Integrand size = 16, antiderivative size = 55 \[ \int \frac {a+b \arccos (c x)}{(d x)^{3/2}} \, dx=-\frac {2 (a+b \arccos (c x))}{d \sqrt {d x}}-\frac {4 b \sqrt {c} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{d^{3/2}} \]
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Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4724, 335, 227} \[ \int \frac {a+b \arccos (c x)}{(d x)^{3/2}} \, dx=-\frac {2 (a+b \arccos (c x))}{d \sqrt {d x}}-\frac {4 b \sqrt {c} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{d^{3/2}} \]
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Rule 227
Rule 335
Rule 4724
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (a+b \arccos (c x))}{d \sqrt {d x}}-\frac {(2 b c) \int \frac {1}{\sqrt {d x} \sqrt {1-c^2 x^2}} \, dx}{d} \\ & = -\frac {2 (a+b \arccos (c x))}{d \sqrt {d x}}-\frac {(4 b c) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {c^2 x^4}{d^2}}} \, dx,x,\sqrt {d x}\right )}{d^2} \\ & = -\frac {2 (a+b \arccos (c x))}{d \sqrt {d x}}-\frac {4 b \sqrt {c} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{d^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.69 \[ \int \frac {a+b \arccos (c x)}{(d x)^{3/2}} \, dx=\frac {2 x \left (-a-b \arccos (c x)+\frac {2 i b \sqrt {-\frac {1}{c}} c^2 \sqrt {1-\frac {1}{c^2 x^2}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {1}{c}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {1-c^2 x^2}}\right )}{(d x)^{3/2}} \]
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Time = 1.11 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.55
method | result | size |
derivativedivides | \(\frac {-\frac {2 a}{\sqrt {d x}}+2 b \left (-\frac {\arccos \left (c x \right )}{\sqrt {d x}}-\frac {2 c \sqrt {-c x +1}\, \sqrt {c x +1}\, \operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{d \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\) | \(85\) |
default | \(\frac {-\frac {2 a}{\sqrt {d x}}+2 b \left (-\frac {\arccos \left (c x \right )}{\sqrt {d x}}-\frac {2 c \sqrt {-c x +1}\, \sqrt {c x +1}\, \operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{d \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\) | \(85\) |
parts | \(-\frac {2 a}{\sqrt {d x}\, d}+\frac {2 b \left (-\frac {\arccos \left (c x \right )}{\sqrt {d x}}-\frac {2 c \sqrt {-c x +1}\, \sqrt {c x +1}\, \operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{d \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\) | \(87\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91 \[ \int \frac {a+b \arccos (c x)}{(d x)^{3/2}} \, dx=\frac {2 \, {\left (2 \, \sqrt {-c^{2} d} b x {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right ) - {\left (b c \arccos \left (c x\right ) + a c\right )} \sqrt {d x}\right )}}{c d^{2} x} \]
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Exception generated. \[ \int \frac {a+b \arccos (c x)}{(d x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {a+b \arccos (c x)}{(d x)^{3/2}} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {a+b \arccos (c x)}{(d x)^{3/2}} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arccos (c x)}{(d x)^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{{\left (d\,x\right )}^{3/2}} \,d x \]
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