Optimal. Leaf size=77 \[ \frac {a^2 \tan ^3(e+f x)}{3 f}-\frac {a^2 \tan (e+f x)}{f}+a^2 x+\frac {b (2 a+b) \tan ^5(e+f x)}{5 f}+\frac {b^2 \tan ^7(e+f x)}{7 f} \]
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Rubi [A] time = 0.10, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4141, 1802, 203} \[ \frac {a^2 \tan ^3(e+f x)}{3 f}-\frac {a^2 \tan (e+f x)}{f}+a^2 x+\frac {b (2 a+b) \tan ^5(e+f x)}{5 f}+\frac {b^2 \tan ^7(e+f x)}{7 f} \]
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 )^2 \tan ^4(e+f x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (a+b \left (1+x^2\right )\right )^2}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a^2+a^2 x^2+b (2 a+b) x^4+b^2 x^6+\frac {a^2}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {a^2 \tan (e+f x)}{f}+\frac {a^2 \tan ^3(e+f x)}{3 f}+\frac {b (2 a+b) \tan ^5(e+f x)}{5 f}+\frac {b^2 \tan ^7(e+f x)}{7 f}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=a^2 x-\frac {a^2 \tan (e+f x)}{f}+\frac {a^2 \tan ^3(e+f x)}{3 f}+\frac {b (2 a+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 [B] time = 1.10, size = 395, normalized size = 5.13 \[ \frac {\sec (e) \sec ^7(e+f x) \left (4480 a^2 \sin (2 e+f x)-3780 a^2 \sin (2 e+3 f x)+2100 a^2 \sin (4 e+3 f x)-1540 a^2 \sin (4 e+5 f x)+420 a^2 \sin (6 e+5 f x)-280 a^2 \sin (6 e+7 f x)+3675 a^2 f x \cos (2 e+f x)+2205 a^2 f x \cos (2 e+3 f x)+2205 a^2 f x \cos (4 e+3 f x)+735 a^2 f x \cos (4 e+5 f x)+735 a^2 f x \cos (6 e+5 f x)+105 a^2 f x \cos (6 e+7 f x)+105 a^2 f x \cos (8 e+7 f x)-5320 a^2 \sin (f x)+3675 a^2 f x \cos (f x)-1260 a b \sin (2 e+f x)+924 a b \sin (2 e+3 f x)-840 a b \sin (4 e+3 f x)+168 a b \sin (4 e+5 f x)-420 a b \sin (6 e+5 f x)+84 a b \sin (6 e+7 f x)+1680 a b \sin (f x)+420 b^2 \sin (2 e+f x)-168 b^2 \sin (2 e+3 f x)-420 b^2 \sin (4 e+3 f x)+84 b^2 \sin (4 e+5 f x)+12 b^2 \sin (6 e+7 f x)+840 b^2 \sin (f x)\right )}{13440 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 113, normalized size = 1.47 \[ \frac {105 \, a^{2} f x \cos \left (f x + e\right )^{7} - {\left (2 \, {\left (70 \, a^{2} - 21 \, a b - 3 \, b^{2}\right )} \cos \left (f x + e\right )^{6} - {\left (35 \, a^{2} - 84 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 6 \, {\left (7 \, a b - 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 15 \, b^{2}\right )} \sin \left (f x + e\right )}{105 \, f \cos \left (f x + e\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 5.15, size = 84, normalized size = 1.09 \[ \frac {15 \, b^{2} \tan \left (f x + e\right )^{7} + 42 \, a b \tan \left (f x + e\right )^{5} + 21 \, b^{2} \tan \left (f x + e\right )^{5} + 35 \, a^{2} \tan \left (f x + e\right )^{3} + 105 \, {\left (f x + e\right )} a^{2} - 105 \, a^{2} \tan \left (f x + e\right )}{105 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 94, normalized size = 1.22 \[ \frac {a^{2} \left (\frac {\left (\tan ^{3}\left (f x +e \right )\right )}{3}-\tan \left (f x +e \right )+f x +e \right )+\frac {2 a b \left (\sin ^{5}\left (f x +e \right )\right )}{5 \cos \left (f x +e \right )^{5}}+b^{2} \left (\frac {\sin ^{5}\left (f x +e \right )}{7 \cos \left (f x +e \right )^{7}}+\frac {2 \left (\sin ^{5}\left (f x +e \right )\right )}{35 \cos \left (f x +e \right )^{5}}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 71, normalized size = 0.92 \[ \frac {15 \, b^{2} \tan \left (f x + e\right )^{7} + 21 \, {\left (2 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{5} + 35 \, a^{2} \tan \left (f x + e\right )^{3} + 105 \, {\left (f x + e\right )} a^{2} - 105 \, a^{2} \tan \left (f x + e\right )}{105 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.58, size = 97, normalized size = 1.26 \[ \frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {{\left (a+b\right )}^2}{3}+\frac {b^2}{3}-\frac {2\,b\,\left (a+b\right )}{3}\right )-\mathrm {tan}\left (e+f\,x\right )\,\left ({\left (a+b\right )}^2+b^2-2\,b\,\left (a+b\right )\right )-{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (\frac {b^2}{5}-\frac {2\,b\,\left (a+b\right )}{5}\right )+\frac {b^2\,{\mathrm {tan}\left (e+f\,x\right )}^7}{7}+a^2\,f\,x}{f} \]
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 \[ \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2} \tan ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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