Optimal. Leaf size=92 \[ \frac {\sqrt {\sin (2 a+2 b x)} F\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}{2 b c^2}+\frac {d}{b c \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}} \]
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Rubi [A] time = 0.14, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2628, 2630, 2573, 2641} \[ \frac {\sqrt {\sin (2 a+2 b x)} F\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}{2 b c^2}+\frac {d}{b c \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2573
Rule 2628
Rule 2630
Rule 2641
Rubi steps
\begin {align*} \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{3/2}} \, dx &=\frac {d}{b c \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}}+\frac {\int \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \, dx}{2 c^2}\\ &=\frac {d}{b c \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}}+\frac {\left (\sqrt {c \cos (a+b x)} \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)}\right ) \int \frac {1}{\sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)}} \, dx}{2 c^2}\\ &=\frac {d}{b c \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}}+\frac {\left (\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx}{2 c^2}\\ &=\frac {d}{b c \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}}+\frac {\sqrt {d \csc (a+b x)} F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}}{2 b c^2}\\ \end {align*}
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Mathematica [C] time = 0.66, size = 84, normalized size = 0.91 \[ \frac {d \sec ^3(a+b x) \left (-\left (-\cot ^2(a+b x)\right )^{3/4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};\csc ^2(a+b x)\right )+\cos (2 (a+b x))+1\right )}{2 b (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \csc \left (b x + a\right )} \sqrt {c \sec \left (b x + a\right )}}{c^{2} \sec \left (b x + a\right )^{2}}, x\right ) \]
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 \[ \int \frac {\sqrt {d \csc \left (b x + a\right )}}{\left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.22, size = 193, normalized size = 2.10 \[ \frac {\left (-\sin \left (b x +a \right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \left (\cos ^{2}\left (b x +a \right )\right )-\cos \left (b x +a \right ) \sqrt {2}\right ) \sqrt {\frac {d}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right ) \sqrt {2}}{2 b \left (-1+\cos \left (b x +a \right )\right ) \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}} \cos \left (b x +a \right )^{2}} \]
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 \[ \int \frac {\sqrt {d \csc \left (b x + a\right )}}{\left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {d \csc {\left (a + b x \right )}}}{\left (c \sec {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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