Optimal. Leaf size=278 \[ \frac {\left (5 a^3-12 a^2 b+24 a b^2-64 b^3\right ) x}{16 a^5}+\frac {b^{7/2} (9 a+8 b) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{2 a^5 (a+b)^{3/2} f}+\frac {\left (15 a^2-26 a b+48 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {(5 a-8 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {b \left (5 a^3-7 a^2 b+12 a b^2+32 b^3\right ) \tan (e+f x)}{16 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )} \]
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Rubi [A]
time = 0.25, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4231, 425, 541,
536, 209, 211} \begin {gather*} \frac {b^{7/2} (9 a+8 b) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{2 a^5 f (a+b)^{3/2}}+\frac {(5 a-8 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \left (a+b \tan ^2(e+f x)+b\right )}+\frac {\left (15 a^2-26 a b+48 b^2\right ) \sin (e+f x) \cos (e+f x)}{48 a^3 f \left (a+b \tan ^2(e+f x)+b\right )}+\frac {x \left (5 a^3-12 a^2 b+24 a b^2-64 b^3\right )}{16 a^5}+\frac {b \left (5 a^3-7 a^2 b+12 a b^2+32 b^3\right ) \tan (e+f x)}{16 a^4 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )}+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 a f \left (a+b \tan ^2(e+f x)+b\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 211
Rule 425
Rule 536
Rule 541
Rule 4231
Rubi steps
\begin {align*} \int \frac {\cos ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^4 \left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {-5 a+b-7 b x^2}{\left (1+x^2\right )^3 \left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{6 a f}\\ &=\frac {(5 a-8 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\text {Subst}\left (\int \frac {15 a^2-a b+8 b^2+5 (5 a-8 b) b x^2}{\left (1+x^2\right )^2 \left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{24 a^2 f}\\ &=\frac {\left (15 a^2-26 a b+48 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {(5 a-8 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {-3 \left (5 a^3+3 a^2 b-2 a b^2-16 b^3\right )-3 b \left (15 a^2-26 a b+48 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{48 a^3 f}\\ &=\frac {\left (15 a^2-26 a b+48 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {(5 a-8 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {b \left (5 a^3-7 a^2 b+12 a b^2+32 b^3\right ) \tan (e+f x)}{16 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {-6 \left (5 a^4-2 a^3 b+5 a^2 b^2-28 a b^3-32 b^4\right )-6 b \left (5 a^3-7 a^2 b+12 a b^2+32 b^3\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{96 a^4 (a+b) f}\\ &=\frac {\left (15 a^2-26 a b+48 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {(5 a-8 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {b \left (5 a^3-7 a^2 b+12 a b^2+32 b^3\right ) \tan (e+f x)}{16 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\left (b^4 (9 a+8 b)\right ) \text {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{2 a^5 (a+b) f}+\frac {\left (5 a^3-12 a^2 b+24 a b^2-64 b^3\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{16 a^5 f}\\ &=\frac {\left (5 a^3-12 a^2 b+24 a b^2-64 b^3\right ) x}{16 a^5}+\frac {b^{7/2} (9 a+8 b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{2 a^5 (a+b)^{3/2} f}+\frac {\left (15 a^2-26 a b+48 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {(5 