Optimal. Leaf size=73 \[ -\frac {2 \cos (a+b x)}{b c \sqrt {c \sin (a+b x)}}-\frac {2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {c \sin (a+b x)}}{b c^2 \sqrt {\sin (a+b x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2716, 2721,
2719} \begin {gather*} -\frac {2 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {c \sin (a+b x)}}{b c^2 \sqrt {\sin (a+b x)}}-\frac {2 \cos (a+b x)}{b c \sqrt {c \sin (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2716
Rule 2719
Rule 2721
Rubi steps
\begin {align*} \int \frac {1}{(c \sin (a+b x))^{3/2}} \, dx &=-\frac {2 \cos (a+b x)}{b c \sqrt {c \sin (a+b x)}}-\frac {\int \sqrt {c \sin (a+b x)} \, dx}{c^2}\\ &=-\frac {2 \cos (a+b x)}{b c \sqrt {c \sin (a+b x)}}-\frac {\sqrt {c \sin (a+b x)} \int \sqrt {\sin (a+b x)} \, dx}{c^2 \sqrt {\sin (a+b x)}}\\ &=-\frac {2 \cos (a+b x)}{b c \sqrt {c \sin (a+b x)}}-\frac {2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {c \sin (a+b x)}}{b c^2 \sqrt {\sin (a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 54, normalized size = 0.74 \begin {gather*} -\frac {2 \left (\cos (a+b x)-E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sqrt {\sin (a+b x)}\right )}{b c \sqrt {c \sin (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 141, normalized size = 1.93
method | result | size |
default | \(\frac {2 \sqrt {-\sin \left (b x +a \right )+1}\, \sqrt {2 \sin \left (b x +a \right )+2}\, \left (\sqrt {\sin }\left (b x +a \right )\right ) \EllipticE \left (\sqrt {-\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {-\sin \left (b x +a \right )+1}\, \sqrt {2 \sin \left (b x +a \right )+2}\, \left (\sqrt {\sin }\left (b x +a \right )\right ) \EllipticF \left (\sqrt {-\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{2}\left (b x +a \right )\right )}{c \cos \left (b x +a \right ) \sqrt {c \sin \left (b x +a \right )}\, b}\) | \(141\) |
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 = 0.10, size = 108, normalized size = 1.48 \begin {gather*} \frac {-i \, \sqrt {2} \sqrt {-i \, c} \sin \left (b x + a\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + i \, \sqrt {2} \sqrt {i \, c} \sin \left (b x + a\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) - 2 \, \sqrt {c \sin \left (b x + a\right )} \cos \left (b x + a\right )}{b c^{2} \sin \left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c \sin {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \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 = -0.01 \begin {gather*} \int \frac {1}{{\left (c\,\sin \left (a+b\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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