Optimal. Leaf size=20 \[ -\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2671, 30}
\begin {gather*} -\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2671
Rubi steps
\begin {align*} \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx &=\frac {d \text {Subst}\left (\int \frac {1}{x^{5/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac {2 d}{3 b (d \tan (a+b x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 20, normalized size = 1.00 \begin {gather*} -\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \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. \(37\) vs.
\(2(16)=32\).
time = 0.53, size = 38, normalized size = 1.90
method | result | size |
default | \(-\frac {2 \cos \left (b x +a \right )}{3 b \sin \left (b x +a \right ) \sqrt {\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 23, normalized size = 1.15 \begin {gather*} -\frac {2}{3 \, \sqrt {d \tan \left (b x + a\right )} b \tan \left (b x + a\right )} \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. 46 vs.
\(2 (16) = 32\).
time = 0.42, size = 46, normalized size = 2.30 \begin {gather*} \frac {2 \, \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )^{2}}{3 \, {\left (b d \cos \left (b x + a\right )^{2} - b d\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 ^{2}{\left (a + b x \right )}}{\sqrt {d \tan {\left (a + b x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.71, size = 23, normalized size = 1.15 \begin {gather*} -\frac {2}{3 \, \sqrt {d \tan \left (b x + a\right )} b \tan \left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.33, size = 102, normalized size = 5.10 \begin {gather*} -\frac {2\,\sqrt {\frac {d\,\sin \left (2\,a+2\,b\,x\right )}{\cos \left (2\,a+2\,b\,x\right )+1}}\,\left (\cos \left (2\,a+2\,b\,x\right )+2\,\cos \left (4\,a+4\,b\,x\right )-\cos \left (6\,a+6\,b\,x\right )-2\right )}{3\,b\,d\,\left (15\,\cos \left (2\,a+2\,b\,x\right )-6\,\cos \left (4\,a+4\,b\,x\right )+\cos \left (6\,a+6\,b\,x\right )-10\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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