Optimal. Leaf size=80 \[ \frac {b (2 a+3 b) \tan ^5(e+f x)}{5 f}+\frac {(a+b) (a+3 b) \tan ^3(e+f x)}{3 f}+\frac {(a+b)^2 \tan (e+f x)}{f}+\frac {b^2 \tan ^7(e+f x)}{7 f} \]
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Rubi [A] time = 0.08, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {4146, 373} \[ \frac {b (2 a+3 b) \tan ^5(e+f x)}{5 f}+\frac {(a+b) (a+3 b) \tan ^3(e+f x)}{3 f}+\frac {(a+b)^2 \tan (e+f x)}{f}+\frac {b^2 \tan ^7(e+f x)}{7 f} \]
Antiderivative was successfully verified.
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Rule 373
Rule 4146
Rubi steps
\begin {align*} \int \sec ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (1+x^2\right ) \left (a+b+b x^2\right )^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left ((a+b)^2+(a+b) (a+3 b) x^2+b (2 a+3 b) x^4+b^2 x^6\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a+b)^2 \tan (e+f x)}{f}+\frac {(a+b) (a+3 b) \tan ^3(e+f x)}{3 f}+\frac {b (2 a+3 b) \tan ^5(e+f x)}{5 f}+\frac {b^2 \tan ^7(e+f x)}{7 f}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 75, normalized size = 0.94 \[ \frac {35 \left (a^2+4 a b+3 b^2\right ) \tan ^3(e+f x)+21 b (2 a+3 b) \tan ^5(e+f x)+105 (a+b)^2 \tan (e+f x)+15 b^2 \tan ^7(e+f x)}{105 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 94, normalized size = 1.18 \[ \frac {{\left (2 \, {\left (35 \, a^{2} + 56 \, a b + 24 \, b^{2}\right )} \cos \left (f x + e\right )^{6} + {\left (35 \, a^{2} + 56 \, a b + 24 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 6 \, {\left (7 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 15 \, b^{2}\right )} \sin \left (f x + e\right )}{105 \, f \cos \left (f x + e\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 123, normalized size = 1.54 \[ \frac {15 \, b^{2} \tan \left (f x + e\right )^{7} + 42 \, a b \tan \left (f x + e\right )^{5} + 63 \, b^{2} \tan \left (f x + e\right )^{5} + 35 \, a^{2} \tan \left (f x + e\right )^{3} + 140 \, a b \tan \left (f x + e\right )^{3} + 105 \, b^{2} \tan \left (f x + e\right )^{3} + 105 \, a^{2} \tan \left (f x + e\right ) + 210 \, a b \tan \left (f x + e\right ) + 105 \, b^{2} \tan \left (f x + e\right )}{105 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.25, size = 104, normalized size = 1.30 \[ \frac {-a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )-2 a b \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (f x +e \right )\right )}{15}\right ) \tan \left (f x +e \right )-b^{2} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (f x +e \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (f x +e \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (f x +e \right )\right )}{35}\right ) \tan \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 81, normalized size = 1.01 \[ \frac {15 \, b^{2} \tan \left (f x + e\right )^{7} + 21 \, {\left (2 \, a b + 3 \, b^{2}\right )} \tan \left (f x + e\right )^{5} + 35 \, {\left (a^{2} + 4 \, a b + 3 \, b^{2}\right )} \tan \left (f x + e\right )^{3} + 105 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )}{105 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.67, size = 70, normalized size = 0.88 \[ \frac {\mathrm {tan}\left (e+f\,x\right )\,{\left (a+b\right )}^2+{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {a^2}{3}+\frac {4\,a\,b}{3}+b^2\right )+\frac {b^2\,{\mathrm {tan}\left (e+f\,x\right )}^7}{7}+\frac {b\,{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (2\,a+3\,b\right )}{5}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2} \sec ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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