Optimal. Leaf size=314 \[ -\frac {5 \sqrt {b} \sqrt {a+b} (a+4 b) (3 a+4 b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^6 f}+\frac {(9 a+10 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \left (a+b \tan ^2(e+f x)+b\right )^2}+\frac {5 x (a+2 b) \left (a^2+16 a b+16 b^2\right )}{16 a^6}-\frac {5 b \left (5 a^2+20 a b+16 b^2\right ) \tan (e+f x)}{16 a^5 f \left (a+b \tan ^2(e+f x)+b\right )}-\frac {5 b \left (9 a^2+32 a b+24 b^2\right ) \tan (e+f x)}{48 a^4 f \left (a+b \tan ^2(e+f x)+b\right )^2}-\frac {\left (33 a^2+110 a b+80 b^2\right ) \sin (e+f x) \cos (e+f x)}{48 a^3 f \left (a+b \tan ^2(e+f x)+b\right )^2}+\frac {\sin ^3(e+f x) \cos ^3(e+f x)}{6 a f \left (a+b \tan ^2(e+f x)+b\right )^2} \]
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Rubi [A] time = 0.50, antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {4132, 470, 578, 527, 522, 203, 205} \[ -\frac {5 b \left (5 a^2+20 a b+16 b^2\right ) \tan (e+f x)}{16 a^5 f \left (a+b \tan ^2(e+f x)+b\right )}-\frac {5 b \left (9 a^2+32 a b+24 b^2\right ) \tan (e+f x)}{48 a^4 f \left (a+b \tan ^2(e+f x)+b\right )^2}-\frac {\left (33 a^2+110 a b+80 b^2\right ) \sin (e+f x) \cos (e+f x)}{48 a^3 f \left (a+b \tan ^2(e+f x)+b\right )^2}+\frac {5 x (a+2 b) \left (a^2+16 a b+16 b^2\right )}{16 a^6}-\frac {5 \sqrt {b} \sqrt {a+b} (a+4 b) (3 a+4 b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^6 f}+\frac {(9 a+10 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \left (a+b \tan ^2(e+f x)+b\right )^2}+\frac {\sin ^3(e+f x) \cos ^3(e+f x)}{6 a f \left (a+b \tan ^2(e+f x)+b\right )^2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 470
Rule 522
Rule 527
Rule 578
Rule 4132
Rubi steps
\begin {align*} \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^4 \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 (a+b)+(-6 a-7 b) x^2\right )}{\left (1+x^2\right )^3 \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{6 a f}\\ &=\frac {(9 a+10 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {(a+b) (9 a+10 b)+\left (-24 a^2-91 a b-70 b^2\right ) x^2}{\left (1+x^2\right )^2 \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{24 a^2 f}\\ &=-\frac {\left (33 a^2+110 a b+80 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 )^2}+\frac {(9 a+10 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {5 (a+b) \left (3 a^2+18 a b+16 b^2\right )-5 b \left (33 a^2+110 a b+80 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{48 a^3 f}\\ &=-\frac {\left (33 a^2+110 a b+80 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 )^2}+\frac {(9 a+10 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {5 b \left (9 a^2+32 a b+24 b^2\right ) \tan (e+f x)}{48 a^4 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {60 (a+b)^2 \left (a^2+8 a b+8 b^2\right )-60 b (a+b) \left (9 a^2+32 a b+24 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 )}{192 a^4 (a+b) f}\\ &=-\frac {\left (33 a^2+110 a b+80 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 )^2}+\frac {(9 a+10 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {5 b \left (9 a^2+32 a b+24 b^2\right ) \tan (e+f x)}{48 a^4 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {5 b \left (5 a^2+20 a b+16 b^2\right ) \tan (e+f x)}{16 a^5 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {120 (a+b)^3 \left (a^2+12 a b+16 b^2\right )-120 b (a+b)^2 \left (5 a^2+20 a b+16 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{384 a^5 (a+b)^2 f}\\ &=-\frac {\left (33 a^2+110 a b+80 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 )^2}+\frac {(9 a+10 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {5 b \left (9 a^2+32 a b+24 b^2\right ) \tan (e+f x)}{48 a^4 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {5 b \left (5 a^2+20 a b+16 b^2\right ) \tan (e+f x)}{16 a^5 f \left (a+b+b \tan ^2(e+f x)\right )}-\frac {(5 b (a+b) (a+4 b) (3 a+4 b)) \operatorname {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{8 a^6 f}+\frac {\left (5 (a+2 b) \left (a^2+16 a b+16 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{16 a^6 f}\\ &=\frac {5 (a+2 b) \left (a^2+16 a b+16 b^2\right ) x}{16 a^6}-\frac {5 \sqrt {b} \sqrt {a+b} (a+4 b) (3 a+4 b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^6 f}-\frac {\left (33 a^2+110 a b+80 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 )^2}+\frac {(9 a+10 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {5 b \left (9 a^2+32 a b+24 b^2\right ) \tan (e+f x)}{48 a^4 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {5 b \left (5 a^2+20 a b+16 b^2\right ) \tan (e+f x)}{16 a^5 f \left (a+b+b \tan ^2(e+f x)\right )}\\ \end {align*}
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Mathematica [C] time = 18.87, size = 1639, normalized size = 5.22 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.63, size = 930, normalized size = 2.96 \[ \left [\frac {30 \, {\left (a^{5} + 18 \, a^{4} b + 48 \, a^{3} b^{2} + 32 \, a^{2} b^{3}\right )} f x \cos \left (f x + e\right )^{4} + 60 \, {\left (a^{4} b + 18 \, a^{3} b^{2} + 48 \, a^{2} b^{3} + 32 \, a b^{4}\right )} f x \cos \left (f x + e\right )^{2} + 30 \, {\left (a^{3} b^{2} + 18 \, a^{2} b^{3} + 48 \, a b^{4} + 32 \, b^{5}\right )} f x + 15 \, {\left ({\left (3 \, a^{4} + 16 \, a^{3} b + 16 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 3 \, a^{2} b^{2} + 16 \, a b^{3} + 16 \, b^{4} + 2 \, {\left (3 \, a^{3} b + 16 \, a^{2} b^{2} + 16 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a b - b^{2}} \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 \, b\right )} \cos \left (f x + e\right )^{3} - b \cos \left (f x + e\right )\right )} \sqrt {-a b - b^{2}} \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 ) - 2 \, {\left (8 \, a^{5} \cos \left (f x + e\right )^{9} - 2 \, {\left (13 \, a^{5} + 10 \, a^{4} b\right )} \cos \left (f x + e\right )^{7} + {\left (33 \, a^{5} + 110 \, a^{4} b + 80 \, a^{3} b^{2}\right )} \cos \left (f x + e\right )^{5} + 20 \, {\left (6 \, a^{4} b + 23 \, a^{3} b^{2} + 18 \, a^{2} b^{3}\right )} \cos \left (f x + e\right )^{3} + 15 \, {\left (5 \, a^{3} b^{2} + 20 \, a^{2} b^{3} + 16 \, a b^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{96 \, {\left (a^{8} f \cos \left (f x + e\right )^{4} + 2 \, a^{7} b f \cos \left (f x + e\right )^{2} + a^{6} b^{2} f\right )}}, \frac {15 \, {\left (a^{5} + 18 \, a^{4} b + 48 \, a^{3} b^{2} + 32 \, a^{2} b^{3}\right )} f x \cos \left (f x + e\right )^{4} + 30 \, {\left (a^{4} b + 18 \, a^{3} b^{2} + 48 \, a^{2} b^{3} + 32 \, a b^{4}\right )} f x \cos \left (f x + e\right )^{2} + 15 \, {\left (a^{3} b^{2} + 18 \, a^{2} b^{3} + 48 \, a b^{4} + 32 \, b^{5}\right )} f x + 15 \, {\left ({\left (3 \, a^{4} + 