Optimal. Leaf size=43 \[ \frac {2 d (d \sec (a+b x))^{7/2}}{7 b}-\frac {2 d^3 (d \sec (a+b x))^{3/2}}{3 b} \]
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Rubi [A] time = 0.05, antiderivative size = 43, 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 = {2622, 14} \[ \frac {2 d (d \sec (a+b x))^{7/2}}{7 b}-\frac {2 d^3 (d \sec (a+b x))^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2622
Rubi steps
\begin {align*} \int (d \sec (a+b x))^{9/2} \sin ^3(a+b x) \, dx &=\frac {d^3 \operatorname {Subst}\left (\int \sqrt {x} \left (-1+\frac {x^2}{d^2}\right ) \, dx,x,d \sec (a+b x)\right )}{b}\\ &=\frac {d^3 \operatorname {Subst}\left (\int \left (-\sqrt {x}+\frac {x^{5/2}}{d^2}\right ) \, dx,x,d \sec (a+b x)\right )}{b}\\ &=-\frac {2 d^3 (d \sec (a+b x))^{3/2}}{3 b}+\frac {2 d (d \sec (a+b x))^{7/2}}{7 b}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 42, normalized size = 0.98 \[ -\frac {d^4 (7 \cos (2 (a+b x))+1) \sec ^3(a+b x) \sqrt {d \sec (a+b x)}}{21 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 44, normalized size = 1.02 \[ -\frac {2 \, {\left (7 \, d^{4} \cos \left (b x + a\right )^{2} - 3 \, d^{4}\right )} \sqrt {\frac {d}{\cos \left (b x + a\right )}}}{21 \, b \cos \left (b x + a\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.83, size = 257, normalized size = 5.98 \[ \frac {16 \, {\left (21 \, {\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}\right )}^{5} \sqrt {-d} d + 7 \, {\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}\right )}^{4} d^{2} - 28 \, {\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}\right )}^{3} \sqrt {-d} d^{2} + 7 \, {\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}\right )} \sqrt {-d} d^{3} + d^{4}\right )} d^{4} \mathrm {sgn}\left (\cos \left (b x + a\right )\right )}{21 \, {\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d} - \sqrt {-d}\right )}^{7} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.68, size = 36, normalized size = 0.84 \[ -\frac {2 \left (7 \left (\cos ^{2}\left (b x +a \right )\right )-3\right ) \cos \left (b x +a \right ) \left (\frac {d}{\cos \left (b x +a \right )}\right )^{\frac {9}{2}}}{21 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 38, normalized size = 0.88 \[ -\frac {2 \, {\left (7 \, d^{2} \left (\frac {d}{\cos \left (b x + a\right )}\right )^{\frac {3}{2}} - 3 \, \left (\frac {d}{\cos \left (b x + a\right )}\right )^{\frac {7}{2}}\right )} d}{21 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.34, size = 95, normalized size = 2.21 \[ -\frac {4\,d^4\,{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\sqrt {\frac {d}{\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}}{2}}}\,\left (2\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+7\,{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+7\right )}{21\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )}^3} \]
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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