a-8 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {b \left (5 a^3-7 a^2 b+12 a b^2+32 b^3\right ) \tan (e+f x)}{16 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 4.88, size = 499, normalized size = 1.79 \begin {gather*} \frac {(a+2 b+a \cos (2 (e+f x))) \sec ^4(e+f x) \left (12 \left (5 a^3-12 a^2 b+24 a b^2-64 b^3\right ) x (a+2 b+a \cos (2 (e+f x)))-\frac {96 b^4 (9 a+8 b) \text {ArcTan}\left (\frac {\sec (f x) (\cos (2 e)-i \sin (2 e)) (-((a+2 b) \sin (f x))+a \sin (2 e+f x))}{2 \sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}}\right ) (a+2 b+a \cos (2 (e+f x))) (\cos (2 e)-i \sin (2 e))}{(a+b)^{3/2} f \sqrt {b (\cos (e)-i \sin (e))^4}}+\frac {3 a \left (15 a^2-32 a b+48 b^2\right ) \cos (2 f x) (a+2 b+a \cos (2 (e+f x))) \sin (2 e)}{f}+\frac {3 a^2 (3 a-4 b) \cos (4 f x) (a+2 b+a \cos (2 (e+f x))) \sin (4 e)}{f}+\frac {a^3 \cos (6 f x) (a+2 b+a \cos (2 (e+f x))) \sin (6 e)}{f}+\frac {3 a \left (15 a^2-32 a b+48 b^2\right ) \cos (2 e) (a+2 b+a \cos (2 (e+f x))) \sin (2 f x)}{f}-\frac {96 b^4 ((a+2 b) \sin (2 e)-a \sin (2 f x))}{(a+b) f (\cos (e)-\sin (e)) (\cos (e)+\sin (e))}+\frac {3 a^2 (3 a-4 b) \cos (4 e) (a+2 b+a \cos (2 (e+f x))) \sin (4 f x)}{f}+\frac {a^3 \cos (6 e) (a+2 b+a \cos (2 (e+f x))) \sin (6 f x)}{f}\right )}{768 a^5 \left (a+b \sec ^2(e+f x)\right )^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.19, size = 210, normalized size = 0.76
method | result | size |
derivativedivides | \(\frac {\frac {b^{4} \left (\frac {a \tan \left (f x +e \right )}{2 \left (a +b \right ) \left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}+\frac {\left (9 a +8 b \right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{2 \left (a +b \right ) \sqrt {\left (a +b \right ) b}}\right )}{a^{5}}+\frac {\frac {\left (\frac {5}{16} a^{3}-\frac {3}{4} a^{2} b +\frac {3}{2} a \,b^{2}\right ) \left (\tan ^{5}\left (f x +e \right )\right )+\left (3 a \,b^{2}+\frac {5}{6} a^{3}-2 a^{2} b \right ) \left (\tan ^{3}\left (f x +e \right )\right )+\left (-\frac {5}{4} a^{2} b +\frac {3}{2} a \,b^{2}+\frac {11}{16} a^{3}\right ) \tan \left (f x +e \right )}{\left (\tan ^{2}\left (f x +e \right )+1\right )^{3}}+\frac {\left (5 a^{3}-12 a^{2} b +24 a \,b^{2}-64 b^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{16}}{a^{5}}}{f}\) | \(210\) |
default | \(\frac {\frac {b^{4} \left (\frac {a \tan \left (f x +e \right )}{2 \left (a +b \right ) \left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}+\frac {\left (9 a +8 b \right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{2 \left (a +b \right ) \sqrt {\left (a +b \right ) b}}\right )}{a^{5}}+\frac {\frac {\left (\frac {5}{16} a^{3}-\frac {3}{4} a^{2} b +\frac {3}{2} a \,b^{2}\right ) \left (\tan ^{5}\left (f x +e \right )\right )+\left (3 a \,b^{2}+\frac {5}{6} a^{3}-2 a^{2} b \right ) \left (\tan ^{3}\left (f x +e \right )\right )+\left (-\frac {5}{4} a^{2} b +\frac {3}{2} a \,b^{2}+\frac {11}{16} a^{3}\right ) \tan \left (f x +e \right )}{\left (\tan ^{2}\left (f x +e \right )+1\right )^{3}}+\frac {\left (5 a^{3}-12 a^{2} b +24 a \,b^{2}-64 b^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{16}}{a^{5}}}{f}\) | \(210\) |
