {d x} (a+b \arccos (c x)) \, dx=-\frac {2 \, {\left (2 \, \sqrt {-c^{2} d} b {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right ) - {\left (3 \, b c^{3} x \arccos \left (c x\right ) + 3 \, a c^{3} x - 2 \, \sqrt {-c^{2} x^{2} + 1} b c^{2}\right )} \sqrt {d x}\right )}}{9 \, c^{3}} \]
[In]
[Out]
Time = 3.84 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97 \[ \int \sqrt {d x} (a+b \arccos (c x)) \, dx=a \left (\begin {cases} \frac {2 \left (d x\right )^{\frac {3}{2}}}{3 d} & \text {for}\: d \neq 0 \\0 & \text {otherwise} \end {cases}\right ) + b c \left (\begin {cases} \frac {\sqrt {d} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {c^{2} x^{2} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac {9}{4}\right )} & \text {for}\: d > -\infty \wedge d < \infty \wedge d \neq 0 \\0 & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} \frac {2 \left (d x\right )^{\frac {3}{2}}}{3 d} & \text {for}\: d \neq 0 \\0 & \text {otherwise} \end {cases}\right ) \operatorname {acos}{\left (c x \right )} \]
[In]
[Out]
\[ \int \sqrt {d x} (a+b \arccos (c x)) \, dx=\int { \sqrt {d x} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \]
[In]
[Out]
\[ \int \sqrt {d x} (a+b \arccos (c x)) \, dx=\int { \sqrt {d x} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sqrt {d x} (a+b \arccos (c x)) \, dx=\int \left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\sqrt {d\,x} \,d x \]
[In]
[Out]