Optimal. Leaf size=89 \[ -\frac {2 c d \sqrt {d \csc (a+b x)}}{b (c \sec (a+b x))^{3/2}}-\frac {2 d^2 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{b \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}} \]
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Rubi [A]
time = 0.10, antiderivative size = 89, 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 = {2705, 2710,
2652, 2719} \begin {gather*} -\frac {2 d^2 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b \sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}-\frac {2 c d \sqrt {d \csc (a+b x)}}{b (c \sec (a+b x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2652
Rule 2705
Rule 2710
Rule 2719
Rubi steps
\begin {align*} \int \frac {(d \csc (a+b x))^{3/2}}{\sqrt {c \sec (a+b x)}} \, dx &=-\frac {2 c d \sqrt {d \csc (a+b x)}}{b (c \sec (a+b x))^{3/2}}-\left (2 d^2\right ) \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx\\ &=-\frac {2 c d \sqrt {d \csc (a+b x)}}{b (c \sec (a+b x))^{3/2}}-\frac {\left (2 d^2\right ) \int \sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)} \, dx}{\sqrt {c \cos (a+b x)} \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)}}\\ &=-\frac {2 c d \sqrt {d \csc (a+b x)}}{b (c \sec (a+b x))^{3/2}}-\frac {\left (2 d^2\right ) \int \sqrt {\sin (2 a+2 b x)} \, dx}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}}\\ &=-\frac {2 c d \sqrt {d \csc (a+b x)}}{b (c \sec (a+b x))^{3/2}}-\frac {2 d^2 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{b \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.50, size = 80, normalized size = 0.90 \begin {gather*} -\frac {2 d^2 \left (\cot ^2(a+b x)+\sqrt [4]{-\cot ^2(a+b x)} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {1}{2};\csc ^2(a+b x)\right )\right ) \tan (a+b x)}{b \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(501\) vs.
\(2(104)=208\).
time = 34.23, size = 502, normalized size = 5.64
method | result | size |
default | \(\frac {\left (2 \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 )}}\, \EllipticE \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 ) \cos \left (b x +a \right )-\cos \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 )+2 \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 )}}\, \EllipticE \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 )-\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 )-\sqrt {2}\, \cos \left (b x +a \right )\right ) \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {3}{2}} \sin \left (b x +a \right ) \sqrt {2}}{b \sqrt {\frac {c}{\cos \left (b x +a \right )}}\, \cos \left (b x +a \right )}\) | \(502\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {\left (d \csc {\left (a + b x \right )}\right )^{\frac {3}{2}}}{\sqrt {c \sec {\left (a + b x \right )}}}\, 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 {{\left (\frac {d}{\sin \left (a+b\,x\right )}\right )}^{3/2}}{\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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