Optimal. Leaf size=230 \[ \frac {x}{a^3}-\frac {b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^3 (a+b)^{9/2} f}+\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \cot (e+f x)}{8 a^2 (a+b)^4 f}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{24 a^2 (a+b)^3 f}-\frac {b \cot ^3(e+f x)}{4 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )} \]
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Rubi [A]
time = 0.33, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4226, 2000,
483, 593, 597, 536, 209, 211} \begin {gather*} \frac {x}{a^3}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{24 a^2 f (a+b)^3}-\frac {b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 f (a+b)^2 \left (a+b \tan ^2(e+f x)+b\right )}-\frac {b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^3 f (a+b)^{9/2}}+\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \cot (e+f x)}{8 a^2 f (a+b)^4}-\frac {b \cot ^3(e+f x)}{4 a f (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 211
Rule 483
Rule 536
Rule 593
Rule 597
Rule 2000
Rule 4226
Rubi steps
\begin {align*} \int \frac {\cot ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^4 \left (1+x^2\right ) \left (a+b \left (1+x^2\right )\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x^4 \left (1+x^2\right ) \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {b \cot ^3(e+f x)}{4 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\text {Subst}\left (\int \frac {4 a-3 b-7 b x^2}{x^4 \left (1+x^2\right ) \left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{4 a (a+b) f}\\ &=-\frac {b \cot ^3(e+f x)}{4 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\text {Subst}\left (\int \frac {8 a^2-39 a b-12 b^2-5 b (11 a+4 b) x^2}{x^4 \left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{8 a^2 (a+b)^2 f}\\ &=-\frac {\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{24 a^2 (a+b)^3 f}-\frac {b \cot ^3(e+f x)}{4 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {3 \left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right )+3 b \left (8 a^2-39 a b-12 b^2\right ) x^2}{x^2 \left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{24 a^2 (a+b)^3 f}\\ &=\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \cot (e+f x)}{8 a^2 (a+b)^4 f}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{24 a^2 (a+b)^3 f}-\frac {b \cot ^3(e+f x)}{4 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\text {Subst}\left (\int \frac {3 \left (8 a^4+40 a^3 b+80 a^2 b^2+17 a b^3+4 b^4\right )+3 b \left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{24 a^2 (a+b)^4 f}\\ &=\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \cot (e+f x)}{8 a^2 (a+b)^4 f}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{24 a^2 (a+b)^3 f}-\frac {b \cot ^3(e+f x)}{4 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{a^3 f}-\frac {\left (b^3 \left (63 a^2+36 a b+8 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{8 a^3 (a+b)^4 f}\\ &=\frac {x}{a^3}-\frac {b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^3 (a+b)^{9/2} f}+\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \cot (e+f x)}{8 a^2 (a+b)^4 f}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{24 a^2 (a+b)^3 f}-\frac {b \cot ^3(e+f x)}{4 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 (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 = 7.83, size = 3340, normalized size = 14.52 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.27, size = 171, normalized size = 0.74
method | result | size |
