Optimal. Leaf size=138 \[ \frac{\left (3 a^2+12 a b+8 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{8 a^3 \sqrt{b} f (a+b)^{3/2}}+\frac{(3 a+4 b) \tan (e+f x)}{8 a^2 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )}-\frac{x}{a^3}+\frac{\tan (e+f x)}{4 a f \left (a+b \tan ^2(e+f x)+b\right )^2} \]
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Rubi [A] time = 0.227172, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4141, 1975, 471, 527, 522, 203, 205} \[ \frac{\left (3 a^2+12 a b+8 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{8 a^3 \sqrt{b} f (a+b)^{3/2}}+\frac{(3 a+4 b) \tan (e+f x)}{8 a^2 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )}-\frac{x}{a^3}+\frac{\tan (e+f x)}{4 a f \left (a+b \tan ^2(e+f x)+b\right )^2} \]
Antiderivative was successfully verified.
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Rule 4141
Rule 1975
Rule 471
Rule 527
Rule 522
Rule 203
Rule 205
Rubi steps
\begin{align*} \int \frac{\tan ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{\left (1+x^2\right ) \left (a+b \left (1+x^2\right )\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\tan (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{1-3 x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{4 a f}\\ &=\frac{\tan (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{(3 a+4 b) \tan (e+f x)}{8 a^2 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{5 a+4 b+(-3 a-4 b) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{8 a^2 (a+b) f}\\ &=\frac{\tan (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{(3 a+4 b) \tan (e+f x)}{8 a^2 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{a^3 f}+\frac{\left (3 a^2+12 a b+8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{8 a^3 (a+b) f}\\ &=-\frac{x}{a^3}+\frac{\left (3 a^2+12 a b+8 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{8 a^3 \sqrt{b} (a+b)^{3/2} f}+\frac{\tan (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{(3 a+4 b) \tan (e+f x)}{8 a^2 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}\\ \end{align*}
Mathematica [C] time = 13.2285, size = 1473, normalized size = 10.67 \[ \frac{(\cos (2 e+2 f x) a+a+2 b)^3 \left (\frac{\left (3 a^2+8 b a+8 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{(a+b)^{5/2}}-\frac{a \sqrt{b} \left (3 a^2+16 b a+3 (a+2 b) \cos (2 (e+f x)) a+16 b^2\right ) \sin (2 (e+f x))}{(a+b)^2 (\cos (2 (e+f x)) a+a+2 b)^2}\right ) \sec ^6(e+f x)}{1024 b^{5/2} f \left (b \sec ^2(e+f x)+a\right )^3}+\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \left (\frac{\sqrt{b} \left (3 a^3+14 b a^2+24 b^2 a+\left (3 a^2+4 b a+4 b^2\right ) \cos (2 (e+f x)) a+16 b^3\right ) \sin (2 (e+f x))}{(a+b)^2 (\cos (2 (e+f x)) a+a+2 b)^2}-\frac{3 a (a+2 b) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{(a+b)^{5/2}}\right ) \sec ^6(e+f x)}{2048 b^{5/2} f \left (b \sec ^2(e+f x)+a\right )^3}-\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \left (\frac{2 \left (3 a^5-10 b a^4+80 b^2 a^3+480 b^3 a^2+640 b^4 a+256 b^5\right ) \tan ^{-1}\left (\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right ) (\cos (2 e)-i \sin (2 e))}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}+\frac{\sec (2 e) \left (-9 \sin (2 e) a^6+9 \sin (2 f x) a^6+3 \sin (2 (e+2 f x)) a^6-3 \sin (4 e+2 f x) a^6+12 b \sin (2 e) a^5-14 b \sin (2 f x) a^5-12 b \sin (2 (e+2 f x)) a^5+10 b \sin (4 e+2 f x) a^5+128 b^2 f x \cos (2 (e+2 f x)) a^4+512 b^2 f x \cos (4 e+2 f x) a^4+128 b^2 f x \cos (6 e+4 f x) a^4+684 b^2 \sin (2 e) a^4-608 b^2 \sin (2 f x) a^4-204 b^2 \sin (2 (e+2 f x)) a^4+304 b^2 \sin (4 e+2 f x) a^4+256 b^3 f x \cos (2 (e+2 f x)) a^3+2048 b^3 f x \cos (4 e+2 f x) a^3+256 b^3 f x \cos (6 e+4 f x) a^3+2880 b^3 \sin (2 e) a^3-2112 b^3 \sin (2 f x) a^3-384 b^3 \sin (2 (e+2 f x)) a^3+1056 b^3 \sin (4 e+2 f x) a^3+128 b^4 f x \cos (2 (e+2 f x)) a^2+2560 b^4 f x \cos (4 e+2 f x) a^2+128 b^4 f x \cos (6 e+4 f x) a^2+5280 b^4 \sin (2 e) a^2-2560 b^4 \sin (2 f x) a^2-192 b^4 \sin (2 (e+2 f x)) a^2+1280 b^4 \sin (4 e+2 f x) a^2+512 b^2 (a+b)^2 (a+2 b) f x \cos (2 f x) a+1024 b^5 f x \cos (4 e+2 f x) a+4608 b^5 \sin (2 e) a-1024 b^5 \sin (2 f x) a+512 b^5 \sin (4 e+2 f x) a+256 b^2 (a+b)^2 \left (3 a^2+8 b a+8 b^2\right ) f x \cos (2 e)+1536 b^6 \sin (2 e)\right )}{(\cos (2 (e+f x)) a+a+2 b)^2}\right ) \sec ^6(e+f x)}{4096 a^3 b^2 (a+b)^2 f \left (b \sec ^2(e+f x)+a\right )^3}-\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \left (\frac{a \sec (2 e) \left (\left (-9 a^4-16 b a^3+48 b^2 a^2+128 b^3 a+64 b^4\right ) \sin (2 f x)+a \left (-3 a^3+2 b a^2+24 b^2 a+16 b^3\right ) \sin (2 (e+2 f x))+\left (3 a^4-64 b^2 a^2-128 b^3 a-64 b^4\right ) \sin (4 e+2 f x)\right )+\left (9 a^5+18 b a^4-64 b^2 a^3-256 b^3 a^2-320 b^4 a-128 b^5\right ) \tan (2 e)}{a^2 (\cos (2 (e+f x)) a+a+2 b)^2}-\frac{6 a^2 \tan ^{-1}\left (\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right ) (\cos (2 e)-i \sin (2 e))}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right ) \sec ^6(e+f x)}{2048 b^2 (a+b)^2 f \left (b \sec ^2(e+f x)+a\right )^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.096, size = 263, normalized size = 1.9 \begin{align*} -{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) }{f{a}^{3}}}+{\frac{3\,b \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{8\,fa \left ( a+b+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2} \left ( a+b \right ) }}+{\frac{{b}^{2} \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{2\,f{a}^{2} \left ( a+b+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2} \left ( a+b \right ) }}+{\frac{5\,\tan \left ( fx+e \right ) }{8\,fa \left ( a+b+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}+{\frac{b\tan \left ( fx+e \right ) }{2\,f{a}^{2} \left ( a+b+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}+{\frac{3}{8\,fa \left ( a+b \right ) }\arctan \left ({b\tan \left ( fx+e \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}}+{\frac{3\,b}{2\,f{a}^{2} \left ( a+b \right ) }\arctan \left ({b\tan \left ( fx+e \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}}+{\frac{{b}^{2}}{f{a}^{3} \left ( a+b \right ) }\arctan \left ({b\tan \left ( fx+e \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.713179, size = 1910, normalized size = 13.84 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.64476, size = 244, normalized size = 1.77 \begin{align*} \frac{\frac{{\left (\pi \left \lfloor \frac{f x + e}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (b\right ) + \arctan \left (\frac{b \tan \left (f x + e\right )}{\sqrt{a b + b^{2}}}\right )\right )}{\left (3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right )}}{{\left (a^{4} + a^{3} b\right )} \sqrt{a b + b^{2}}} + \frac{3 \, a b \tan \left (f x + e\right )^{3} + 4 \, b^{2} \tan \left (f x + e\right )^{3} + 5 \, a^{2} \tan \left (f x + e\right ) + 9 \, a b \tan \left (f x + e\right ) + 4 \, b^{2} \tan \left (f x + e\right )}{{\left (a^{3} + a^{2} b\right )}{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{2}} - \frac{8 \,{\left (f x + e\right )}}{a^{3}}}{8 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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