Optimal. Leaf size=45 \[ \frac {2}{9 b d (d \cos (a+b x))^{9/2}}-\frac {2}{5 b d^3 (d \cos (a+b x))^{5/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2645, 14}
\begin {gather*} \frac {2}{9 b d (d \cos (a+b x))^{9/2}}-\frac {2}{5 b d^3 (d \cos (a+b x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2645
Rubi steps
\begin {align*} \int \frac {\sin ^3(a+b x)}{(d \cos (a+b x))^{11/2}} \, dx &=-\frac {\text {Subst}\left (\int \frac {1-\frac {x^2}{d^2}}{x^{11/2}} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac {\text {Subst}\left (\int \left (\frac {1}{x^{11/2}}-\frac {1}{d^2 x^{7/2}}\right ) \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=\frac {2}{9 b d (d \cos (a+b x))^{9/2}}-\frac {2}{5 b d^3 (d \cos (a+b x))^{5/2}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(94\) vs. \(2(45)=90\).
time = 0.37, size = 94, normalized size = 2.09 \begin {gather*} \frac {2 \left (4 \sqrt [4]{\cos ^2(a+b x)}+\left (9-8 \sqrt [4]{\cos ^2(a+b x)}\right ) \csc ^2(a+b x)+4 \left (-1+\sqrt [4]{\cos ^2(a+b x)}\right ) \csc ^4(a+b x)\right ) \tan ^4(a+b x)}{45 b d^5 \sqrt {d \cos (a+b x)}} \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. \(123\) vs.
\(2(37)=74\).
time = 0.46, size = 124, normalized size = 2.76
method | result | size |
default | \(\frac {8 \sqrt {-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +d}\, \left (9 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-9 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1\right )}{45 d^{6} \left (32 \left (\sin ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-80 \left (\sin ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+80 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-40 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+10 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) b}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 37, normalized size = 0.82 \begin {gather*} -\frac {2 \, {\left (9 \, d^{2} \cos \left (b x + a\right )^{2} - 5 \, d^{2}\right )}}{45 \, \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} b d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 38, normalized size = 0.84 \begin {gather*} -\frac {2 \, \sqrt {d \cos \left (b x + a\right )} {\left (9 \, \cos \left (b x + a\right )^{2} - 5\right )}}{45 \, b d^{6} \cos \left (b x + a\right )^{5}} \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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.33, size = 279, normalized size = 6.20 \begin {gather*} \frac {16\,{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\sqrt {d\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}}{2}\right )}}{5\,b\,d^6\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^2}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\sqrt {d\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}}{2}\right )}\,464{}\mathrm {i}}{45\,b\,d^6\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^3}-\frac {128\,{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\sqrt {d\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}}{2}\right )}}{9\,b\,d^6\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^4}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\sqrt {d\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}}{2}\right )}\,64{}\mathrm {i}}{9\,b\,d^6\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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