derivativedivides | \(\frac {\frac {\arctan \left (\tan \left (f x +e \right )\right )}{a^{3}}-\frac {1}{3 \left (a +b \right )^{3} \tan \left (f x +e \right )^{3}}-\frac {-a -4 b}{\left (a +b \right )^{4} \tan \left (f x +e \right )}-\frac {b^{3} \left (\frac {\left (\frac {15}{8} a^{2} b +\frac {1}{2} a \,b^{2}\right ) \left (\tan ^{3}\left (f x +e \right )\right )+\frac {a \left (17 a^{2}+21 a b +4 b^{2}\right ) \tan \left (f x +e \right )}{8}}{\left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {\left (63 a^{2}+36 a b +8 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{8 \sqrt {\left (a +b \right ) b}}\right )}{a^{3} \left (a +b \right )^{4}}}{f}\) | \(171\) |
default | \(\frac {\frac {\arctan \left (\tan \left (f x +e \right )\right )}{a^{3}}-\frac {1}{3 \left (a +b \right )^{3} \tan \left (f x +e \right )^{3}}-\frac {-a -4 b}{\left (a +b \right )^{4} \tan \left (f x +e \right )}-\frac {b^{3} \left (\frac {\left (\frac {15}{8} a^{2} b +\frac {1}{2} a \,b^{2}\right ) \left (\tan ^{3}\left (f x +e \right )\right )+\frac {a \left (17 a^{2}+21 a b +4 b^{2}\right ) \tan \left (f x +e \right )}{8}}{\left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {\left (63 a^{2}+36 a b +8 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{8 \sqrt {\left (a +b \right ) b}}\right )}{a^{3} \left (a +b \right )^{4}}}{f}\) | \(171\) |
risch | \(\frac {x}{a^{3}}-\frac {i \left (-48 a^{6} {\mathrm e}^{4 i \left (f x +e \right )}-144 b^{6} {\mathrm e}^{4 i \left (f x +e \right )}-80 a^{6} {\mathrm e}^{2 i \left (f x +e \right )}-48 a^{6} {\mathrm e}^{12 i \left (f x +e \right )}-144 a^{6} {\mathrm e}^{10 i \left (f x +e \right )}+144 b^{6} {\mathrm e}^{10 i \left (f x +e \right )}-128 a^{6} {\mathrm e}^{8 i \left (f x +e \right )}-432 b^{6} {\mathrm e}^{8 i \left (f x +e \right )}-32 a^{6} {\mathrm e}^{6 i \left (f x +e \right )}+432 b^{6} {\mathrm e}^{6 i \left (f x +e \right )}-832 a^{4} b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+51 a^{3} b^{3} {\mathrm e}^{12 i \left (f x +e \right )}-1511 a^{3} b^{3} {\mathrm e}^{4 i \left (f x +e \right )}-104 a^{5} b -51 a^{3} b^{3}-18 a^{2} b^{4}-32 a^{6}-120 a^{5} b \,{\mathrm e}^{12 i \left (f x +e \right )}+132 a^{2} b^{4} {\mathrm e}^{12 i \left (f x +e \right )}+48 a \,b^{5} {\mathrm e}^{12 i \left (f x +e \right )}-672 a^{5} b \,{\mathrm e}^{10 i \left (f x +e \right )}-960 a^{4} b^{2} {\mathrm e}^{10 i \left (f x +e \right )}+66 a^{2} b^{4} {\mathrm e}^{10 i \left (f x +e \right )}-264 a \,b^{5} {\mathrm e}^{4 i \left (f x +e \right )}-480 a^{5} b \,{\mathrm e}^{2 i \left (f x +e \right )}+408 a \,b^{5} {\mathrm e}^{10 i \left (f x +e \right )}-440 a^{5} b \,{\mathrm e}^{8 i \left (f x +e \right )}-1152 a^{4} b^{2} {\mathrm e}^{8 i \left (f x +e \right )}-2073 a^{3} b^{3} {\mathrm e}^{8 i \left (f x +e \right )}-642 a^{2} b^{4} {\mathrm e}^{8 i \left (f x +e \right )}-1416 a \,b^{5} {\mathrm e}^{8 i \left (f x +e \right )}+384 a^{5} b \,{\mathrm e}^{6 i \left (f x +e \right )}+2048 a^{4} b^{2} {\mathrm e}^{6 i \left (f x +e \right )}-640 a^{4} b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+528 a^{2} b^{4} {\mathrm e}^{4 i \left (f x +e \right )}-294 a^{2} b^{4} {\mathrm e}^{2 i \left (f x +e \right )}-96 a \,b^{5} {\mathrm e}^{2 i \left (f x +e \right )}+3072 a^{3} b^{3} {\mathrm e}^{6 i \left (f x +e \right )}+228 a^{2} b^{4} {\mathrm e}^{6 i \left (f x +e \right )}+1320 a \,b^{5} {\mathrm e}^{6 i \left (f x +e \right )}-104 a^{5} b \,{\mathrm e}^{4 i \left (f x +e \right )}\right )}{12 a^{3} f \left (a +b \right )^{4} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3} \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 {63 \sqrt {-\left (a +b \right ) b}\, b^{2} \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 )^{5} f a}+\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 )^{5} f \,a^{2}}+\frac {\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 )}{2 \left (a +b \right )^{5} f \,a^{3}}-\frac {63 \sqrt {-\left (a +b \right ) b}\, b^{2} \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 )^{5} f a}-\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 )^{5} f \,a^{2}}-\frac {\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 )}{2 \left (a +b \right )^{5} f \,a^{3}}\) | \(1015\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 417, normalized size = 1.81 \begin {gather*} -\frac {\frac {3 \, {\left (63 \, a^{2} b^{3} + 36 \, 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^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} \sqrt {{\left (a + b\right )} b}} - \frac {3 \, {\left (8 \, a^{3} b^{2} + 32 \, a^{2} b^{3} - 15 \, a b^{4} - 4 \, b^{5}\right )} \tan \left (f x + e\right )^{6} - 8 \, a^{5} - 24 \, a^{4} b - 24 \, a^{3} b^{2} - 8 \, a^{2} b^{3} + {\left (48 \, a^{4} b + 232 \, a^{3} b^{2} + 133 \, a^{2} b^{3} - 63 \, a b^{4} - 12 \, b^{5}\right )} \tan \left (f x + e\right )^{4} + 8 \, {\left (3 \, a^{5} + 16 \, a^{4} b + 23 \, a^{3} b^{2} + 10 \, a^{2} b^{3}\right )} \tan \left (f x + e\right )^{2}}{{\left (a^{6} b^{2} + 4 \, a^{5} b^{3} + 6 \, a^{4} b^{4} + 4 \, a^{3} b^{5} + a^{2} b^{6}\right )} \tan \left (f x + e\right )^{7} + 2 \, {\left (a^{7} b + 5 \, a^{6} b^{2} + 10 \, a^{5} b^{3} + 10 \, a^{4} b^{4} + 5 \, a^{3} b^{5} + a^{2} b^{6}\right )} \tan \left (f x + e\right )^{5} + {\left (a^{8} + 6 \, a^{7} b + 15 \, a^{6} b^{2} + 20 \, a^{5} b^{3} + 15 \, a^{4} b^{4} + 6 \, a^{3} b^{5} + a^{2} b^{6}\right )} \tan \left (f x + e\right )^{3}} - \frac {24 \, {\left (f x + e\right )}}{a^{3}}}{24 \, 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. 800 vs.
\(2 (219) = 438\).
time = 4.07, size = 1691, normalized size = 7.35 \begin {gather*} \left [\frac {4 \, {\left (32 \, a^{6} + 104 \, a^{5} b + 51 \, a^{3} b^{3} + 18 \, a^{2} b^{4}\right )} \cos \left (f x + e\right )^{7} - 4 \, {\left (24 \, a^{6} + 32 \, a^{5} b - 208 \, a^{4} b^{2} + 102 \, a^{3} b^{3} - 9 \, a^{2} b^{4} - 12 \, a b^{5}\right )} \cos \left (f x + e\right )^{5} - 4 \, {\left (48 \, a^{5} b + 160 \, a^{4} b^{2} - 155 \, a^{3} b^{3} + 72 \, a^{2} b^{4} + 24 \, a b^{5}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left ({\left (63 \, a^{4} b^{2} + 36 \, a^{3} b^{3} + 8 \, a^{2} b^{4}\right )} \cos \left (f x + e\right )^{6} - 63 \, a^{2} b^{4} - 36 \, a b^{5} - 8 \, b^{6} - {\left (63 \, a^{4} b^{2} - 90 \, a^{3} b^{3} - 64 \, a^{2} b^{4} - 16 \, a b^{5}\right )} \cos \left (f x + e\right )^{4} - {\left (126 \, a^{3} b^{3} + 9 \, a^{2} b^{4} - 20 \, a b^{5} - 8 \, 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 ) \sin \left (f x + e\right ) - 12 \, {\left (8 \, a^{4} b^{2} + 32 \, a^{3} b^{3} - 15 \, a^{2} b^{4} - 4 \, a b^{5}\right )} \cos \left (f x + e\right ) + 96 \, {\left ({\left (a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right )} f x \cos \left (f x + e\right )^{6} - {\left (a^{6} + 2 \, a^{5} b - 2 \, a^{4} b^{2} - 8 \, a^{3} b^{3} - 7 \, a^{2} b^{4} - 2 \, a b^{5}\right )} f x \cos \left (f x + e\right )^{4} - {\left (2 \, a^{5} b + 7 \, a^{4} b^{2} + 8 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right )} f x \cos \left (f x + e\right )^{2} - {\left (a^{4} b^{2} + 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} + 4 \, a b^{5} + b^{6}\right )} f x\right )} \sin \left (f x + e\right )}{96 \, {\left ({\left (a^{9} + 4 \, a^{8} b + 6 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + a^{5} b^{4}\right )} f \cos \left (f x + e\right )^{6} - {\left (a^{9} + 2 \, a^{8} b - 2 \, a^{7} b^{2} - 8 \, a^{6} b^{3} - 7 \, a^{5} b^{4} - 2 \, a^{4} b^{5}\right )} f \cos \left (f x + e\right )^{4} - {\left (2 \, a^{8} b + 7 \, a^{7} b^{2} + 8 \, a^{6} b^{3} + 2 \, a^{5} b^{4} - 2 \, a^{4} b^{5} - a^{3} b^{6}\right )} f \cos \left (f x + e\right )^{2} - {\left (a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 4 \, a^{4} b^{5} + a^{3} b^{6}\right )} f\right )} \sin \left (f x + e\right )}, \frac {2 \, {\left (32 \, a^{6} + 104 \, a^{5} b + 51 \, a^{3} b^{3} + 18 \, a^{2} b^{4}\right )} \cos \left (f x + e\right )^{7} - 2 \, {\left (24 \, a^{6} + 32 \, a^{5} b - 208 \, a^{4} b^{2} + 102 \, a^{3} b^{3} - 9 \, a^{2} b^{4} - 12 \, a b^{5}\right )} \cos \left (f x + e\right )^{5} - 2 \, {\left (48 \, a^{5} b + 160 \, a^{4} b^{2} - 155 \, a^{3} b^{3} + 72 \, a^{2} b^{4} + 24 \, a b^{5}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left ({\left (63 \, a^{4} b^{2} + 36 \, a^{3} b^{3} + 8 \, a^{2} b^{4}\right )} \cos \left (f x + e\right )^{6} - 63 \, a^{2} b^{4} - 36 \, a b^{5} - 8 \, b^{6} - {\left (63 \, a^{4} b^{2} - 90 \, a^{3} b^{3} - 64 \, a^{2} b^{4} - 16 \, a b^{5}\right )} \cos \left (f x + e\right )^{4} - {\left (126 \, a^{3} b^{3} + 9 \, a^{2} b^{4} - 20 \, a b^{5} - 8 \, 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 ) \sin \left (f x + e\right ) - 6 \, {\left (8 \, a^{4} b^{2} + 32 \, a^{3} b^{3} - 15 \, a^{2} b^{4} - 4 \, a b^{5}\right )} \cos \left (f x + e\right ) + 48 \, {\left ({\left (a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right )} f x \cos \left (f x + e\right )^{6} - {\left (a^{6} + 2 \, a^{5} b - 2 \, a^{4} b^{2} - 8 \, a^{3} b^{3} - 7 \, a^{2} b^{4} - 2 \, a b^{5}\right )} f x \cos \left (f x + e\right )^{4} - {\left (2 \, a^{5} b + 7 \, a^{4} b^{2} + 8 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right )} f x \cos \left (f x + e\right )^{2} - {\left (a^{4} b^{2} + 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} + 4 \, a b^{5} + b^{6}\right )} f x\right )} \sin \left (f x + e\right )}{48 \, {\left ({\left (a^{9} + 4 \, a^{8} b + 6 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + a^{5} b^{4}\right )} f \cos \left (f x + e\right )^{6} - {\left (a^{9} + 2 \, a^{8} b - 2 \, a^{7} b^{2} - 8 \, a^{6} b^{3} - 7 \, a^{5} b^{4} - 2 \, a^{4} b^{5}\right )} f \cos \left (f x + e\right )^{4} - {\left (2 \, a^{8} b + 7 \, a^{7} b^{2} + 8 \, a^{6} b^{3} + 2 \, a^{5} b^{4} - 2 \, a^{4} b^{5} - a^{3} b^{6}\right )} f \cos \left (f x + e\right )^{2} - {\left (a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 4 \, a^{4} b^{5} + a^{3} b^{6}\right )} f\right )} \sin \left (f x + e\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 = 0.60, size = 302, normalized size = 1.31 \begin {gather*} -\frac {\frac {3 \, {\left (63 \, a^{2} b^{3} + 36 \, 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^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} \sqrt {a b + b^{2}}} + \frac {3 \, {\left (15 \, a b^{4} \tan \left (f x + e\right )^{3} + 4 \, b^{5} \tan \left (f x + e\right )^{3} + 17 \, a^{2} b^{3} \tan \left (f x + e\right ) + 21 \, a b^{4} \tan \left (f x + e\right ) + 4 \, b^{5} \tan \left (f x + e\right )\right )}}{{\left (a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right )} {\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{2}} - \frac {24 \, {\left (f x + e\right )}}{a^{3}} - \frac {8 \, {\left (3 \, a \tan \left (f x + e\right )^{2} + 12 \, b \tan \left (f x + e\right )^{2} - a - b\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \tan \left (f x + e\right )^{3}}}{24 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 11.98, size = 2500, normalized size = 10.87 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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