Optimal. Leaf size=352 \[ \frac {\left (5 a^3-18 a^2 b+48 a b^2-160 b^3\right ) x}{16 a^6}+\frac {b^{7/2} \left (99 a^2+176 a b+80 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^6 (a+b)^{5/2} f}+\frac {\left (15 a^2-34 a b+80 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {5 (a-2 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (15 a^3-29 a^2 b+64 a b^2+120 b^3\right ) \tan (e+f x)}{48 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (5 a^4-8 a^3 b+17 a^2 b^2+116 a b^3+80 b^4\right ) \tan (e+f x)}{16 a^5 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )} \]
[Out]
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Rubi [A]
time = 0.34, antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps
used = 9, 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 {5 (a-2 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 {b^{7/2} \left (99 a^2+176 a b+80 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^6 f (a+b)^{5/2}}+\frac {\left (15 a^2-34 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 {x \left (5 a^3-18 a^2 b+48 a b^2-160 b^3\right )}{16 a^6}+\frac {b \left (15 a^3-29 a^2 b+64 a b^2+120 b^3\right ) \tan (e+f x)}{48 a^4 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}+\frac {b \left (5 a^4-8 a^3 b+17 a^2 b^2+116 a b^3+80 b^4\right ) \tan (e+f x)}{16 a^5 f (a+b)^2 \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 )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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 )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^4 \left (a+b+b x^2\right )^3} \, 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 )^2}-\frac {\text {Subst}\left (\int \frac {-5 a+b-9 b x^2}{\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 {5 (a-2 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 ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\text {Subst}\left (\int \frac {15 a^2+a b+10 b^2+35 (a-2 b) b 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 (15 a^2-34 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 {5 (a-2 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 ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-15 a^3-21 a^2 b+26 a b^2+80 b^3-5 b \left (15 a^2-34 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 (15 a^2-34 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 {5 (a-2 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 ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (15 a^3-29 a^2 b+64 a b^2+120 b^3\right ) \tan (e+f x)}{48 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-12 \left (5 a^4+2 a^3 b+a^2 b^2-48 a b^3-40 b^4\right )-12 b \left (15 a^3-29 a^2 b+64 a b^2+120 b^3\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 (15 a^2-34 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 {5 (a-2 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 ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (15 a^3-29 a^2 b+64 a b^2+120 b^3\right ) \tan (e+f x)}{48 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (5 a^4-8 a^3 b+17 a^2 b^2+116 a b^3+80 b^4\right ) \tan (e+f x)}{16 a^5 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {-24 \left (5 a^5-3 a^4 b+9 a^3 b^2-65 a^2 b^3-156 a b^4-80 