Optimal. Leaf size=112 \[ \frac {2 \csc (a+b x)}{21 b d \sqrt {d \tan (a+b x)}}-\frac {2 \csc ^3(a+b x)}{7 b d \sqrt {d \tan (a+b x)}}-\frac {2 \csc (a+b x) F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{21 b d^2} \]
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Rubi [A]
time = 0.10, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2677, 2679,
2681, 2653, 2720} \begin {gather*} -\frac {2 \sqrt {\sin (2 a+2 b x)} \csc (a+b x) F\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (a+b x)}}{21 b d^2}-\frac {2 \csc ^3(a+b x)}{7 b d \sqrt {d \tan (a+b x)}}+\frac {2 \csc (a+b x)}{21 b d \sqrt {d \tan (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2653
Rule 2677
Rule 2679
Rule 2681
Rule 2720
Rubi steps
\begin {align*} \int \frac {\csc ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx &=-\frac {2 \csc ^3(a+b x)}{7 b d \sqrt {d \tan (a+b x)}}-\frac {\int \csc ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx}{7 d^2}\\ &=\frac {2 \csc (a+b x)}{21 b d \sqrt {d \tan (a+b x)}}-\frac {2 \csc ^3(a+b x)}{7 b d \sqrt {d \tan (a+b x)}}-\frac {2 \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx}{21 d^2}\\ &=\frac {2 \csc (a+b x)}{21 b d \sqrt {d \tan (a+b x)}}-\frac {2 \csc ^3(a+b x)}{7 b d \sqrt {d \tan (a+b x)}}-\frac {\left (2 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}} \, dx}{21 d^2 \sqrt {\sin (a+b x)}}\\ &=\frac {2 \csc (a+b x)}{21 b d \sqrt {d \tan (a+b x)}}-\frac {2 \csc ^3(a+b x)}{7 b d \sqrt {d \tan (a+b x)}}-\frac {\left (2 \csc (a+b x) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx}{21 d^2}\\ &=\frac {2 \csc (a+b x)}{21 b d \sqrt {d \tan (a+b x)}}-\frac {2 \csc ^3(a+b x)}{7 b d \sqrt {d \tan (a+b x)}}-\frac {2 \csc (a+b x) F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{21 b d^2}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 11.82, size = 136, normalized size = 1.21 \begin {gather*} \frac {\csc ^3(a+b x) \left ((1+10 \cos (2 (a+b x))+\cos (4 (a+b x))) \sec ^2(a+b x)^{3/2}-8 \sqrt [4]{-1} \cos (2 (a+b x)) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt {\tan (a+b x)}\right )\right |-1\right ) \tan ^{\frac {7}{2}}(a+b x)\right )}{42 b d \sqrt {\sec ^2(a+b x)} \sqrt {d \tan (a+b x)} \left (-1+\tan ^2(a+b x)\right )} \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. \(556\) vs.
\(2(123)=246\).
time = 0.36, size = 557, normalized size = 4.97
method | result | size |
default | \(-\frac {\left (-1+\cos \left (b x +a \right )\right )^{2} \left (-2 \sqrt {\frac {-1+\cos \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 )}{\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 )}}\, \left (\cos ^{3}\left (b x +a \right )\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 )-2 \sqrt {\frac {-1+\cos \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 )}{\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 )}}\, \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 ) \left (\cos ^{2}\left (b x +a \right )\right )+2 \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 {\frac {-1+\cos \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 )}{\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 )}}\, \sin \left (b x +a \right ) \cos \left (b x +a \right )+2 \sqrt {\frac {-1+\cos \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 )}{\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 )}}\, \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 )+\left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {2}+2 \cos \left (b x +a \right ) \sqrt {2}\right ) \left (\cos \left (b x +a \right )+1\right )^{2} \sqrt {2}}{21 b \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{6} \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}}}\) | \(557\) |
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 = 0.10, size = 161, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left ({\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \sqrt {i \, d} {\rm ellipticF}\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ), -1\right ) + {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \sqrt {-i \, d} {\rm ellipticF}\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ), -1\right ) - {\left (\cos \left (b x + a\right )^{3} + 2 \, \cos \left (b x + a\right )\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}\right )}}{21 \, {\left (b d^{2} \cos \left (b x + a\right )^{4} - 2 \, b d^{2} \cos \left (b x + a\right )^{2} + b d^{2}\right )}} \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 {\csc ^{3}{\left (a + b x \right )}}{\left (d \tan {\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 {1}{{\sin \left (a+b\,x\right )}^3\,{\left (d\,\mathrm {tan}\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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