Optimal. Leaf size=55 \[ -\frac {2 (a+b \text {ArcCos}(c x))}{d \sqrt {d x}}-\frac {4 b \sqrt {c} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{d^{3/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4724, 335, 227}
\begin {gather*} -\frac {2 (a+b \text {ArcCos}(c x))}{d \sqrt {d x}}-\frac {4 b \sqrt {c} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{d^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 335
Rule 4724
Rubi steps
\begin {align*} \int \frac {a+b \cos ^{-1}(c x)}{(d x)^{3/2}} \, dx &=-\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{d \sqrt {d x}}-\frac {(2 b c) \int \frac {1}{\sqrt {d x} \sqrt {1-c^2 x^2}} \, dx}{d}\\ &=-\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{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 \left (a+b \cos ^{-1}(c x)\right )}{d \sqrt {d x}}-\frac {4 b \sqrt {c} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{d^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.10, size = 93, normalized size = 1.69 \begin {gather*} \frac {2 x \left (-a-b \text {ArcCos}(c x)+\frac {2 i b \sqrt {-\frac {1}{c}} c^2 \sqrt {1-\frac {1}{c^2 x^2}} x^{3/2} F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {1}{c}}}{\sqrt {x}}\right )\right |-1\right )}{\sqrt {1-c^2 x^2}}\right )}{(d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 85, normalized size = 1.55
method | result | size |
derivativedivides | \(\frac {-\frac {2 a}{\sqrt {d x}}+2 b \left (-\frac {\arccos \left (c x \right )}{\sqrt {d x}}-\frac {2 c \sqrt {-c x +1}\, \sqrt {c x +1}\, \EllipticF \left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{d \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\) | \(85\) |
default | \(\frac {-\frac {2 a}{\sqrt {d x}}+2 b \left (-\frac {\arccos \left (c x \right )}{\sqrt {d x}}-\frac {2 c \sqrt {-c x +1}\, \sqrt {c x +1}\, \EllipticF \left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{d \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.55, size = 50, normalized size = 0.91 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{{\left (d\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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