Optimal. Leaf size=189 \[ -\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (a^2+7 b^2\right )}{6 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]
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Rubi [A]
time = 0.23, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3646, 3709,
3610, 3612, 3611} \begin {gather*} -\frac {a^2 \left (a^2+7 b^2\right )}{6 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac {a \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {4 a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^4}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 3610
Rule 3611
Rule 3612
Rule 3646
Rule 3709
Rubi steps
\begin {align*} \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^4} \, dx &=-\frac {a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {\int \frac {a^2-3 a b \tan (c+d x)+\left (a^2+3 b^2\right ) \tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx}{3 b \left (a^2+b^2\right )}\\ &=-\frac {a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (a^2+7 b^2\right )}{6 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\int \frac {-6 a b^2-3 b \left (a^2-b^2\right ) \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx}{3 b \left (a^2+b^2\right )^2}\\ &=-\frac {a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (a^2+7 b^2\right )}{6 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \frac {-3 b^2 \left (3 a^2-b^2\right )-3 a b \left (a^2-3 b^2\right ) \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{3 b \left (a^2+b^2\right )^3}\\ &=-\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}-\frac {a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (a^2+7 b^2\right )}{6 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^4}\\ &=-\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (a^2+7 b^2\right )}{6 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 6.24, size = 387, normalized size = 2.05 \begin {gather*} -\frac {\tan (c+d x)}{2 b d (a+b \tan (c+d x))^3}-\frac {\frac {a}{3 b d (a+b \tan (c+d x))^3}+\frac {2 b \left (-\frac {a \left (-\frac {i \log (i-\tan (c+d x))}{2 (a+i b)^4}+\frac {i \log (i+\tan (c+d x))}{2 (a-i b)^4}+\frac {4 a (a-b) b (a+b) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}-\frac {b}{3 \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac {a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}\right )}{b}+\frac {-\frac {\log (i-\tan (c+d x))}{2 (i a-b)^3}+\frac {\log (i+\tan (c+d x))}{2 (i a+b)^3}+\frac {b \left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}-\frac {b}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {2 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}}{b}\right )}{d}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 206, normalized size = 1.09
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 a^{3} b +4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {a \left (a^{2}-3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a^{2} \left (a^{2}+3 b^{2}\right )}{2 \left (a^{2}+b^{2}\right )^{2} b^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{3}}{3 b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}}{d}\) | \(206\) |
default | \(\frac {\frac {\frac {\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 a^{3} b +4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {a \left (a^{2}-3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a^{2} \left (a^{2}+3 b^{2}\right )}{2 \left (a^{2}+b^{2}\right )^{2} b^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{3}}{3 b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}}{d}\) | \(206\) |
norman | \(\frac {\frac {\left (a^{6}+6 a^{4} b^{2}-3 a^{2} b^{4}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 a d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {a \left (-3 a^{4} b +a^{2} b^{3}\right )}{3 d b \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {b \left (a^{6}+8 a^{4} b^{2}-9 a^{2} b^{4}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{6 a^{2} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {4 \left (a^{2}-b^{2}\right ) a^{4} b x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {12 b^{2} \left (a^{2}-b^{2}\right ) a^{3} x \tan \left (d x +c \right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {12 b^{3} \left (a^{2}-b^{2}\right ) a^{2} x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {4 b^{4} \left (a^{2}-b^{2}\right ) a x \left (\tan ^{3}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\) | \(540\) |
risch | \(-\frac {i x}{4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 a^{2} b^{2}-b^{4}}-\frac {2 i a^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}+\frac {12 i a^{2} b^{2} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {2 i b^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {2 i a^{4} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {12 i a^{2} b^{2} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {2 i b^{4} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {2 i a \left (-18 i a \,b^{4}+9 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-18 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+9 a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}-6 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+9 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+9 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-3 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}+6 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+24 a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-18 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+26 i a^{3} b^{2}-3 i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}+13 a^{4} b -22 a^{2} b^{3}+9 b^{5}\right )}{3 \left (-i a +b \right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{3} d \left (i a +b \right )^{4}}+\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {6 a^{2} b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{4}}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\) | \(790\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 402 vs.
\(2 (185) = 370\).
time = 0.50, size = 402, normalized size = 2.13 \begin {gather*} -\frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {a^{7} + 14 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, {\left (a^{3} b^{4} - 3 \, a b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{6} b + 8 \, a^{4} b^{3} - 9 \, a^{2} b^{5}\right )} \tan \left (d x + c\right )}{a^{9} b^{2} + 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} + a^{3} b^{8} + {\left (a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{4} + 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} + a b^{10}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b^{3} + 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} + a^{2} b^{9}\right )} \tan \left (d x + c\right )}}{6 \, d} \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. 526 vs.
\(2 (185) = 370\).
time = 1.06, size = 526, normalized size = 2.78 \begin {gather*} \frac {3 \, a^{7} - 30 \, a^{5} b^{2} + 11 \, a^{3} b^{4} + {\left (a^{6} b + 18 \, a^{4} b^{3} - 27 \, a^{2} b^{5} - 24 \, {\left (a^{3} b^{4} - a b^{6}\right )} d x\right )} \tan \left (d x + c\right )^{3} - 24 \, {\left (a^{6} b - a^{4} b^{3}\right )} d x + 3 \, {\left (a^{7} + 16 \, a^{5} b^{2} - 23 \, a^{3} b^{4} + 6 \, a b^{6} - 24 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{7} - 6 \, a^{5} b^{2} + a^{3} b^{4} + {\left (a^{4} b^{3} - 6 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{5} b^{2} - 6 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{6} b - 6 \, a^{4} b^{3} + a^{2} b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 3 \, {\left (9 \, a^{6} b - 26 \, a^{4} b^{3} + 9 \, a^{2} b^{5} - 24 \, {\left (a^{5} b^{2} - a^{3} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{6 \, {\left ({\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b + 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 400 vs.
\(2 (185) = 370\).
time = 1.10, size = 400, normalized size = 2.12 \begin {gather*} -\frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{4} b - 6 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} + \frac {11 \, a^{4} b^{5} \tan \left (d x + c\right )^{3} - 66 \, a^{2} b^{7} \tan \left (d x + c\right )^{3} + 11 \, b^{9} \tan \left (d x + c\right )^{3} + 39 \, a^{5} b^{4} \tan \left (d x + c\right )^{2} - 210 \, a^{3} b^{6} \tan \left (d x + c\right )^{2} + 15 \, a b^{8} \tan \left (d x + c\right )^{2} + 3 \, a^{8} b \tan \left (d x + c\right ) + 60 \, a^{6} b^{3} \tan \left (d x + c\right ) - 201 \, a^{4} b^{5} \tan \left (d x + c\right ) + 6 \, a^{2} b^{7} \tan \left (d x + c\right ) + a^{9} + 26 \, a^{7} b^{2} - 63 \, a^{5} b^{4}}{{\left (a^{8} b^{2} + 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.64, size = 359, normalized size = 1.90 \begin {gather*} \frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {1}{{\left (a^2+b^2\right )}^2}-\frac {8\,b^2}{{\left (a^2+b^2\right )}^3}+\frac {8\,b^4}{{\left (a^2+b^2\right )}^4}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4+a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2-a\,b^3\,4{}\mathrm {i}+b^4\right )}-\frac {\frac {a\,\left (a^6+14\,a^4\,b^2-11\,a^2\,b^4\right )}{6\,b^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (3\,a\,b^4-a^3\,b^2\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (a^6+8\,a^4\,b^2-9\,a^2\,b^4\right )}{2\,b\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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