b^5\right )-24 b \left (5 a^4-8 a^3 b+17 a^2 b^2+116 a b^3+80 b^4\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 (15 a^2-34 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 {5 (a-2 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 ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (15 a^3-29 a^2 b+64 a b^2+120 b^3\right ) \tan (e+f x)}{48 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (5 a^4-8 a^3 b+17 a^2 b^2+116 a b^3+80 b^4\right ) \tan (e+f x)}{16 a^5 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\left (b^4 \left (99 a^2+176 a b+80 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{8 a^6 (a+b)^2 f}+\frac {\left (5 a^3-18 a^2 b+48 a b^2-160 b^3\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{16 a^6 f}\\ &=\frac {\left (5 a^3-18 a^2 b+48 a b^2-160 b^3\right ) x}{16 a^6}+\frac {b^{7/2} \left (99 a^2+176 a b+80 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^6 (a+b)^{5/2} f}+\frac {\left (15 a^2-34 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 {5 (a-2 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 ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (15 a^3-29 a^2 b+64 a b^2+120 b^3\right ) \tan (e+f x)}{48 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (5 a^4-8 a^3 b+17 a^2 b^2+116 a b^3+80 b^4\right ) \tan (e+f x)}{16 a^5 (a+b)^2 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 = 6.69, size = 1770, normalized size = 5.03 \begin {gather*} \frac {\left (99 a^2+176 a b+80 b^2\right ) (a+2 b+a \cos (2 e+2 f x))^3 \sec ^6(e+f x) \left (-\frac {b^4 \text {ArcTan}\left (\sec (f x) \left (\frac {\cos (2 e)}{2 \sqrt {a+b} \sqrt {b \cos (4 e)-i b \sin (4 e)}}-\frac {i \sin (2 e)}{2 \sqrt {a+b} \sqrt {b \cos (4 e)-i b \sin (4 e)}}\right ) (-a \sin (f x)-2 b \sin (f x)+a \sin (2 e+f x))\right ) \cos (2 e)}{64 a^6 \sqrt {a+b} f \sqrt {b \cos (4 e)-i b \sin (4 e)}}+\frac {i b^4 \text {ArcTan}\left (\sec (f x) \left (\frac {\cos (2 e)}{2 \sqrt {a+b} \sqrt {b \cos (4 e)-i b \sin (4 e)}}-\frac {i \sin (2 e)}{2 \sqrt {a+b} \sqrt {b \cos (4 e)-i b \sin (4 e)}}\right ) (-a \sin (f x)-2 b \sin (f x)+a \sin (2 e+f x))\right ) \sin (2 e)}{64 a^6 \sqrt {a+b} f \sqrt {b \cos (4 e)-i b \sin (4 e)}}\right )}{(a+b)^2 \left (a+b \sec ^2(e+f x)\right )^3}+\frac {(a+2 b+a \cos (2 e+2 f x)) \sec (2 e) \sec ^6(e+f x) \left (720 a^7 f x \cos (2 e)+768 a^6 b f x \cos (2 e)+1296 a^5 b^2 f x \cos (2 e)-8352 a^4 b^3 f x \cos (2 e)-64128 a^3 b^4 f x \cos (2 e)-158976 a^2 b^5 f x \cos (2 e)-165888 a b^6 f x \cos (2 e)-61440 b^7 f x \cos (2 e)+480 a^7 f x \cos (2 f x)+192 a^6 b f x \cos (2 f x)+96 a^5 b^2 f x \cos (2 f x)-4608 a^4 b^3 f x \cos (2 f x)-41856 a^3 b^4 f x \cos (2 f x)-67584 a^2 b^5 f x \cos (2 f x)-30720 a b^6 f x \cos (2 f x)+480 a^7 f x \cos (4 e+2 f x)+192 a^6 b f x \cos (4 e+2 f x)+96 a^5 b^2 f x \cos (4 e+2 f x)-4608 a^4 b^3 f x \cos (4 e+2 f x)-41856 a^3 b^4 f x \cos (4 e+2 f x)-67584 a^2 b^5 f x \cos (4 e+2 f x)-30720 a b^6 f x \cos (4 e+2 f x)+120 a^7 f x \cos (2 e+4 f x)-192 a^6 b f x \cos (2 e+4 f x)+408 a^5 b^2 f x \cos (2 e+4 f x)-1968 a^4 b^3 f x \cos (2 e+4 f x)-6528 a^3 b^4 f x \cos (2 e+4 f x)-3840 a^2 b^5 f x \cos (2 e+4 f x)+120 a^7 f x \cos (6 e+4 f x)-192 a^6 b f x \cos (6 e+4 f