Optimal. Leaf size=33 \[ \frac {2 c d (c \sec (a+b x))^{3/2}}{3 b (d \csc (a+b x))^{3/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2699}
\begin {gather*} \frac {2 c d (c \sec (a+b x))^{3/2}}{3 b (d \csc (a+b x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2699
Rubi steps
\begin {align*} \int \frac {(c \sec (a+b x))^{5/2}}{\sqrt {d \csc (a+b x)}} \, dx &=\frac {2 c d (c \sec (a+b x))^{3/2}}{3 b (d \csc (a+b x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 33, normalized size = 1.00 \begin {gather*} \frac {2 c d (c \sec (a+b x))^{3/2}}{3 b (d \csc (a+b x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 58.52, size = 42, normalized size = 1.27
method | result | size |
default | \(\frac {2 \cos \left (b x +a \right ) \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {5}{2}} \sin \left (b x +a \right )}{3 b \sqrt {\frac {d}{\sin \left (b x +a \right )}}}\) | \(42\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (27) = 54\).
time = 2.51, size = 58, normalized size = 1.76 \begin {gather*} -\frac {2 \, {\left (c^{2} \cos \left (b x + a\right )^{2} - c^{2}\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}}}{3 \, b d \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 [B]
time = 0.86, size = 66, normalized size = 2.00 \begin {gather*} \frac {c^2\,\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}\,\left (\cos \left (a+b\,x\right )-\cos \left (3\,a+3\,b\,x\right )\right )}{3\,b\,d\,\left (\cos \left (2\,a+2\,b\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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