Optimal. Leaf size=20 \[ -\frac {2 d}{5 b (d \tan (a+b x))^{5/2}} \]
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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} \[ -\frac {2 d}{5 b (d \tan (a+b x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2591
Rubi steps
\begin {align*} \int \frac {\csc ^2(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx &=\frac {d \operatorname {Subst}\left (\int \frac {1}{x^{7/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac {2 d}{5 b (d \tan (a+b x))^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 20, normalized size = 1.00 \[ -\frac {2 d}{5 b (d \tan (a+b x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 58, normalized size = 2.90 \[ \frac {2 \, \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )^{3}}{5 \, {\left (b d^{2} \cos \left (b x + a\right )^{2} - b d^{2}\right )} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 26, normalized size = 1.30 \[ -\frac {2}{5 \, \sqrt {d \tan \left (b x + a\right )} b d \tan \left (b x + a\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.50, size = 38, normalized size = 1.90 \[ -\frac {2 \cos \left (b x +a \right )}{5 b \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}} \sin \left (b x +a \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 23, normalized size = 1.15 \[ -\frac {2}{5 \, \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} b \tan \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.45, size = 381, normalized size = 19.05 \[ -\frac {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}\,14{}\mathrm {i}}{5\,b\,d^2\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}-\frac {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}\,8{}\mathrm {i}}{15\,b\,d^2\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^2}-\frac {16\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}}{5\,b\,d^2\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}-\frac {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}\,32{}\mathrm {i}}{15\,b\,d^2\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}^2}+\frac {8\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}}{5\,b\,d^2\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}^3} \]
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 \[ \int \frac {\csc ^{2}{\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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