x)+408 a^5 b^2 f x \cos (6 e+4 f x)-1968 a^4 b^3 f x \cos (6 e+4 f x)-6528 a^3 b^4 f x \cos (6 e+4 f x)-3840 a^2 b^5 f x \cos (6 e+4 f x)-6048 a^3 b^4 \sin (2 e)-21312 a^2 b^5 \sin (2 e)-29952 a b^6 \sin (2 e)-13824 b^7 \sin (2 e)+262 a^7 \sin (2 f x)+524 a^6 b \sin (2 f x)-26 a^5 b^2 \sin (2 f x)+1728 a^4 b^3 \sin (2 f x)+14976 a^3 b^4 \sin (2 f x)+28416 a^2 b^5 \sin (2 f x)+14592 a b^6 \sin (2 f x)+262 a^7 \sin (4 e+2 f x)+524 a^6 b \sin (4 e+2 f x)-26 a^5 b^2 \sin (4 e+2 f x)+1728 a^4 b^3 \sin (4 e+2 f x)+6912 a^3 b^4 \sin (4 e+2 f x)+5376 a^2 b^5 \sin (4 e+2 f x)+768 a b^6 \sin (4 e+2 f x)+238 a^7 \sin (2 e+4 f x)+304 a^6 b \sin (2 e+4 f x)-250 a^5 b^2 \sin (2 e+4 f x)+1556 a^4 b^3 \sin (2 e+4 f x)+5904 a^3 b^4 \sin (2 e+4 f x)+3744 a^2 b^5 \sin (2 e+4 f x)+238 a^7 \sin (6 e+4 f x)+304 a^6 b \sin (6 e+4 f x)-250 a^5 b^2 \sin (6 e+4 f x)+1556 a^4 b^3 \sin (6 e+4 f x)+3888 a^3 b^4 \sin (6 e+4 f x)+2016 a^2 b^5 \sin (6 e+4 f x)+87 a^7 \sin (4 e+6 f x)+46 a^6 b \sin (4 e+6 f x)-9 a^5 b^2 \sin (4 e+6 f x)+192 a^4 b^3 \sin (4 e+6 f x)+160 a^3 b^4 \sin (4 e+6 f x)+87 a^7 \sin (8 e+6 f x)+46 a^6 b \sin (8 e+6 f x)-9 a^5 b^2 \sin (8 e+6 f x)+192 a^4 b^3 \sin (8 e+6 f x)+160 a^3 b^4 \sin (8 e+6 f x)+13 a^7 \sin (6 e+8 f x)+16 a^6 b \sin (6 e+8 f x)-7 a^5 b^2 \sin (6 e+8 f x)-10 a^4 b^3 \sin (6 e+8 f x)+13 a^7 \sin (10 e+8 f x)+16 a^6 b \sin (10 e+8 f x)-7 a^5 b^2 \sin (10 e+8 f x)-10 a^4 b^3 \sin (10 e+8 f x)+a^7 \sin (8 e+10 f x)+2 a^6 b \sin (8 e+10 f x)+a^5 b^2 \sin (8 e+10 f x)+a^7 \sin (12 e+10 f x)+2 a^6 b \sin (12 e+10 f x)+a^5 b^2 \sin (12 e+10 f x)\right )}{12288 a^6 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^3} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
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Maple [A]
time = 0.24, size = 267, normalized size = 0.76
method | result | size |
derivativedivides | \(\frac {\frac {b^{4} \left (\frac {\frac {a b \left (19 a +16 b \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{8 a^{2}+16 a b +8 b^{2}}+\frac {\left (21 a +16 b \right ) a \tan \left (f x +e \right )}{8 a +8 b}}{\left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {\left (99 a^{2}+176 a b +80 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \sqrt {\left (a +b \right ) b}}\right )}{a^{6}}+\frac {\frac {\left (\frac {5}{16} a^{3}-\frac {9}{8} a^{2} b +3 a \,b^{2}\right ) \left (\tan ^{5}\left (f x +e \right )\right )+\left (6 a \,b^{2}+\frac {5}{6} a^{3}-3 a^{2} b \right ) \left (\tan ^{3}\left (f x +e \right )\right )+\left (-\frac {15}{8} a^{2} b +3 a \,b^{2}+\frac {11}{16} a^{3}\right ) \tan \left (f x +e \right )}{\left (\tan ^{2}\left (f x +e \right )+1\right )^{3}}+\frac {\left (5 a^{3}-18 a^{2} b +48 a \,b^{2}-160 b^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{16}}{a^{6}}}{f}\) | \(267\) |
