Optimal. Leaf size=129 \[ \frac {\left (8 a^2+12 a b+5 b^2\right ) \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \tan (e+f x) \sec (e+f x)}{16 f}+\frac {b (8 a+5 b) \tan (e+f x) \sec ^3(e+f x)}{24 f}+\frac {b \tan (e+f x) \sec ^5(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{6 f} \]
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Rubi [A] time = 0.13, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4147, 413, 385, 199, 206} \[ \frac {\left (8 a^2+12 a b+5 b^2\right ) \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \tan (e+f x) \sec (e+f x)}{16 f}+\frac {b (8 a+5 b) \tan (e+f x) \sec ^3(e+f x)}{24 f}+\frac {b \tan (e+f x) \sec ^5(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{6 f} \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 385
Rule 413
Rule 4147
Rubi steps
\begin {align*} \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b-a x^2\right )^2}{\left (1-x^2\right )^4} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {b \sec ^5(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{6 f}-\frac {\operatorname {Subst}\left (\int \frac {-(a+b) (6 a+5 b)+3 a (2 a+b) x^2}{\left (1-x^2\right )^3} \, dx,x,\sin (e+f x)\right )}{6 f}\\ &=\frac {b (8 a+5 b) \sec ^3(e+f x) \tan (e+f x)}{24 f}+\frac {b \sec ^5(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{6 f}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{8 f}\\ &=\frac {\left (8 a^2+12 a b+5 b^2\right ) \sec (e+f x) \tan (e+f x)}{16 f}+\frac {b (8 a+5 b) \sec ^3(e+f x) \tan (e+f x)}{24 f}+\frac {b \sec ^5(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{6 f}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (e+f x)\right )}{16 f}\\ &=\frac {\left (8 a^2+12 a b+5 b^2\right ) \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \sec (e+f x) \tan (e+f x)}{16 f}+\frac {b (8 a+5 b) \sec ^3(e+f x) \tan (e+f x)}{24 f}+\frac {b \sec ^5(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{6 f}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 94, normalized size = 0.73 \[ \frac {3 \left (8 a^2+12 a b+5 b^2\right ) \tanh ^{-1}(\sin (e+f x))+\tan (e+f x) \sec (e+f x) \left (3 \left (8 a^2+12 a b+5 b^2\right )+2 b (12 a+5 b) \sec ^2(e+f x)+8 b^2 \sec ^4(e+f x)\right )}{48 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 143, normalized size = 1.11 \[ \frac {3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{6} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{6} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (12 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, b^{2}\right )} \sin \left (f x + e\right )}{96 \, f \cos \left (f x + e\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.31, size = 191, normalized size = 1.48 \[ \frac {a^{2} \tan \left (f x +e \right ) \sec \left (f x +e \right )}{2 f}+\frac {a^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2 f}+\frac {a b \tan \left (f x +e \right ) \left (\sec ^{3}\left (f x +e \right )\right )}{2 f}+\frac {3 a b \tan \left (f x +e \right ) \sec \left (f x +e \right )}{4 f}+\frac {3 a b \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{4 f}+\frac {b^{2} \tan \left (f x +e \right ) \left (\sec ^{5}\left (f x +e \right )\right )}{6 f}+\frac {5 b^{2} \tan \left (f x +e \right ) \left (\sec ^{3}\left (f x +e \right )\right )}{24 f}+\frac {5 b^{2} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{16 f}+\frac {5 b^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 166, normalized size = 1.29 \[ \frac {3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \log \left (\sin \left (f x + e\right ) - 1\right ) - \frac {2 \, {\left (3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \sin \left (f x + e\right )^{5} - 8 \, {\left (6 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \sin \left (f x + e\right )^{3} + 3 \, {\left (8 \, a^{2} + 20 \, a b + 11 \, b^{2}\right )} \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1}}{96 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.55, size = 134, normalized size = 1.04 \[ \frac {\mathrm {atanh}\left (\sin \left (e+f\,x\right )\right )\,\left (\frac {a^2}{2}+\frac {3\,a\,b}{4}+\frac {5\,b^2}{16}\right )}{f}-\frac {\left (\frac {a^2}{2}+\frac {3\,a\,b}{4}+\frac {5\,b^2}{16}\right )\,{\sin \left (e+f\,x\right )}^5+\left (-a^2-2\,a\,b-\frac {5\,b^2}{6}\right )\,{\sin \left (e+f\,x\right )}^3+\left (\frac {a^2}{2}+\frac {5\,a\,b}{4}+\frac {11\,b^2}{16}\right )\,\sin \left (e+f\,x\right )}{f\,\left ({\sin \left (e+f\,x\right )}^6-3\,{\sin \left (e+f\,x\right )}^4+3\,{\sin \left (e+f\,x\right )}^2-1\right )} \]
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 \[ \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2} \sec ^{3}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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