Optimal. Leaf size=31 \[ \frac {\sec ^8(a+b x)}{8 b}-\frac {\sec ^6(a+b x)}{6 b} \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2606, 14} \[ \frac {\sec ^8(a+b x)}{8 b}-\frac {\sec ^6(a+b x)}{6 b} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2606
Rubi steps
\begin {align*} \int \sec ^6(a+b x) \tan ^3(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x^5 \left (-1+x^2\right ) \, dx,x,\sec (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (-x^5+x^7\right ) \, dx,x,\sec (a+b x)\right )}{b}\\ &=-\frac {\sec ^6(a+b x)}{6 b}+\frac {\sec ^8(a+b x)}{8 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 28, normalized size = 0.90 \[ -\frac {4 \sec ^6(a+b x)-3 \sec ^8(a+b x)}{24 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 25, normalized size = 0.81 \[ -\frac {4 \, \cos \left (b x + a\right )^{2} - 3}{24 \, b \cos \left (b x + a\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 25, normalized size = 0.81 \[ -\frac {4 \, \cos \left (b x + a\right )^{2} - 3}{24 \, b \cos \left (b x + a\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 60, normalized size = 1.94 \[ \frac {\frac {\sin ^{4}\left (b x +a \right )}{8 \cos \left (b x +a \right )^{8}}+\frac {\sin ^{4}\left (b x +a \right )}{12 \cos \left (b x +a \right )^{6}}+\frac {\sin ^{4}\left (b x +a \right )}{24 \cos \left (b x +a \right )^{4}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 59, normalized size = 1.90 \[ \frac {4 \, \sin \left (b x + a\right )^{2} - 1}{24 \, {\left (\sin \left (b x + a\right )^{8} - 4 \, \sin \left (b x + a\right )^{6} + 6 \, \sin \left (b x + a\right )^{4} - 4 \, \sin \left (b x + a\right )^{2} + 1\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 35, normalized size = 1.13 \[ \frac {{\mathrm {tan}\left (a+b\,x\right )}^4\,\left (3\,{\mathrm {tan}\left (a+b\,x\right )}^4+8\,{\mathrm {tan}\left (a+b\,x\right )}^2+6\right )}{24\,b} \]
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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