Optimal. Leaf size=18 \[ \frac {2 d \sqrt {d \tan (a+b x)}}{b} \]
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Rubi [A]
time = 0.03, antiderivative size = 18, 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 \sqrt {d \tan (a+b x)}}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2671
Rubi steps
\begin {align*} \int \csc ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx &=\frac {d \text {Subst}\left (\int \frac {1}{\sqrt {x}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=\frac {2 d \sqrt {d \tan (a+b x)}}{b}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 d \sqrt {d \tan (a+b x)}}{b} \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. \(57\) vs.
\(2(16)=32\).
time = 0.33, size = 58, normalized size = 3.22
method | result | size |
default | \(\frac {2 \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}} \cos \left (b x +a \right ) \left (-1+\cos \left (b x +a \right )\right )^{2} \left (\cos \left (b x +a \right )+1\right )^{2}}{b \sin \left (b x +a \right )^{5}}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 23, normalized size = 1.28 \begin {gather*} \frac {2 \, \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}}{b \tan \left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 24, normalized size = 1.33 \begin {gather*} \frac {2 \, d \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{b} \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 [A]
time = 0.66, size = 16, normalized size = 0.89 \begin {gather*} \frac {2 \, \sqrt {d \tan \left (b x + a\right )} d}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.77, size = 43, normalized size = 2.39 \begin {gather*} \frac {2\,d\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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