Optimal. Leaf size=77 \[ -\frac {2 \cos (a+b x)}{3 b c \sqrt {c \csc (a+b x)}}+\frac {2 \sqrt {c \csc (a+b x)} F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{3 b c^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 77, 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 = {3854, 3856,
2720} \begin {gather*} \frac {2 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {c \csc (a+b x)}}{3 b c^2}-\frac {2 \cos (a+b x)}{3 b c \sqrt {c \csc (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3854
Rule 3856
Rubi steps
\begin {align*} \int \frac {1}{(c \csc (a+b x))^{3/2}} \, dx &=-\frac {2 \cos (a+b x)}{3 b c \sqrt {c \csc (a+b x)}}+\frac {\int \sqrt {c \csc (a+b x)} \, dx}{3 c^2}\\ &=-\frac {2 \cos (a+b x)}{3 b c \sqrt {c \csc (a+b x)}}+\frac {\left (\sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{3 c^2}\\ &=-\frac {2 \cos (a+b x)}{3 b c \sqrt {c \csc (a+b x)}}+\frac {2 \sqrt {c \csc (a+b x)} F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{3 b c^2}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 63, normalized size = 0.82 \begin {gather*} -\frac {\csc ^2(a+b x) \left (2 F\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sqrt {\sin (a+b x)}+\sin (2 (a+b x))\right )}{3 b (c \csc (a+b x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 1.13, size = 189, normalized size = 2.45
method | result | size |
default | \(-\frac {\left (i \sqrt {-\frac {i \cos \left (x b +a \right )-i-\sin \left (x b +a \right )}{\sin \left (x b +a \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (x b +a \right )-i+\sin \left (x b +a \right )}{\sin \left (x b +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sin \left (x b +a \right ) \sqrt {-\frac {i \left (-1+\cos \left (x b +a \right )\right )}{\sin \left (x b +a \right )}}\, \sqrt {\frac {i \cos \left (x b +a \right )-i+\sin \left (x b +a \right )}{\sin \left (x b +a \right )}}+\left (\cos ^{2}\left (x b +a \right )\right ) \sqrt {2}-\cos \left (x b +a \right ) \sqrt {2}\right ) \sqrt {2}}{3 b \left (-1+\cos \left (x b +a \right )\right ) \left (\frac {c}{\sin \left (x b +a \right )}\right )^{\frac {3}{2}} \sin \left (x b +a \right )}\) | \(189\) |
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 = 0.80, size = 85, normalized size = 1.10 \begin {gather*} -\frac {2 \, \sqrt {\frac {c}{\sin \left (b x + a\right )}} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + i \, \sqrt {2 i \, c} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - i \, \sqrt {-2 i \, c} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{3 \, b c^{2}} \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 \csc {\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 (\frac {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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