Optimal. Leaf size=31 \[ \frac {\tan ^8(a+b x)}{8 b}+\frac {\tan ^6(a+b x)}{6 b} \]
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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 = {2607, 14} \[ \frac {\tan ^8(a+b x)}{8 b}+\frac {\tan ^6(a+b x)}{6 b} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2607
Rubi steps
\begin {align*} \int \sec ^4(a+b x) \tan ^5(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x^5 \left (1+x^2\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (x^5+x^7\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {\tan ^6(a+b x)}{6 b}+\frac {\tan ^8(a+b x)}{8 b}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 38, normalized size = 1.23 \[ \frac {3 \sec ^8(a+b x)-8 \sec ^6(a+b x)+6 \sec ^4(a+b x)}{24 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 35, normalized size = 1.13 \[ \frac {6 \, \cos \left (b x + a\right )^{4} - 8 \, \cos \left (b x + a\right )^{2} + 3}{24 \, b \cos \left (b x + a\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 93, normalized size = 3.00 \[ -\frac {32 \, {\left (\frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}}\right )}}{3 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 42, normalized size = 1.35 \[ \frac {\frac {\sin ^{6}\left (b x +a \right )}{8 \cos \left (b x +a \right )^{8}}+\frac {\sin ^{6}\left (b x +a \right )}{24 \cos \left (b x +a \right )^{6}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 69, normalized size = 2.23 \[ \frac {6 \, \sin \left (b x + a\right )^{4} - 4 \, \sin \left (b x + a\right )^{2} + 1}{24 \, {\left (\sin \left (b x + a\right )^{8} - 4 \, \sin \left (b x + a\right )^{6} + 6 \, \sin \left (b x + a\right )^{4} - 4 \, \sin \left (b x + a\right )^{2} + 1\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 25, normalized size = 0.81 \[ \frac {{\mathrm {tan}\left (a+b\,x\right )}^6\,\left (3\,{\mathrm {tan}\left (a+b\,x\right )}^2+4\right )}{24\,b} \]
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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