16 \, a^{3} b + 16 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 3 \, a^{2} b^{2} + 16 \, a b^{3} + 16 \, b^{4} + 2 \, {\left (3 \, a^{3} b + 16 \, a^{2} b^{2} + 16 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {a b + b^{2}} \arctan \left (\frac {{\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b}{2 \, \sqrt {a b + b^{2}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) - {\left (8 \, a^{5} \cos \left (f x + e\right )^{9} - 2 \, {\left (13 \, a^{5} + 10 \, a^{4} b\right )} \cos \left (f x + e\right )^{7} + {\left (33 \, a^{5} + 110 \, a^{4} b + 80 \, a^{3} b^{2}\right )} \cos \left (f x + e\right )^{5} + 20 \, {\left (6 \, a^{4} b + 23 \, a^{3} b^{2} + 18 \, a^{2} b^{3}\right )} \cos \left (f x + e\right )^{3} + 15 \, {\left (5 \, a^{3} b^{2} + 20 \, a^{2} b^{3} + 16 \, a b^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, {\left (a^{8} f \cos \left (f x + e\right )^{4} + 2 \, a^{7} b f \cos \left (f x + e\right )^{2} + a^{6} b^{2} f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 372, normalized size = 1.18 \[ \frac {\frac {15 \, {\left (a^{3} + 18 \, a^{2} b + 48 \, a b^{2} + 32 \, b^{3}\right )} {\left (f x + e\right )}}{a^{6}} - \frac {30 \, {\left (3 \, a^{3} b + 19 \, a^{2} b^{2} + 32 \, a b^{3} + 16 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )\right )}}{\sqrt {a b + b^{2}} a^{6}} - \frac {6 \, {\left (7 \, a^{2} b^{2} \tan \left (f x + e\right )^{3} + 23 \, a b^{3} \tan \left (f x + e\right )^{3} + 16 \, b^{4} \tan \left (f x + e\right )^{3} + 9 \, a^{3} b \tan \left (f x + e\right ) + 34 \, a^{2} b^{2} \tan \left (f x + e\right ) + 41 \, a b^{3} \tan \left (f x + e\right ) + 16 \, b^{4} \tan \left (f x + e\right )\right )}}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{2} a^{5}} - \frac {33 \, a^{2} \tan \left (f x + e\right )^{5} + 162 \, a b \tan \left (f x + e\right )^{5} + 144 \, b^{2} \tan \left (f x + e\right )^{5} + 40 \, a^{2} \tan \left (f x + e\right )^{3} + 288 \, a b \tan \left (f x + e\right )^{3} + 288 \, b^{2} \tan \left (f x + e\right )^{3} + 15 \, a^{2} \tan \left (f x + e\right ) + 126 \, a b \tan \left (f x + e\right ) + 144 \, b^{2} \tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{3} a^{5}}}{48 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.28, size = 689, normalized size = 2.19 \[ -\frac {7 b^{2} \left (\tan ^{3}\left (f x +e \right )\right )}{8 f \,a^{3} \left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {23 b^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{8 f \,a^{4} \left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {9 b \tan \left (f x +e \right )}{8 a^{2} f \left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {17 b^{2} \tan \left (f x +e \right )}{4 f \,a^{3} \left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {41 b^{3} \tan \left (f x +e \right )}{8 f \,a^{4} \left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {15 b \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{8 f \,a^{3} \sqrt {\left (a +b \right ) b}}-\frac {95 b^{2} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{8 f \,a^{4} \sqrt {\left (a +b \right ) b}}-\frac {20 b^{3} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{f \,a^{5} \sqrt {\left (a +b \right ) b}}-\frac {2 b^{4} \left (\tan ^{3}\left (f x +e \right )\right )}{f \,a^{5} \left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {2 b^{4} \tan \left (f x +e \right )}{f \,a^{5} \left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {10 b^{4} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{f \,a^{6} \sqrt {\left (a +b \right ) b}}-\frac {27 \left (\tan ^{5}\left (f x +e \right )\right ) b}{8 f \,a^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}-\frac {3 \left (\tan ^{5}\left (f x +e \right )\right ) b^{2}}{f \,a^{5} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}-\frac {11 \left (\tan ^{5}\left (f x +e \right )\right )}{16 f \,a^{3} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}-\frac {6 \left (\tan ^{3}\left (f x +e \right )\right ) b}{f \,a^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}-\frac {6 \left (\tan ^{3}\left (f x +e \right )\right ) b^{2}}{f \,a^{5} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}-\frac {5 \left (\tan ^{3}\left (f x +e \right )\right )}{6 f \,a^{3} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}-\frac {5 \tan \left (f x +e \right )}{16 f \,a^{3} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}-\frac {21 \tan \left (f x +e \right ) b}{8 f \,a^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}-\frac {3 \tan \left (f x +e \right ) b^{2}}{f \,a^{5} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}+\frac {45 \arctan \left (\tan \left (f x +e \right )\right ) b}{8 f \,a^{4}}+\frac {15 \arctan \left (\tan \left (f x +e \right )\right ) b^{2}}{f \,a^{5}}+\frac {10 \arctan \left (\tan \left (f x +e \right )\right ) b^{3}}{f \,a^{6}}+\frac {5 \arctan \left (\tan \left (f x +e \right )\right )}{16 f \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 418, normalized size = 1.33 \[ -\frac {\frac {15 \, {\left (5 \, a^{2} b^{2} + 20 \, a b^{3} + 16 \, b^{4}\right )} \tan \left (f x + e\right )^{9} + 40 \, {\left (3 \, a^{3} b + 19 \, a^{2} b^{2} + 39 \, a b^{3} + 24 \, b^{4}\right )} \tan \left (f x + e\right )^{7} + {\left (33 \, a^{4} + 470 \, a^{3} b + 1910 \, a^{2} b^{2} + 2880 \, a b^{3} + 1440 \, b^{4}\right )} \tan \left (f x + e\right )^{5} + 40 \, {\left (a^{4} + 14 \, a^{3} b + 46 \, a^{2} b^{2} + 57 \, a b^{3} + 24 \, b^{4}\right )} \tan \left (f x + e\right )^{3} + 15 \, {\left (a^{4} + 14 \, a^{3} b + 41 \, a^{2} b^{2} + 44 \, a b^{3} + 16 \, b^{4}\right )} \tan \left (f x + e\right )}{a^{5} b^{2} \tan \left (f x + e\right )^{10} + {\left (2 \, a^{6} b + 5 \, a^{5} b^{2}\right )} \tan \left (f x + e\right )^{8} + a^{7} + 2 \, a^{6} b + a^{5} b^{2} + {\left (a^{7} + 8 \, a^{6} b + 10 \, a^{5} b^{2}\right )} \tan \left (f x + e\right )^{6} + {\left (3 \, a^{7} + 12 \, a^{6} b + 10 \, a^{5} b^{2}\right )} \tan \left (f x + e\right )^{4} + {\left (3 \, a^{7} + 8 \, a^{6} b + 5 \, a^{5} b^{2}\right )} \tan \left (f x + e\right )^{2}} - \frac {15 \, {\left (a^{3} + 18 \, a^{2} b + 48 \, a b^{2} + 32 \, b^{3}\right )} {\left (f x + e\right )}}{a^{6}} + \frac {30 \, {\left (3 \, a^{3} b + 19 \, a^{2} b^{2} + 32 \, a b^{3} + 16 \, b^{4}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {{\left (a + b\right )} b} a^{6}}}{48 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.72, size = 2117, normalized size = 6.74 \[ \text {result too large to display} \]
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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