Optimal. Leaf size=69 \[ -\frac {8 c d^3 (d \csc (a+b x))^{3/2}}{21 b (c \sec (a+b x))^{3/2}}-\frac {2 c d (d \csc (a+b x))^{7/2}}{7 b (c \sec (a+b x))^{3/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2705, 2699}
\begin {gather*} -\frac {8 c d^3 (d \csc (a+b x))^{3/2}}{21 b (c \sec (a+b x))^{3/2}}-\frac {2 c d (d \csc (a+b x))^{7/2}}{7 b (c \sec (a+b x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2699
Rule 2705
Rubi steps
\begin {align*} \int \frac {(d \csc (a+b x))^{9/2}}{\sqrt {c \sec (a+b x)}} \, dx &=-\frac {2 c d (d \csc (a+b x))^{7/2}}{7 b (c \sec (a+b x))^{3/2}}+\frac {1}{7} \left (4 d^2\right ) \int \frac {(d \csc (a+b x))^{5/2}}{\sqrt {c \sec (a+b x)}} \, dx\\ &=-\frac {8 c d^3 (d \csc (a+b x))^{3/2}}{21 b (c \sec (a+b x))^{3/2}}-\frac {2 c d (d \csc (a+b x))^{7/2}}{7 b (c \sec (a+b x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 45, normalized size = 0.65 \begin {gather*} \frac {2 c d (-5+2 \cos (2 (a+b x))) (d \csc (a+b x))^{7/2}}{21 b (c \sec (a+b x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 36.54, size = 54, normalized size = 0.78
method | result | size |
default | \(\frac {2 \left (4 \left (\cos ^{2}\left (b x +a \right )\right )-7\right ) \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {9}{2}} \cos \left (b x +a \right ) \sin \left (b x +a \right )}{21 b \sqrt {\frac {c}{\cos \left (b x +a \right )}}}\) | \(54\) |
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 [A]
time = 3.30, size = 79, normalized size = 1.14 \begin {gather*} -\frac {2 \, {\left (4 \, d^{4} \cos \left (b x + a\right )^{4} - 7 \, d^{4} \cos \left (b x + a\right )^{2}\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}}}{21 \, {\left (b c \cos \left (b x + a\right )^{2} - b c\right )} \sin \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 [B]
time = 1.83, size = 99, normalized size = 1.43 \begin {gather*} \frac {8\,d^4\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}\,\left (11\,\sin \left (2\,a+2\,b\,x\right )-7\,\sin \left (4\,a+4\,b\,x\right )+\sin \left (6\,a+6\,b\,x\right )\right )}{21\,b\,\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\left (15\,\cos \left (2\,a+2\,b\,x\right )-6\,\cos \left (4\,a+4\,b\,x\right )+\cos \left (6\,a+6\,b\,x\right )-10\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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