Optimal. Leaf size=46 \[ \frac {\sec ^{12}(a+b x)}{12 b}-\frac {\sec ^{10}(a+b x)}{5 b}+\frac {\sec ^8(a+b x)}{8 b} \]
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Rubi [A] time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2606, 266, 43} \[ \frac {\sec ^{12}(a+b x)}{12 b}-\frac {\sec ^{10}(a+b x)}{5 b}+\frac {\sec ^8(a+b x)}{8 b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2606
Rubi steps
\begin {align*} \int \sec ^8(a+b x) \tan ^5(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x^7 \left (-1+x^2\right )^2 \, dx,x,\sec (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int (-1+x)^2 x^3 \, dx,x,\sec ^2(a+b x)\right )}{2 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (x^3-2 x^4+x^5\right ) \, dx,x,\sec ^2(a+b x)\right )}{2 b}\\ &=\frac {\sec ^8(a+b x)}{8 b}-\frac {\sec ^{10}(a+b x)}{5 b}+\frac {\sec ^{12}(a+b x)}{12 b}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 38, normalized size = 0.83 \[ \frac {10 \sec ^{12}(a+b x)-24 \sec ^{10}(a+b x)+15 \sec ^8(a+b x)}{120 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 35, normalized size = 0.76 \[ \frac {15 \, \cos \left (b x + a\right )^{4} - 24 \, \cos \left (b x + a\right )^{2} + 10}{120 \, b \cos \left (b x + a\right )^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 183, normalized size = 3.98 \[ -\frac {32 \, {\left (\frac {5 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - \frac {15 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac {39 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} - \frac {42 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} + \frac {39 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} - \frac {15 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{8}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{8}} + \frac {5 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{9}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{9}}\right )}}{15 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 78, normalized size = 1.70 \[ \frac {\frac {\sin ^{6}\left (b x +a \right )}{12 \cos \left (b x +a \right )^{12}}+\frac {\sin ^{6}\left (b x +a \right )}{20 \cos \left (b x +a \right )^{10}}+\frac {\sin ^{6}\left (b x +a \right )}{40 \cos \left (b x +a \right )^{8}}+\frac {\sin ^{6}\left (b x +a \right )}{120 \cos \left (b x +a \right )^{6}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 89, normalized size = 1.93 \[ \frac {15 \, \sin \left (b x + a\right )^{4} - 6 \, \sin \left (b x + a\right )^{2} + 1}{120 \, {\left (\sin \left (b x + a\right )^{12} - 6 \, \sin \left (b x + a\right )^{10} + 15 \, \sin \left (b x + a\right )^{8} - 20 \, \sin \left (b x + a\right )^{6} + 15 \, \sin \left (b x + a\right )^{4} - 6 \, \sin \left (b x + a\right )^{2} + 1\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 45, normalized size = 0.98 \[ \frac {\frac {{\mathrm {tan}\left (a+b\,x\right )}^{12}}{12}+\frac {3\,{\mathrm {tan}\left (a+b\,x\right )}^{10}}{10}+\frac {3\,{\mathrm {tan}\left (a+b\,x\right )}^8}{8}+\frac {{\mathrm {tan}\left (a+b\,x\right )}^6}{6}}{b} \]
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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