Optimal. Leaf size=48 \[ \frac {a \tan ^3(e+f x)}{3 f}-\frac {a \tan (e+f x)}{f}+a x+\frac {b \tan ^5(e+f x)}{5 f} \]
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Rubi [A] time = 0.06, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4141, 1802, 203} \[ \frac {a \tan ^3(e+f x)}{3 f}-\frac {a \tan (e+f x)}{f}+a x+\frac {b \tan ^5(e+f x)}{5 f} \]
Antiderivative was successfully verified.
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Rule 203
Rule 1802
Rule 4141
Rubi steps
\begin {align*} \int \left (a+b \sec ^2(e+f x)\right ) \tan ^4(e+f x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (a+b \left (1+x^2\right )\right )}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a+a x^2+b x^4+\frac {a}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {a \tan (e+f x)}{f}+\frac {a \tan ^3(e+f x)}{3 f}+\frac {b \tan ^5(e+f x)}{5 f}+\frac {a \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=a x-\frac {a \tan (e+f x)}{f}+\frac {a \tan ^3(e+f x)}{3 f}+\frac {b \tan ^5(e+f x)}{5 f}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 57, normalized size = 1.19 \[ \frac {a \tan ^{-1}(\tan (e+f x))}{f}+\frac {a \tan ^3(e+f x)}{3 f}-\frac {a \tan (e+f x)}{f}+\frac {b \tan ^5(e+f x)}{5 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 72, normalized size = 1.50 \[ \frac {15 \, a f x \cos \left (f x + e\right )^{5} - {\left ({\left (20 \, a - 3 \, b\right )} \cos \left (f x + e\right )^{4} - {\left (5 \, a - 6 \, b\right )} \cos \left (f x + e\right )^{2} - 3 \, b\right )} \sin \left (f x + e\right )}{15 \, f \cos \left (f x + e\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.67, size = 49, normalized size = 1.02 \[ \frac {3 \, b \tan \left (f x + e\right )^{5} + 5 \, a \tan \left (f x + e\right )^{3} + 15 \, {\left (f x + e\right )} a - 15 \, a \tan \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 50, normalized size = 1.04 \[ \frac {a \left (\frac {\left (\tan ^{3}\left (f x +e \right )\right )}{3}-\tan \left (f x +e \right )+f x +e \right )+\frac {b \left (\sin ^{5}\left (f x +e \right )\right )}{5 \cos \left (f x +e \right )^{5}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 45, normalized size = 0.94 \[ \frac {3 \, b \tan \left (f x + e\right )^{5} + 5 \, a \tan \left (f x + e\right )^{3} + 15 \, {\left (f x + e\right )} a - 15 \, a \tan \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.68, size = 40, normalized size = 0.83 \[ \frac {\frac {b\,{\mathrm {tan}\left (e+f\,x\right )}^5}{5}+\frac {a\,{\mathrm {tan}\left (e+f\,x\right )}^3}{3}-a\,\mathrm {tan}\left (e+f\,x\right )+a\,f\,x}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.21, size = 54, normalized size = 1.12 \[ a \left (\begin {cases} x + \frac {\tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {\tan {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \tan ^{4}{\relax (e )} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} x \tan ^{4}{\relax (e )} \sec ^{2}{\relax (e )} & \text {for}\: f = 0 \\\frac {\tan ^{5}{\left (e + f x \right )}}{5 f} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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