risch | \(\frac {5 x}{16 a^{2}}-\frac {3 x b}{4 a^{3}}+\frac {3 x \,b^{2}}{2 a^{4}}-\frac {4 x \,b^{3}}{a^{5}}-\frac {i {\mathrm e}^{-2 i \left (f x +e \right )} b}{4 a^{3} f}+\frac {i {\mathrm e}^{4 i \left (f x +e \right )} b}{32 a^{3} f}+\frac {3 i {\mathrm e}^{-2 i \left (f x +e \right )} b^{2}}{8 a^{4} f}-\frac {3 i {\mathrm e}^{4 i \left (f x +e \right )}}{128 a^{2} f}-\frac {15 i {\mathrm e}^{2 i \left (f x +e \right )}}{128 a^{2} f}-\frac {3 i {\mathrm e}^{2 i \left (f x +e \right )} b^{2}}{8 a^{4} f}+\frac {15 i {\mathrm e}^{-2 i \left (f x +e \right )}}{128 a^{2} f}+\frac {3 i {\mathrm e}^{-4 i \left (f x +e \right )}}{128 a^{2} f}+\frac {i {\mathrm e}^{2 i \left (f x +e \right )} b}{4 a^{3} f}-\frac {i {\mathrm e}^{-4 i \left (f x +e \right )} b}{32 a^{3} f}+\frac {i b^{4} \left (a \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )}{a^{5} \left (a +b \right ) f \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )}+\frac {9 \sqrt {-\left (a +b \right ) b}\, b^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-\left (a +b \right ) b}-a -2 b}{a}\right )}{4 \left (a +b \right )^{2} f \,a^{4}}+\frac {2 \sqrt {-\left (a +b \right ) b}\, b^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-\left (a +b \right ) b}-a -2 b}{a}\right )}{\left (a +b \right )^{2} f \,a^{5}}-\frac {9 \sqrt {-\left (a +b \right ) b}\, b^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-\left (a +b \right ) b}+a +2 b}{a}\right )}{4 \left (a +b \right )^{2} f \,a^{4}}-\frac {2 \sqrt {-\left (a +b \right ) b}\, b^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-\left (a +b \right ) b}+a +2 b}{a}\right )}{\left (a +b \right )^{2} f \,a^{5}}+\frac {\sin \left (6 f x +6 e \right )}{192 a^{2} f}\) | \(547\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 379, normalized size = 1.36 \begin {gather*} \frac {\frac {24 \, {\left (9 \, a b^{4} + 8 \, b^{5}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{{\left (a^{6} + a^{5} b\right )} \sqrt {{\left (a + b\right )} b}} + \frac {3 \, {\left (5 \, a^{3} b - 7 \, a^{2} b^{2} + 12 \, a b^{3} + 32 \, b^{4}\right )} \tan \left (f x + e\right )^{7} + {\left (15 \, a^{4} + 34 \, a^{3} b - 41 \, a^{2} b^{2} + 156 \, a b^{3} + 288 \, b^{4}\right )} \tan \left (f x + e\right )^{5} + {\left (40 \, a^{4} + 17 \, a^{3} b - 35 \, a^{2} b^{2} + 204 \, a b^{3} + 288 \, b^{4}\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (11 \, a^{4} + 2 \, a^{3} b - 5 \, a^{2} b^{2} + 28 \, a b^{3} + 32 \, b^{4}\right )} \tan \left (f x + e\right )}{{\left (a^{5} b + a^{4} b^{2}\right )} \tan \left (f x + e\right )^{8} + {\left (a^{6} + 5 \, a^{5} b + 4 \, a^{4} b^{2}\right )} \tan \left (f x + e\right )^{6} + a^{6} + 2 \, a^{5} b + a^{4} b^{2} + 3 \, {\left (a^{6} + 3 \, a^{5} b + 2 \, a^{4} b^{2}\right )} \tan \left (f x + e\right )^{4} + {\left (3 \, a^{6} + 7 \, a^{5} b + 4 \, a^{4} b^{2}\right )} \tan \left (f x + e\right )^{2}} + \frac {3 \, {\left (5 \, a^{3} - 12 \, a^{2} b + 24 \, a b^{2} - 64 \, b^{3}\right )} {\left (f x + e\right )}}{a^{5}}}{48 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.51, size = 815, normalized size = 2.93 \begin {gather*} \left [\frac {3 \, {\left (5 \, a^{5} - 7 \, a^{4} b + 12 \, a^{3} b^{2} - 40 \, a^{2} b^{3} - 64 \, a b^{4}\right )} f x \cos \left (f x + e\right )^{2} + 3 \, {\left (5 \, a^{4} b - 7 \, a^{3} b^{2} + 12 \, a^{2} b^{3} - 40 \, a b^{4} - 64 \, b^{5}\right )} f x + 6 \, {\left (9 \, a b^{4} + 8 \, b^{5} + {\left (9 \, a^{2} b^{3} + 8 \, a b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-\frac {b}{a + b}} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left ({\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (a b + b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {b}{a + b}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right ) + {\left (8 \, {\left (a^{5} + a^{4} b\right )} \cos \left (f x + e\right )^{7} + 2 \, {\left (5 \, a^{5} - 3 \, a^{4} b - 8 \, a^{3} b^{2}\right )} \cos \left (f x + e\right )^{5} + {\left (15 \, a^{5} - 11 \, a^{4} b + 22 \, a^{3} b^{2} + 48 \, a^{2} b^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (5 \, a^{4} b - 7 \, a^{3} b^{2} + 12 \, a^{2} b^{3} + 32 \, a b^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, {\left ({\left (a^{7} + a^{6} b\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{6} b + a^{5} b^{2}\right )} f\right )}}, \frac {3 \, {\left (5 \, a^{5} - 7 \, a^{4} b + 12 \, a^{3} b^{2} - 40 \, a^{2} b^{3} - 64 \, a b^{4}\right )} f x \cos \left (f x + e\right )^{2} + 3 \, {\left (5 \, a^{4} b - 7 \, a^{3} b^{2} + 12 \, a^{2} b^{3} - 40 \, a b^{4} - 64 \, b^{5}\right )} f x - 12 \, {\left (9 \, a b^{4} + 8 \, b^{5} + {\left (9 \, a^{2} b^{3} + 8 \, a b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{a + b}} \arctan \left (\frac {{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {b}{a + b}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) + {\left (8 \, {\left (a^{5} + a^{4} b\right )} \cos \left (f x + e\right )^{7} + 2 \, {\left (5 \, a^{5} - 3 \, a^{4} b - 8 \, a^{3} b^{2}\right )} \cos \left (f x + e\right )^{5} + {\left (15 \, a^{5} - 11 \, a^{4} b + 22 \, a^{3} b^{2} + 48 \, a^{2} b^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (5 \, a^{4} b - 7 \, a^{3} b^{2} + 12 \, a^{2} b^{3} + 32 \, a b^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, {\left ({\left (a^{7} + a^{6} b\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{6} b + a^{5} b^{2}\right )} f\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 270, normalized size = 0.97 \begin {gather*} \frac {\frac {24 \, b^{4} \tan \left (f x + e\right )}{{\left (a^{5} + a^{4} b\right )} {\left (b \tan \left (f x + e\right )^{2} + a + b\right )}} + \frac {24 \, {\left (9 \, a b^{4} + 8 \, b^{5}\right )} {\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 (a^{6} + a^{5} b\right )} \sqrt {a b + b^{2}}} + \frac {3 \, {\left (5 \, a^{3} - 12 \, a^{2} b + 24 \, a b^{2} - 64 \, b^{3}\right )} {\left (f x + e\right )}}{a^{5}} + \frac {15 \, a^{2} \tan \left (f x + e\right )^{5} - 36 \, a b \tan \left (f x + e\right )^{5} + 72 \, b^{2} \tan \left (f x + e\right )^{5} + 40 \, a^{2} \tan \left (f x + e\right )^{3} - 96 \, a b \tan \left (f x + e\right )^{3} + 144 \, b^{2} \tan \left (f x + e\right )^{3} + 33 \, a^{2} \tan \left (f x + e\right ) - 60 \, a b \tan \left (f x + e\right ) + 72 \, b^{2} \tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{3} a^{4}}}{48 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.56, size = 2500, normalized size = 8.99 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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