default | \(\frac {\frac {b^{4} \left (\frac {\frac {a b \left (19 a +16 b \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{8 a^{2}+16 a b +8 b^{2}}+\frac {\left (21 a +16 b \right ) a \tan \left (f x +e \right )}{8 a +8 b}}{\left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {\left (99 a^{2}+176 a b +80 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \sqrt {\left (a +b \right ) b}}\right )}{a^{6}}+\frac {\frac {\left (\frac {5}{16} a^{3}-\frac {9}{8} a^{2} b +3 a \,b^{2}\right ) \left (\tan ^{5}\left (f x +e \right )\right )+\left (6 a \,b^{2}+\frac {5}{6} a^{3}-3 a^{2} b \right ) \left (\tan ^{3}\left (f x +e \right )\right )+\left (-\frac {15}{8} a^{2} b +3 a \,b^{2}+\frac {11}{16} a^{3}\right ) \tan \left (f x +e \right )}{\left (\tan ^{2}\left (f x +e \right )+1\right )^{3}}+\frac {\left (5 a^{3}-18 a^{2} b +48 a \,b^{2}-160 b^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{16}}{a^{6}}}{f}\) | \(267\) |
risch | \(\frac {5 x}{16 a^{3}}-\frac {9 x b}{8 a^{4}}+\frac {3 x \,b^{2}}{a^{5}}-\frac {10 x \,b^{3}}{a^{6}}+\frac {3 i {\mathrm e}^{4 i \left (f x +e \right )} b}{64 a^{4} f}-\frac {15 i {\mathrm e}^{2 i \left (f x +e \right )}}{128 f \,a^{3}}+\frac {3 i {\mathrm e}^{-2 i \left (f x +e \right )} b^{2}}{4 a^{5} f}-\frac {3 i {\mathrm e}^{-2 i \left (f x +e \right )} b}{8 a^{4} f}-\frac {3 i {\mathrm e}^{4 i \left (f x +e \right )}}{128 a^{3} f}-\frac {3 i {\mathrm e}^{2 i \left (f x +e \right )} b^{2}}{4 a^{5} f}+\frac {15 i {\mathrm e}^{-2 i \left (f x +e \right )}}{128 f \,a^{3}}-\frac {i {\mathrm e}^{6 i \left (f x +e \right )}}{384 a^{3} f}-\frac {3 i {\mathrm e}^{-4 i \left (f x +e \right )} b}{64 a^{4} f}+\frac {i b^{4} \left (21 a^{3} {\mathrm e}^{6 i \left (f x +e \right )}+64 a^{2} b \,{\mathrm e}^{6 i \left (f x +e \right )}+40 a \,b^{2} {\mathrm e}^{6 i \left (f x +e \right )}+63 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}+222 a^{2} b \,{\mathrm e}^{4 i \left (f x +e \right )}+312 a \,b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+144 b^{3} {\mathrm e}^{4 i \left (f x +e \right )}+63 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}+176 a^{2} b \,{\mathrm e}^{2 i \left (f x +e \right )}+104 a \,b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+21 a^{3}+18 a^{2} b \right )}{4 a^{6} \left (a +b \right )^{2} 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 )^{2}}+\frac {3 i {\mathrm e}^{-4 i \left (f x +e \right )}}{128 a^{3} f}+\frac {3 i {\mathrm e}^{2 i \left (f x +e \right )} b}{8 a^{4} f}+\frac {i {\mathrm e}^{-6 i \left (f x +e \right )}}{384 a^{3} f}+\frac {99 \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 )}{16 \left (a +b \right )^{3} f \,a^{4}}+\frac {11 \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 )^{3} f \,a^{5}}+\frac {5 \sqrt {-\left (a +b \right ) b}\, b^{5} \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 )^{3} f \,a^{6}}-\frac {99 \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 )}{16 \left (a +b \right )^{3} f \,a^{4}}-\frac {11 \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 )^{3} f \,a^{5}}-\frac {5 \sqrt {-\left (a +b \right ) b}\, b^{5} \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 )^{3} f \,a^{6}}\) | \(812\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 623, normalized size = 1.77 \begin {gather*} \frac {\frac {6 \, {\left (99 \, a^{2} b^{4} + 176 \, a b^{5} + 80 \, b^{6}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{{\left (a^{8} + 2 \, a^{7} b + a^{6} b^{2}\right )} \sqrt {{\left (a + b\right )} b}} + \frac {3 \, {\left (5 \, a^{4} b^{2} - 8 \, a^{3} b^{3} + 17 \, a^{2} b^{4} + 