Optimal. Leaf size=43 \[ \frac {2 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {c \sin (a+b x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2642, 2641} \[ \frac {2 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {c \sin (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 2642
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {c \sin (a+b x)}} \, dx &=\frac {\sqrt {\sin (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{\sqrt {c \sin (a+b x)}}\\ &=\frac {2 F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{b \sqrt {c \sin (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.98 \[ -\frac {2 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )}{b \sqrt {c \sin (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c \sin \left (b x + a\right )}}{c \sin \left (b x + a\right )}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {c \sin \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 74, normalized size = 1.72 \[ -\frac {\sqrt {-\sin \left (b x +a \right )+1}\, \sqrt {2 \sin \left (b x +a \right )+2}\, \left (\sqrt {\sin }\left (b x +a \right )\right ) \EllipticF \left (\sqrt {-\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{\cos \left (b x +a \right ) \sqrt {c \sin \left (b x +a \right )}\, b} \]
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 \[ \int \frac {1}{\sqrt {c \sin \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 36, normalized size = 0.84 \[ -\frac {2\,\sqrt {\sin \left (a+b\,x\right )}\,\mathrm {F}\left (\frac {\pi }{4}-\frac {a}{2}-\frac {b\,x}{2}\middle |2\right )}{b\,\sqrt {c\,\sin \left (a+b\,x\right )}} \]
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 \[ \int \frac {1}{\sqrt {c \sin {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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