Optimal. Leaf size=114 \[ -\frac {\left (a^2-8 a b+4 b^2\right ) \tan (e+f x)}{4 f}+\frac {1}{8} x \left (3 a^2-24 a b+8 b^2\right )+\frac {a^2 \sin ^4(e+f x) \tan (e+f x)}{4 f}-\frac {a (a-8 b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {b^2 \tan ^3(e+f x)}{3 f} \]
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Rubi [A] time = 0.12, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4132, 463, 455, 1153, 203} \[ -\frac {\left (a^2-8 a b+4 b^2\right ) \tan (e+f x)}{4 f}+\frac {1}{8} x \left (3 a^2-24 a b+8 b^2\right )+\frac {a^2 \sin ^4(e+f x) \tan (e+f x)}{4 f}-\frac {a (a-8 b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {b^2 \tan ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Rule 203
Rule 455
Rule 463
Rule 1153
Rule 4132
Rubi steps
\begin {align*} \int \left (a+b \sec ^2(e+f x)\right )^2 \sin ^4(e+f x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (a+b+b x^2\right )^2}{\left (1+x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a^2 \sin ^4(e+f x) \tan (e+f x)}{4 f}-\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (5 a^2-4 (a+b)^2-4 b^2 x^2\right )}{\left (1+x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{4 f}\\ &=-\frac {a (a-8 b) \cos (e+f x) \sin (e+f x)}{8 f}+\frac {a^2 \sin ^4(e+f x) \tan (e+f x)}{4 f}+\frac {\operatorname {Subst}\left (\int \frac {a (a-8 b)-2 a (a-8 b) x^2+8 b^2 x^4}{1+x^2} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=-\frac {a (a-8 b) \cos (e+f x) \sin (e+f x)}{8 f}+\frac {a^2 \sin ^4(e+f x) \tan (e+f x)}{4 f}+\frac {\operatorname {Subst}\left (\int \left (-2 \left (a^2-8 a b+4 b^2\right )+8 b^2 x^2+\frac {3 a^2-24 a b+8 b^2}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=-\frac {a (a-8 b) \cos (e+f x) \sin (e+f x)}{8 f}-\frac {\left (a^2-8 a b+4 b^2\right ) \tan (e+f x)}{4 f}+\frac {a^2 \sin ^4(e+f x) \tan (e+f x)}{4 f}+\frac {b^2 \tan ^3(e+f x)}{3 f}+\frac {\left (3 a^2-24 a b+8 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=\frac {1}{8} \left (3 a^2-24 a b+8 b^2\right ) x-\frac {a (a-8 b) \cos (e+f x) \sin (e+f x)}{8 f}-\frac {\left (a^2-8 a b+4 b^2\right ) \tan (e+f x)}{4 f}+\frac {a^2 \sin ^4(e+f x) \tan (e+f x)}{4 f}+\frac {b^2 \tan ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [A] time = 1.62, size = 153, normalized size = 1.34 \[ \frac {\sec ^3(e+f x) \left (a \cos ^2(e+f x)+b\right )^2 \left (3 \cos ^3(e+f x) \left (4 f x \left (3 a^2-24 a b+8 b^2\right )+a^2 \sin (4 (e+f x))-8 a (a-2 b) \sin (2 (e+f x))\right )+64 b (3 a-2 b) \sec (e) \sin (f x) \cos ^2(e+f x)+32 b^2 \tan (e) \cos (e+f x)+32 b^2 \sec (e) \sin (f x)\right )}{24 f (a \cos (2 (e+f x))+a+2 b)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 107, normalized size = 0.94 \[ \frac {3 \, {\left (3 \, a^{2} - 24 \, a b + 8 \, b^{2}\right )} f x \cos \left (f x + e\right )^{3} + {\left (6 \, a^{2} \cos \left (f x + e\right )^{6} - 3 \, {\left (5 \, a^{2} - 8 \, a b\right )} \cos \left (f x + e\right )^{4} + 16 \, {\left (3 \, a b - 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, b^{2}\right )} \sin \left (f x + e\right )}{24 \, f \cos \left (f x + e\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 132, normalized size = 1.16 \[ \frac {8 \, b^{2} \tan \left (f x + e\right )^{3} + 48 \, a b \tan \left (f x + e\right ) - 24 \, b^{2} \tan \left (f x + e\right ) + 3 \, {\left (3 \, a^{2} - 24 \, a b + 8 \, b^{2}\right )} {\left (f x + e\right )} - \frac {3 \, {\left (5 \, a^{2} \tan \left (f x + e\right )^{3} - 8 \, a b \tan \left (f x + e\right )^{3} + 3 \, a^{2} \tan \left (f x + e\right ) - 8 \, a b \tan \left (f x + e\right )\right )}}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{2}}}{24 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.90, size = 123, normalized size = 1.08 \[ \frac {a^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+2 a b \left (\frac {\sin ^{5}\left (f x +e \right )}{\cos \left (f x +e \right )}+\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )-\frac {3 f x}{2}-\frac {3 e}{2}\right )+b^{2} \left (\frac {\left (\tan ^{3}\left (f x +e \right )\right )}{3}-\tan \left (f x +e \right )+f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 120, normalized size = 1.05 \[ \frac {8 \, b^{2} \tan \left (f x + e\right )^{3} + 3 \, {\left (3 \, a^{2} - 24 \, a b + 8 \, b^{2}\right )} {\left (f x + e\right )} + 24 \, {\left (2 \, a b - b^{2}\right )} \tan \left (f x + e\right ) - \frac {3 \, {\left ({\left (5 \, a^{2} - 8 \, a b\right )} \tan \left (f x + e\right )^{3} + {\left (3 \, a^{2} - 8 \, a b\right )} \tan \left (f x + e\right )\right )}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}}{24 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.33, size = 116, normalized size = 1.02 \[ x\,\left (\frac {3\,a^2}{8}-3\,a\,b+b^2\right )+\frac {\left (a\,b-\frac {5\,a^2}{8}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^3+\left (a\,b-\frac {3\,a^2}{8}\right )\,\mathrm {tan}\left (e+f\,x\right )}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^4+2\,{\mathrm {tan}\left (e+f\,x\right )}^2+1\right )}+\frac {b^2\,{\mathrm {tan}\left (e+f\,x\right )}^3}{3\,f}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (3\,b^2-2\,b\,\left (a+b\right )\right )}{f} \]
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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