Optimal. Leaf size=92 \[ \frac {d}{b c \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}}+\frac {\sqrt {d \csc (a+b x)} F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}}{2 b c^2} \]
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Rubi [A]
time = 0.10, 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 = {2708, 2710,
2653, 2720} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2653
Rule 2708
Rule 2710
Rule 2720
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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.70, size = 84, normalized size = 0.91 \begin {gather*} \frac {d \left (1+\cos (2 (a+b x))-\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 )\right ) \sec ^3(a+b x)}{2 b \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 35.59, size = 195, normalized size = 2.12
method | result | size |
default | \(\frac {\left (-\sin \left (b x +a \right ) \sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+\left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}-\sqrt {2}\, \cos \left (b x +a \right )\right ) \sqrt {\frac {d}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right ) \sqrt {2}}{2 b \left (-1+\cos \left (b x +a \right )\right ) \cos \left (b x +a \right )^{2} \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}}}\) | \(195\) |
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d \csc {\left (a + b x \right )}}}{\left (c \sec {\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 {\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}}{{\left (\frac {c}{\cos \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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