Optimal. Leaf size=20 \[ -\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \]
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Rubi [A] time = 0.04, 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 = {2591, 30} \[ -\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2591
Rubi steps
\begin {align*} \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx &=\frac {d \operatorname {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 = 0.10, size = 20, normalized size = 1.00 \[ -\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 46, normalized size = 2.30 \[ \frac {2 \, \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )^{2}}{3 \, {\left (b d \cos \left (b x + a\right )^{2} - b d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.79, size = 23, normalized size = 1.15 \[ -\frac {2}{3 \, \sqrt {d \tan \left (b x + a\right )} b \tan \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.55, size = 38, normalized size = 1.90 \[ -\frac {2 \cos \left (b x +a \right )}{3 b \sin \left (b x +a \right ) \sqrt {\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 23, normalized size = 1.15 \[ -\frac {2}{3 \, \sqrt {d \tan \left (b x + a\right )} b \tan \left (b x + a\right )} \]
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 \[ -\frac {2\,\sqrt {\frac {d\,\sin \left (2\,a+2\,b\,x\right )}{\cos \left (2\,a+2\,b\,x\right )+1}}\,\left (\cos \left (2\,a+2\,b\,x\right )+2\,\cos \left (4\,a+4\,b\,x\right )-\cos \left (6\,a+6\,b\,x\right )-2\right )}{3\,b\,d\,\left (15\,\cos \left (2\,a+2\,b\,x\right )-6\,\cos \left (4\,a+4\,b\,x\right )+\cos \left (6\,a+6\,b\,x\right )-10\right )} \]
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 \[ \int \frac {\csc ^{2}{\left (a + b x \right )}}{\sqrt {d \tan {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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