116 \, a b^{5} + 80 \, b^{6}\right )} \tan \left (f x + e\right )^{9} + 2 \, {\left (15 \, a^{5} b + 11 \, a^{4} b^{2} - 5 \, a^{3} b^{3} + 368 \, a^{2} b^{4} + 876 \, a b^{5} + 480 \, b^{6}\right )} \tan \left (f x + e\right )^{7} + {\left (15 \, a^{6} + 86 \, a^{5} b + 3 \, a^{4} b^{2} + 240 \, a^{3} b^{3} + 1982 \, a^{2} b^{4} + 3168 \, a b^{5} + 1440 \, b^{6}\right )} \tan \left (f x + e\right )^{5} + 2 \, {\left (20 \, a^{6} + 41 \, a^{5} b - 15 \, a^{4} b^{2} + 197 \, a^{3} b^{3} + 980 \, a^{2} b^{4} + 1236 \, a b^{5} + 480 \, b^{6}\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (11 \, a^{6} + 14 \, a^{5} b - 6 \, a^{4} b^{2} + 56 \, a^{3} b^{3} + 221 \, a^{2} b^{4} + 236 \, a b^{5} + 80 \, b^{6}\right )} \tan \left (f x + e\right )}{{\left (a^{7} b^{2} + 2 \, a^{6} b^{3} + a^{5} b^{4}\right )} \tan \left (f x + e\right )^{10} + a^{9} + 4 \, a^{8} b + 6 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + a^{5} b^{4} + {\left (2 \, a^{8} b + 9 \, a^{7} b^{2} + 12 \, a^{6} b^{3} + 5 \, a^{5} b^{4}\right )} \tan \left (f x + e\right )^{8} + {\left (a^{9} + 10 \, a^{8} b + 27 \, a^{7} b^{2} + 28 \, a^{6} b^{3} + 10 \, a^{5} b^{4}\right )} \tan \left (f x + e\right )^{6} + {\left (3 \, a^{9} + 18 \, a^{8} b + 37 \, a^{7} b^{2} + 32 \, a^{6} b^{3} + 10 \, a^{5} b^{4}\right )} \tan \left (f x + e\right )^{4} + {\left (3 \, a^{9} + 14 \, a^{8} b + 24 \, a^{7} b^{2} + 18 \, a^{6} b^{3} + 5 \, a^{5} b^{4}\right )} \tan \left (f x + e\right )^{2}} + \frac {3 \, {\left (5 \, a^{3} - 18 \, a^{2} b + 48 \, a b^{2} - 160 \, b^{3}\right )} {\left (f x + e\right )}}{a^{6}}}{48 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.11, size = 1330, normalized size = 3.78 \begin {gather*} \left [\frac {6 \, {\left (5 \, a^{7} - 8 \, a^{6} b + 17 \, a^{5} b^{2} - 82 \, a^{4} b^{3} - 272 \, a^{3} b^{4} - 160 \, a^{2} b^{5}\right )} f x \cos \left (f x + e\right )^{4} + 12 \, {\left (5 \, a^{6} b - 8 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 82 \, a^{3} b^{4} - 272 \, a^{2} b^{5} - 160 \, a b^{6}\right )} f x \cos \left (f x + e\right )^{2} + 6 \, {\left (5 \, a^{5} b^{2} - 8 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 82 \, a^{2} b^{5} - 272 \, a b^{6} - 160 \, b^{7}\right )} f x + 3 \, {\left (99 \, a^{2} b^{5} + 176 \, a b^{6} + 80 \, b^{7} + {\left (99 \, a^{4} b^{3} + 176 \, a^{3} b^{4} + 80 \, a^{2} b^{5}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (99 \, a^{3} b^{4} + 176 \, a^{2} b^{5} + 80 \, a b^{6}\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 ) + 2 \, {\left (8 \, {\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} \cos \left (f x + e\right )^{9} + 10 \, {\left (a^{7} - 3 \, a^{5} b^{2} - 2 \, a^{4} b^{3}\right )} \cos \left (f x + e\right )^{7} + {\left (15 \, a^{7} - 4 \, a^{6} b + 27 \, a^{5} b^{2} + 126 \, a^{4} b^{3} + 80 \, a^{3} b^{4}\right )} \cos \left (f x + e\right )^{5} + 2 \, {\left (15 \, a^{6} b - 19 \, a^{5} b^{2} + 43 \, a^{4} b^{3} + 266 \, a^{3} b^{4} + 180 \, a^{2} b^{5}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (5 \, a^{5} b^{2} - 8 \, a^{4} b^{3} + 17 \, a^{3} b^{4} + 116 \, a^{2} b^{5} + 80 \, a b^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{96 \, {\left ({\left (a^{10} + 2 \, a^{9} b + a^{8} b^{2}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{9} b + 2 \, a^{8} b^{2} + a^{7} b^{3}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{8} b^{2} + 2 \, a^{7} b^{3} + a^{6} b^{4}\right )} f\right )}}, \frac {3 \, {\left (5 \, a^{7} - 8 \, a^{6} b + 17 \, a^{5} b^{2} - 82 \, a^{4} b^{3} - 272 \, a^{3} b^{4} - 160 \, a^{2} b^{5}\right )} f x \cos \left (f x + e\right )^{4} + 6 \, {\left (5 \, a^{6} b - 8 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 82 \, a^{3} b^{4} - 272 \, a^{2} b^{5} - 160 \, a b^{6}\right )} f x \cos \left (f x + e\right )^{2} + 3 \, {\left (5 \, a^{5} b^{2} - 8 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 82 \, a^{2} b^{5} - 272 \, a b^{6} - 160 \, b^{7}\right )} f x - 3 \, {\left (99 \, a^{2} b^{5} + 176 \, a b^{6} + 80 \, b^{7} + {\left (99 \, a^{4} b^{3} + 176 \, a^{3} b^{4} + 80 \, a^{2} b^{5}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (99 \, a^{3} b^{4} + 176 \, a^{2} b^{5} + 80 \, a b^{6}\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^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} \cos \left (f x + e\right )^{9} + 10 \, {\left (a^{7} - 3 \, a^{5} b^{2} - 2 \, a^{4} b^{3}\right )} \cos \left (f x + e\right )^{7} + {\left (15 \, a^{7} - 4 \, a^{6} b + 27 \, a^{5} b^{2} + 126 \, a^{4} b^{3} + 80 \, a^{3} b^{4}\right )} \cos \left (f x + e\right )^{5} + 2 \, {\left (15 \, a^{6} b - 19 \, a^{5} b^{2} + 43 \, a^{4} b^{3} + 266 \, a^{3} b^{4} + 180 \, a^{2} b^{5}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (5 \, a^{5} b^{2} - 8 \, a^{4} b^{3} + 17 \, a^{3} b^{4} + 116 \, a^{2} b^{5} + 80 \, a b^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, {\left ({\left (a^{10} + 2 \, a^{9} b + a^{8} b^{2}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{9} b + 2 \, a^{8} b^{2} + a^{7} b^{3}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{8} b^{2} + 2 \, a^{7} b^{3} + a^{6} b^{4}\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 = 0.50, size = 350, normalized size = 0.99 \begin {gather*} \frac {\frac {6 \, {\left (99 \, a^{2} b^{4} + 176 \, a b^{5} + 80 \, b^{6}\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^{8} + 2 \, a^{7} b + a^{6} b^{2}\right )} \sqrt {a b + b^{2}}} + \frac {6 \, {\left (19 \, a b^{5} \tan \left (f x + e\right )^{3} + 16 \, b^{6} \tan \left (f x + e\right )^{3} + 21 \, a^{2} b^{4} \tan \left (f x + e\right ) + 37 \, a b^{5} \tan \left (f x + e\right ) + 16 \, b^{6} \tan \left (f x + e\right )\right )}}{{\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} {\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{2}} + \frac {3 \, {\left (5 \, a^{3} - 18 \, a^{2} b + 48 \, a b^{2} - 160 \, b^{3}\right )} {\left (f x + e\right )}}{a^{6}} + \frac {15 \, a^{2} \tan \left (f x + e\right )^{5} - 54 \, 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} - 144 \, a b \tan \left (f x + e\right )^{3} + 288 \, b^{2} \tan \left (f x + e\right )^{3} + 33 \, a^{2} \tan \left (f x + e\right ) - 90 \, 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.11, size = 2500, normalized size = 7.10 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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