Optimal. Leaf size=110 \[ -\frac {4 d^3 \csc (a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {4 d^2 \csc (a+b x) F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{3 b}+\frac {2 d \csc ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b} \]
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Rubi [A]
time = 0.11, antiderivative size = 110, 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 = {2673, 2679,
2681, 2653, 2720} \begin {gather*} -\frac {4 d^3 \csc (a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {4 d^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)}}{3 b}+\frac {2 d \csc ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2653
Rule 2673
Rule 2679
Rule 2681
Rule 2720
Rubi steps
\begin {align*} \int \csc ^5(a+b x) (d \tan (a+b x))^{5/2} \, dx &=\frac {2 d \csc ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\left (2 d^2\right ) \int \csc ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx\\ &=-\frac {4 d^3 \csc (a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {2 d \csc ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac {1}{3} \left (4 d^2\right ) \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx\\ &=-\frac {4 d^3 \csc (a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {2 d \csc ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac {\left (4 d^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}{3 \sqrt {\sin (a+b x)}}\\ &=-\frac {4 d^3 \csc (a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {2 d \csc ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac {1}{3} \left (4 d^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\\ &=-\frac {4 d^3 \csc (a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {4 d^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)}}{3 b}+\frac {2 d \csc ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.54, size = 110, normalized size = 1.00 \begin {gather*} -\frac {2 d \csc ^3(a+b x) \left (\cos (2 (a+b x)) \sqrt {\sec ^2(a+b x)}+2 \sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt {\tan (a+b x)}\right )\right |-1\right ) \sin (2 (a+b x)) \sqrt {\tan (a+b x)}\right ) (d \tan (a+b x))^{3/2}}{3 b \sqrt {\sec ^2(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. \(315\) vs.
\(2(121)=242\).
time = 0.36, size = 316, normalized size = 2.87
method | result | size |
default | \(\frac {\left (-1+\cos \left (b x +a \right )\right )^{2} \left (4 \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right ) \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \left (\cos ^{2}\left (b x +a \right )\right )+4 \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right ) \cos \left (b x +a \right )-2 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}+\sqrt {2}\right ) \cos \left (b x +a \right ) \left (\cos \left (b x +a \right )+1\right )^{2} \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {5}{2}} \sqrt {2}}{3 b \sin \left (b x +a \right )^{8}}\) | \(316\) |
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 = 160, normalized size = 1.45 \begin {gather*} -\frac {2 \, {\left (2 \, {\left (d^{2} \cos \left (b x + a\right )^{3} - d^{2} \cos \left (b x + a\right )\right )} \sqrt {i \, d} {\rm ellipticF}\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ), -1\right ) + 2 \, {\left (d^{2} \cos \left (b x + a\right )^{3} - d^{2} \cos \left (b x + a\right )\right )} \sqrt {-i \, d} {\rm ellipticF}\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ), -1\right ) - {\left (2 \, d^{2} \cos \left (b x + a\right )^{2} - d^{2}\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}\right )}}{3 \, {\left (b \cos \left (b x + a\right )^{3} - b \cos \left (b x + a\right )\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 \frac {{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{5/2}}{{\sin \left (a+b\,x\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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