Optimal. Leaf size=18 \[ \frac {2 d \sqrt {d \tan (a+b x)}}{b} \]
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Rubi [A] time = 0.04, 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 = {2591, 30} \[ \frac {2 d \sqrt {d \tan (a+b x)}}{b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2591
Rubi steps
\begin {align*} \int \csc ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx &=\frac {d \operatorname {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 \[ \frac {2 d \sqrt {d \tan (a+b x)}}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 24, normalized size = 1.33 \[ \frac {2 \, d \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.37, size = 16, normalized size = 0.89 \[ \frac {2 \, \sqrt {d \tan \left (b x + a\right )} d}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.50, size = 58, normalized size = 3.22 \[ \frac {2 \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}} \cos \left (b x +a \right ) \left (-1+\cos \left (b x +a \right )\right )^{2} \left (\cos \left (b x +a \right )+1\right )^{2}}{b \sin \left (b x +a \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 23, normalized size = 1.28 \[ \frac {2 \, \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}}{b \tan \left (b x + a\right )} \]
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 \[ \frac {2\,d\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}}{b} \]
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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