Optimal. Leaf size=93 \[ \frac {2 c d (c \sec (a+b x))^{3/2}}{3 b \sqrt {d \csc (a+b x)}}+\frac {2 c^2 \sqrt {d \csc (a+b x)} F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}}{3 b} \]
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Rubi [A]
time = 0.10, antiderivative size = 93, 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 = {2706, 2710,
2653, 2720} \begin {gather*} \frac {2 c^2 \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)}}{3 b}+\frac {2 c d (c \sec (a+b x))^{3/2}}{3 b \sqrt {d \csc (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2653
Rule 2706
Rule 2710
Rule 2720
Rubi steps
\begin {align*} \int \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{5/2} \, dx &=\frac {2 c d (c \sec (a+b x))^{3/2}}{3 b \sqrt {d \csc (a+b x)}}+\frac {1}{3} \left (2 c^2\right ) \int \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \, dx\\ &=\frac {2 c d (c \sec (a+b x))^{3/2}}{3 b \sqrt {d \csc (a+b x)}}+\frac {1}{3} \left (2 c^2 \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\\ &=\frac {2 c d (c \sec (a+b x))^{3/2}}{3 b \sqrt {d \csc (a+b x)}}+\frac {1}{3} \left (2 c^2 \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\\ &=\frac {2 c d (c \sec (a+b x))^{3/2}}{3 b \sqrt {d \csc (a+b x)}}+\frac {2 c^2 \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)}}{3 b}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.65, size = 68, normalized size = 0.73 \begin {gather*} -\frac {2 c d \left (-1+\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 ) (c \sec (a+b x))^{3/2}}{3 b \sqrt {d \csc (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 55.88, size = 188, normalized size = 2.02
method | result | size |
default | \(-\frac {\left (2 \sin \left (b x +a \right ) \cos \left (b x +a \right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\, \cos \left (b x +a \right )+\sqrt {2}\right ) \cos \left (b x +a \right ) \sqrt {\frac {d}{\sin \left (b x +a \right )}}\, \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {5}{2}} \sin \left (b x +a \right ) \sqrt {2}}{3 b \left (-1+\cos \left (b x +a \right )\right )}\) | \(188\) |
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 complex when optimal does not.
time = 1.01, size = 117, normalized size = 1.26 \begin {gather*} \frac {-i \, \sqrt {-4 i \, c d} c^{2} \cos \left (b x + a\right ) {\rm ellipticF}\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ), -1\right ) + i \, \sqrt {4 i \, c d} c^{2} \cos \left (b x + a\right ) {\rm ellipticF}\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ), -1\right ) + 2 \, c^{2} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} \sin \left (b x + a\right )}{3 \, b \cos \left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{5/2}\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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