Optimal. Leaf size=250 \[ \frac {3 \left (a^2+10 a b+5 b^2\right ) x}{8 (a-b)^5}-\frac {3 \sqrt {b} \left (5 a^2+10 a b+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 \sqrt {a} (a-b)^5 f}-\frac {(5 a+3 b) \cos (e+f x) \sin (e+f x)}{8 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b (7 a+5 b) \tan (e+f x)}{8 (a-b)^3 f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {3 b (a+b) \tan (e+f x)}{2 (a-b)^4 f \left (a+b \tan ^2(e+f x)\right )} \]
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Rubi [A]
time = 0.23, antiderivative size = 250, 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 = {3744, 481, 541,
536, 209, 211} \begin {gather*} -\frac {3 \sqrt {b} \left (5 a^2+10 a b+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 \sqrt {a} f (a-b)^5}+\frac {3 x \left (a^2+10 a b+5 b^2\right )}{8 (a-b)^5}-\frac {3 b (a+b) \tan (e+f x)}{2 f (a-b)^4 \left (a+b \tan ^2(e+f x)\right )}-\frac {b (7 a+5 b) \tan (e+f x)}{8 f (a-b)^3 \left (a+b \tan ^2(e+f x)\right )^2}+\frac {\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}-\frac {(5 a+3 b) \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 211
Rule 481
Rule 536
Rule 541
Rule 3744
Rubi steps
\begin {align*} \int \frac {\sin ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^3 \left (a+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\cos ^3(e+f x) \sin (e+f x)}{4 (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {a+(-4 a-3 b) x^2}{\left (1+x^2\right )^2 \left (a+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{4 (a-b) f}\\ &=-\frac {(5 a+3 b) \cos (e+f x) \sin (e+f x)}{8 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {\text {Subst}\left (\int \frac {a (3 a+5 b)-5 b (5 a+3 b) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{8 (a-b)^2 f}\\ &=-\frac {(5 a+3 b) \cos (e+f x) \sin (e+f x)}{8 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b (7 a+5 b) \tan (e+f x)}{8 (a-b)^3 f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {\text {Subst}\left (\int \frac {12 a^2 (a+3 b)-12 a b (7 a+5 b) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{32 a (a-b)^3 f}\\ &=-\frac {(5 a+3 b) \cos (e+f x) \sin (e+f x)}{8 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b (7 a+5 b) \tan (e+f x)}{8 (a-b)^3 f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {3 b (a+b) \tan (e+f x)}{2 (a-b)^4 f \left (a+b \tan ^2(e+f x)\right )}+\frac {\text {Subst}\left (\int \frac {24 a^2 \left (a^2+6 a b+b^2\right )-96 a^2 b (a+b) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{64 a^2 (a-b)^4 f}\\ &=-\frac {(5 a+3 b) \cos (e+f x) \sin (e+f x)}{8 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b (7 a+5 b) \tan (e+f x)}{8 (a-b)^3 f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {3 b (a+b) \tan (e+f x)}{2 (a-b)^4 f \left (a+b \tan ^2(e+f x)\right )}-\frac {\left (3 b \left (5 a^2+10 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{8 (a-b)^5 f}+\frac {\left (3 \left (a^2+10 a b+5 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{8 (a-b)^5 f}\\ &=\frac {3 \left (a^2+10 a b+5 b^2\right ) x}{8 (a-b)^5}-\frac {3 \sqrt {b} \left (5 a^2+10 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 \sqrt {a} (a-b)^5 f}-\frac {(5 a+3 b) \cos (e+f x) \sin (e+f x)}{8 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b (7 a+5 b) \tan (e+f x)}{8 (a-b)^3 f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {3 b (a+b) \tan (e+f x)}{2 (a-b)^4 f \left (a+b \tan ^2(e+f x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 194, normalized size = 0.78 \begin {gather*} \frac {12 \left (a^2+10 a b+5 b^2\right ) (e+f x)-\frac {12 \sqrt {b} \left (5 a^2+10 a