Optimal. Leaf size=31 \[ -\frac {\sec ^5(a+b x)}{5 b}+\frac {\sec ^7(a+b x)}{7 b} \]
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Rubi [A]
time = 0.02, 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 = {2686, 14}
\begin {gather*} \frac {\sec ^7(a+b x)}{7 b}-\frac {\sec ^5(a+b x)}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2686
Rubi steps
\begin {align*} \int \sec ^5(a+b x) \tan ^3(a+b x) \, dx &=\frac {\text {Subst}\left (\int x^4 \left (-1+x^2\right ) \, dx,x,\sec (a+b x)\right )}{b}\\ &=\frac {\text {Subst}\left (\int \left (-x^4+x^6\right ) \, dx,x,\sec (a+b x)\right )}{b}\\ &=-\frac {\sec ^5(a+b x)}{5 b}+\frac {\sec ^7(a+b x)}{7 b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 31, normalized size = 1.00 \begin {gather*} -\frac {\sec ^5(a+b x)}{5 b}+\frac {\sec ^7(a+b x)}{7 b} \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. \(95\) vs.
\(2(27)=54\).
time = 0.09, size = 96, normalized size = 3.10
method | result | size |
risch | \(-\frac {32 \left (7 \,{\mathrm e}^{9 i \left (b x +a \right )}-6 \,{\mathrm e}^{7 i \left (b x +a \right )}+7 \,{\mathrm e}^{5 i \left (b x +a \right )}\right )}{35 b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{7}}\) | \(53\) |
derivativedivides | \(\frac {\frac {\sin ^{4}\left (b x +a \right )}{7 \cos \left (b x +a \right )^{7}}+\frac {3 \left (\sin ^{4}\left (b x +a \right )\right )}{35 \cos \left (b x +a \right )^{5}}+\frac {\sin ^{4}\left (b x +a \right )}{35 \cos \left (b x +a \right )^{3}}-\frac {\sin ^{4}\left (b x +a \right )}{35 \cos \left (b x +a \right )}-\frac {\left (2+\sin ^{2}\left (b x +a \right )\right ) \cos \left (b x +a \right )}{35}}{b}\) | \(96\) |
default | \(\frac {\frac {\sin ^{4}\left (b x +a \right )}{7 \cos \left (b x +a \right )^{7}}+\frac {3 \left (\sin ^{4}\left (b x +a \right )\right )}{35 \cos \left (b x +a \right )^{5}}+\frac {\sin ^{4}\left (b x +a \right )}{35 \cos \left (b x +a \right )^{3}}-\frac {\sin ^{4}\left (b x +a \right )}{35 \cos \left (b x +a \right )}-\frac {\left (2+\sin ^{2}\left (b x +a \right )\right ) \cos \left (b x +a \right )}{35}}{b}\) | \(96\) |
norman | \(\frac {-\frac {4 \left (\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {8 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {4}{35 b}-\frac {4 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 b}-\frac {4 \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {8 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 b}}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{7}}\) | \(103\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 25, normalized size = 0.81 \begin {gather*} -\frac {7 \, \cos \left (b x + a\right )^{2} - 5}{35 \, b \cos \left (b x + a\right )^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 25, normalized size = 0.81 \begin {gather*} -\frac {7 \, \cos \left (b x + a\right )^{2} - 5}{35 \, b \cos \left (b x + a\right )^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.82, size = 25, normalized size = 0.81 \begin {gather*} -\frac {7 \, \cos \left (b x + a\right )^{2} - 5}{35 \, b \cos \left (b x + a\right )^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.63, size = 25, normalized size = 0.81 \begin {gather*} -\frac {7\,{\cos \left (a+b\,x\right )}^2-5}{35\,b\,{\cos \left (a+b\,x\right )}^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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