Optimal. Leaf size=41 \[ \frac {2 d (d \tan (a+b x))^{3/2}}{3 b}-\frac {2 d^3}{b \sqrt {d \tan (a+b x)}} \]
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Rubi [A] time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2591, 14} \[ \frac {2 d (d \tan (a+b x))^{3/2}}{3 b}-\frac {2 d^3}{b \sqrt {d \tan (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2591
Rubi steps
\begin {align*} \int \csc ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx &=\frac {d \operatorname {Subst}\left (\int \frac {d^2+x^2}{x^{3/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=\frac {d \operatorname {Subst}\left (\int \left (\frac {d^2}{x^{3/2}}+\sqrt {x}\right ) \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac {2 d^3}{b \sqrt {d \tan (a+b x)}}+\frac {2 d (d \tan (a+b x))^{3/2}}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 32, normalized size = 0.78 \[ -\frac {2 d \left (3 \cot ^2(a+b x)-1\right ) (d \tan (a+b x))^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 58, normalized size = 1.41 \[ -\frac {2 \, {\left (4 \, d^{2} \cos \left (b x + a\right )^{2} - d^{2}\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{3 \, b \cos \left (b x + a\right ) \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 42, normalized size = 1.02 \[ \frac {2}{3} \, d^{2} {\left (\frac {\sqrt {d \tan \left (b x + a\right )} \tan \left (b x + a\right )}{b} - \frac {3 \, d}{\sqrt {d \tan \left (b x + a\right )} b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 50, normalized size = 1.22 \[ -\frac {2 \left (4 \left (\cos ^{2}\left (b x +a \right )\right )-1\right ) \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {5}{2}} \cos \left (b x +a \right )}{3 b \sin \left (b x +a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 36, normalized size = 0.88 \[ -\frac {2 \, d^{3} {\left (\frac {3}{\sqrt {d \tan \left (b x + a\right )}} - \frac {\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}}{d^{2}}\right )}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.72, size = 64, normalized size = 1.56 \[ -\frac {4\,d^2\,\left (\sin \left (2\,a+2\,b\,x\right )+\sin \left (4\,a+4\,b\,x\right )\right )\,\sqrt {\frac {d\,\sin \left (2\,a+2\,b\,x\right )}{\cos \left (2\,a+2\,b\,x\right )+1}}}{3\,b\,{\sin \left (2\,a+2\,b\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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