b+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{\sqrt {a}}-8 (a-b) (a+2 b) \sin (2 (e+f x))+\frac {16 a (a-b) b^2 \sin (2 (e+f x))}{(a+b+(a-b) \cos (2 (e+f x)))^2}-\frac {4 (a-b) b (9 a+5 b) \sin (2 (e+f x))}{a+b+(a-b) \cos (2 (e+f x))}+(a-b)^2 \sin (4 (e+f x))}{32 (a-b)^5 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 209, normalized size = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (-\frac {1}{4} a b +\frac {7}{8} b^{2}-\frac {5}{8} a^{2}\right ) \left (\tan ^{3}\left (f x +e \right )\right )+\left (-\frac {3}{8} a^{2}-\frac {3}{4} a b +\frac {9}{8} b^{2}\right ) \tan \left (f x +e \right )}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{2}}+\frac {3 \left (a^{2}+10 a b +5 b^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{8}}{\left (a -b \right )^{5}}-\frac {b \left (\frac {\left (\frac {7}{8} a^{2} b -\frac {1}{4} a \,b^{2}-\frac {5}{8} b^{3}\right ) \left (\tan ^{3}\left (f x +e \right )\right )+\frac {3 a \left (3 a^{2}-2 a b -b^{2}\right ) \tan \left (f x +e \right )}{8}}{\left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {3 \left (5 a^{2}+10 a b +b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{\left (a -b \right )^{5}}}{f}\) | \(209\) |
default | \(\frac {\frac {\frac {\left (-\frac {1}{4} a b +\frac {7}{8} b^{2}-\frac {5}{8} a^{2}\right ) \left (\tan ^{3}\left (f x +e \right )\right )+\left (-\frac {3}{8} a^{2}-\frac {3}{4} a b +\frac {9}{8} b^{2}\right ) \tan \left (f x +e \right )}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{2}}+\frac {3 \left (a^{2}+10 a b +5 b^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{8}}{\left (a -b \right )^{5}}-\frac {b \left (\frac {\left (\frac {7}{8} a^{2} b -\frac {1}{4} a \,b^{2}-\frac {5}{8} b^{3}\right ) \left (\tan ^{3}\left (f x +e \right )\right )+\frac {3 a \left (3 a^{2}-2 a b -b^{2}\right ) \tan \left (f x +e \right )}{8}}{\left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {3 \left (5 a^{2}+10 a b +b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{\left (a -b \right )^{5}}}{f}\) | \(209\) |
risch | \(\frac {3 x \,a^{2}}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \left (a -b \right )^{2}}+\frac {15 x a b}{4 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \left (a -b \right )^{2}}+\frac {15 x \,b^{2}}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \left (a -b \right )^{2}}-\frac {i {\mathrm e}^{-2 i \left (f x +e \right )} b}{4 \left (a -b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) f}-\frac {i {\mathrm e}^{4 i \left (f x +e \right )}}{64 \left (a -b \right )^{3} f}+\frac {i {\mathrm e}^{2 i \left (f x +e \right )} a}{8 \left (a^{2}-2 a b +b^{2}\right ) \left (a -b \right )^{2} f}-\frac {i {\mathrm e}^{-2 i \left (f x +e \right )} a}{8 \left (a -b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) f}-\frac {i \left (-9 a^{3} {\mathrm e}^{6 i \left (f x +e \right )}-9 a^{2} b \,{\mathrm e}^{6 i \left (f x +e \right )}+13 a \,b^{2} {\mathrm e}^{6 i \left (f x +e \right )}+5 b^{3} {\mathrm e}^{6 i \left (f x +e \right )}-27 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}-33 a^{2} b \,{\mathrm e}^{4 i \left (f x +e \right )}-37 a \,b^{2} {\mathrm e}^{4 i \left (f x +e \right )}-15 b^{3} {\mathrm e}^{4 i \left (f x +e \right )}-27 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}-11 a^{2} b \,{\mathrm e}^{2 i \left (f x +e \right )}+23 a \,b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+15 b^{3} {\mathrm e}^{2 i \left (f x +e \right )}-9 a^{3}+13 a^{2} b +a \,b^{2}-5 b^{3}\right ) b}{4 \left (-a^{3}+3 a^{2} b -3 a \,b^{2}+b^{3}\right ) \left (-a \,{\mathrm e}^{4 i \left (f x +e \right )}+b \,{\mathrm e}^{4 i \left (f x +e \right )}-2 a \,{\mathrm e}^{2 i \left (f x +e \right )}-2 b \,{\mathrm e}^{2 i \left (f x +e \right )}-a +b \right )^{2} f \left (a^{2}-2 a b +b^{2}\right )}+\frac {i {\mathrm e}^{2 i \left (f x +e \right )} b}{4 \left (a^{2}-2 a b +b^{2}\right ) \left (a -b \right )^{2} f}+\frac {i {\mathrm e}^{-4 i \left (f x +e \right )}}{64 \left (a^{2}-2 a b +b^{2}\right ) \left (a -b \right ) f}+\frac {15 a \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right )}{16 \left (a -b \right )^{5} f}+\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right ) b}{8 \left (a -b \right )^{5} f}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right ) b^{2}}{16 a \left (a -b \right )^{5} f}-\frac {15 a \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right )}{16 \left (a -b \right )^{5} f}-\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right ) b}{8 \left (a -b \right )^{5} f}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right ) b^{2}}{16 a \left (a -b \right )^{5} f}\) | \(948\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 470, normalized size = 1.88 \begin {gather*} \frac {\frac {3 \, {\left (a^{2} + 10 \, a b + 5 \, b^{2}\right )} {\left (f x + e\right )}}{a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}} - \frac {3 \, {\left (5 \, a^{2} b + 10 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} \sqrt {a b}} - \frac {12 \, {\left (a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{7} + {\left (19 \, a^{2} b + 34 \, a b^{2} + 19 \, b^{3}\right )} \tan \left (f x + e\right )^{5} + {\left (5 \, a^{3} + 31 \, a^{2} b + 31 \, a b^{2} + 5 \, b^{3}\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (a^{3} + 6 \, a^{2} b + a b^{2}\right )} \tan \left (f x + e\right )}{{\left (a^{4} b^{2} - 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}\right )} \tan \left (f x + e\right )^{8} + 2 \, {\left (a^{5} b - 3 \, a^{4} b^{2} + 2 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 3 \, a b^{5} + b^{6}\right )} \tan \left (f x + e\right )^{6} + a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4} + {\left (a^{6} - 9 \, a^{4} b^{2} + 16 \, a^{3} b^{3} - 9 \, a^{2} b^{4} + b^{6}\right )} \tan \left (f x + e\right )^{4} + 2 \, {\left (a^{6} - 3 \, a^{5} b + 2 \, a^{4} b^{2} + 2 \, a^{3} b^{3} - 3 \, a^{2} b^{4} + a b^{5}\right )} \tan \left (f x + e\right )^{2}}}{8 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 565 vs.
\(2 (241) = 482\).
time = 2.97, size = 1223, normalized size = 4.89 \begin {gather*} \left [\frac {12 \, {\left (a^{4} + 8 \, a^{3} b - 14 \, a^{2} b^{2} + 5 \, b^{4}\right )} f x \cos \left (f x + e\right )^{4} + 24 \, {\left (a^{3} b + 9 \, a^{2} b^{2} - 5 \, a b^{3} - 5 \, b^{4}\right )} f x \cos \left (f x + e\right )^{2} + 12 \, {\left (a^{2} b^{2} + 10 \, a b^{3} + 5 \, b^{4}\right )} f x - 3 \, {\left ({\left (5 \, a^{4} - 14 \, a^{2} b^{2} + 8 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{4} + 5 \, a^{2} b^{2} + 10 \, a b^{3} + b^{4} + 2 \, {\left (5 \, a^{3} b + 5 \, a^{2} b^{2} - 9 \, a b^{3} - b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {{\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left ({\left (a^{2} + a b\right )} \cos \left (f x + e\right )^{3} - a b \cos \left (f x + e\right )\right )} \sqrt {-\frac {b}{a}} \sin \left (f x + e\right ) + b^{2}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}}\right ) + 4 \, {\left (2 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{7} - {\left (5 \, a^{4} - 12 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} - 3 \, b^{4}\right )} \cos \left (f x + e\right )^{5} - {\left (19 \, a^{3} b - 21 \, a^{2} b^{2} - 15 \, a b^{3} + 17 \, b^{4}\right )} \cos \left (f x + e\right )^{3} - 12 \, {\left (a^{2} b^{2} - b^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{32 \, {\left ({\left (a^{7} - 7 \, a^{6} b + 21 \, a^{5} b^{2} - 35 \, a^{4} b^{3} + 35 \, a^{3} b^{4} - 21 \, a^{2} b^{5} + 7 \, a b^{6} - b^{7}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{6} b - 6 \, a^{5} b^{2} + 15 \, a^{4} b^{3} - 20 \, a^{3} b^{4} + 15 \, a^{2} b^{5} - 6 \, a b^{6} + b^{7}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{5} b^{2} - 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} - 10 \, a^{2} b^{5} + 5 \, a b^{6} - b^{7}\right )} f\right )}}, \frac {6 \, {\left (a^{4} + 8 \, a^{3} b - 14 \, a^{2} b^{2} + 5 \, b^{4}\right )} f x \cos \left (f x + e\right )^{4} + 12 \, {\left (a^{3} b + 9 \, a^{2} b^{2} - 5 \, a b^{3} - 5 \, b^{4}\right )} f x \cos \left (f x + e\right )^{2} + 6 \, {\left (a^{2} b^{2} + 10 \, a b^{3} + 5 \, b^{4}\right )} f x + 3 \, {\left ({\left (5 \, a^{4} - 14 \, a^{2} b^{2} + 8 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{4} + 5 \, a^{2} b^{2} + 10 \, a b^{3} + b^{4} + 2 \, {\left (5 \, a^{3} b + 5 \, a^{2} b^{2} - 9 \, a b^{3} - b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {b}{a}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) + 2 \, {\left (2 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{7} - {\left (5 \, a^{4} - 12 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} - 3 \, b^{4}\right )} \cos \left (f x + e\right )^{5} - {\left (19 \, a^{3} b - 21 \, a^{2} b^{2} - 15 \, a b^{3} + 17 \, b^{4}\right )} \cos \left (f x + e\right )^{3} - 12 \, {\left (a^{2} b^{2} - b^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{16 \, {\left ({\left (a^{7} - 7 \, a^{6} b + 21 \, a^{5} b^{2} - 35 \, a^{4} b^{3} + 35 \, a^{3} b^{4} - 21 \, a^{2} b^{5} + 7 \, a b^{6} - b^{7}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{6} b - 6 \, a^{5} b^{2} + 15 \, a^{4} b^{3} - 20 \, a^{3} b^{4} + 15 \, a^{2} b^{5} - 6 \, a b^{6} + b^{7}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{5} b^{2} - 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} - 10 \, a^{2} b^{5} + 5 \, a b^{6} - b^{7}\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 = 1.18, size = 399, normalized size = 1.60 \begin {gather*} \frac {\frac {3 \, {\left (a^{2} + 10 \, a b + 5 \, b^{2}\right )} {\left (f x + e\right )}}{a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}} - \frac {3 \, {\left (5 \, a^{2} b + 10 \, a b^{2} + b^{3}\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}}\right )\right )}}{{\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} \sqrt {a b}} - \frac {12 \, a b^{2} \tan \left (f x + e\right )^{7} + 12 \, b^{3} \tan \left (f x + e\right )^{7} + 19 \, a^{2} b \tan \left (f x + e\right )^{5} + 34 \, a b^{2} \tan \left (f x + e\right )^{5} + 19 \, b^{3} \tan \left (f x + e\right )^{5} + 5 \, a^{3} \tan \left (f x + e\right )^{3} + 31 \, a^{2} b \tan \left (f x + e\right )^{3} + 31 \, a b^{2} \tan \left (f x + e\right )^{3} + 5 \, b^{3} \tan \left (f x + e\right )^{3} + 3 \, a^{3} \tan \left (f x + e\right ) + 18 \, a^{2} b \tan \left (f x + e\right ) + 3 \, a b^{2} \tan \left (f x + e\right )}{{\left (b \tan \left (f x + e\right )^{4} + a \tan \left (f x + e\right )^{2} + b \tan \left (f x + e\right )^{2} + a\right )}^{2} {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )}}}{8 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 16.39, size = 2500, normalized size = 10.00 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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