Optimal. Leaf size=50 \[ \frac {(a+b) \tan ^5(e+f x)}{5 f}+\frac {(2 a+b) \tan ^3(e+f x)}{3 f}+\frac {a \tan (e+f x)}{f} \]
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Rubi [A] time = 0.04, antiderivative size = 50, 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 = {3191, 373} \[ \frac {(a+b) \tan ^5(e+f x)}{5 f}+\frac {(2 a+b) \tan ^3(e+f x)}{3 f}+\frac {a \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 373
Rule 3191
Rubi steps
\begin {align*} \int \sec ^6(e+f x) \left (a+b \sin ^2(e+f x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \left (1+x^2\right ) \left (a+(a+b) x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (a+(2 a+b) x^2+(a+b) x^4\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a \tan (e+f x)}{f}+\frac {(2 a+b) \tan ^3(e+f x)}{3 f}+\frac {(a+b) \tan ^5(e+f x)}{5 f}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 64, normalized size = 1.28 \[ \frac {\tan (e+f x) \left (3 a \tan ^4(e+f x)+10 a \tan ^2(e+f x)+15 a+3 b \sec ^4(e+f x)-b \sec ^2(e+f x)-2 b\right )}{15 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 59, normalized size = 1.18 \[ \frac {{\left (2 \, {\left (4 \, a - b\right )} \cos \left (f x + e\right )^{4} + {\left (4 \, a - b\right )} \cos \left (f x + e\right )^{2} + 3 \, a + 3 \, b\right )} \sin \left (f x + e\right )}{15 \, f \cos \left (f x + e\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 64, normalized size = 1.28 \[ \frac {3 \, a \tan \left (f x + e\right )^{5} + 3 \, b \tan \left (f x + e\right )^{5} + 10 \, a \tan \left (f x + e\right )^{3} + 5 \, b \tan \left (f x + e\right )^{3} + 15 \, a \tan \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 76, normalized size = 1.52 \[ \frac {-a \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (f x +e \right )\right )}{15}\right ) \tan \left (f x +e \right )+b \left (\frac {\sin ^{3}\left (f x +e \right )}{5 \cos \left (f x +e \right )^{5}}+\frac {2 \left (\sin ^{3}\left (f x +e \right )\right )}{15 \cos \left (f x +e \right )^{3}}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 43, normalized size = 0.86 \[ \frac {3 \, {\left (a + b\right )} \tan \left (f x + e\right )^{5} + 5 \, {\left (2 \, a + b\right )} \tan \left (f x + e\right )^{3} + 15 \, a \tan \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.90, size = 45, normalized size = 0.90 \[ \frac {\left (\frac {a}{5}+\frac {b}{5}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^5+\left (\frac {2\,a}{3}+\frac {b}{3}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^3+a\,\mathrm {tan}\left (e+f\,x\right )}{f} \]
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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