Optimal. Leaf size=315 \[ -\frac {\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}-\frac {4 a b \left (a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {4 a^3 \left (a^6+4 a^4 b^2+6 a^2 b^4+5 b^6\right ) \log (a+b \tan (c+d x))}{b^5 \left (a^2+b^2\right )^4 d}+\frac {\left (4 a^6+12 a^4 b^2+13 a^2 b^4+b^6\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right )^3 d}-\frac {a^2 \tan ^4(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (2 a^2+5 b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (a^4+3 a^2 b^2+4 b^4\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]
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Rubi [A]
time = 0.61, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3646, 3726,
3728, 3707, 3698, 31, 3556} \begin {gather*} -\frac {a^2 \tan ^4(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac {a^2 \left (2 a^2+5 b^2\right ) \tan ^3(c+d x)}{3 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {4 a b \left (a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^4}-\frac {x \left (a^4-6 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4}-\frac {2 a^2 \left (a^4+3 a^2 b^2+4 b^4\right ) \tan ^2(c+d x)}{b^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac {\left (4 a^6+12 a^4 b^2+13 a^2 b^4+b^6\right ) \tan (c+d x)}{b^4 d \left (a^2+b^2\right )^3}-\frac {4 a^3 \left (a^6+4 a^4 b^2+6 a^2 b^4+5 b^6\right ) \log (a+b \tan (c+d x))}{b^5 d \left (a^2+b^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 3556
Rule 3646
Rule 3698
Rule 3707
Rule 3726
Rule 3728
Rubi steps
\begin {align*} \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^4} \, dx &=-\frac {a^2 \tan ^4(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {\int \frac {\tan ^3(c+d x) \left (4 a^2-3 a b \tan (c+d x)+\left (4 a^2+3 b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 b \left (a^2+b^2\right )}\\ &=-\frac {a^2 \tan ^4(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (2 a^2+5 b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\int \frac {\tan ^2(c+d x) \left (6 a^2 \left (2 a^2+5 b^2\right )-12 a b^3 \tan (c+d x)+6 \left (2 a^4+4 a^2 b^2+b^4\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{6 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac {a^2 \tan ^4(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (2 a^2+5 b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (a^4+3 a^2 b^2+4 b^4\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \frac {\tan (c+d x) \left (24 a^2 \left (a^4+3 a^2 b^2+4 b^4\right )+6 a b^3 \left (a^2-3 b^2\right ) \tan (c+d x)+6 \left (4 a^6+12 a^4 b^2+13 a^2 b^4+b^6\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{6 b^3 \left (a^2+b^2\right )^3}\\ &=\frac {\left (4 a^6+12 a^4 b^2+13 a^2 b^4+b^6\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right )^3 d}-\frac {a^2 \tan ^4(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (2 a^2+5 b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (a^4+3 a^2 b^2+4 b^4\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \frac {-6 a \left (4 a^6+12 a^4 b^2+13 a^2 b^4+b^6\right )+6 b^5 \left (3 a^2-b^2\right ) \tan (c+d x)-24 a \left (a^2+b^2\right )^3 \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{6 b^4 \left (a^2+b^2\right )^3}\\ &=-\frac {\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}+\frac {\left (4 a^6+12 a^4 b^2+13 a^2 b^4+b^6\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right )^3 d}-\frac {a^2 \tan ^4(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (2 a^2+5 b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (a^4+3 a^2 b^2+4 b^4\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\left (4 a b \left (a^2-b^2\right )\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^4}-\frac {\left (4 a^3 \left (a^6+4 a^4 b^2+6 a^2 b^4+5 b^6\right )\right ) \int \frac {1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^4 \left (a^2+b^2\right )^4}\\ &=-\frac {\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}-\frac {4 a b \left (a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {\left (4 a^6+12 a^4 b^2+13 a^2 b^4+b^6\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right )^3 d}-\frac {a^2 \tan ^4(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (2 a^2+5 b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (a^4+3 a^2 b^2+4 b^4\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {\left (4 a^3 \left (a^6+4 a^4 b^2+6 a^2 b^4+5 b^6\right )\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^5 \left (a^2+b^2\right )^4 d}\\ &=-\frac {\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}-\frac {4 a b \left (a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {4 a^3 \left (a^6+4 a^4 b^2+6 a^2 b^4+5 b^6\right ) \log (a+b \tan (c+d x))}{b^5 \left (a^2+b^2\right )^4 d}+\frac {\left (4 a^6+12 a^4 b^2+13 a^2 b^4+b^6\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right )^3 d}-\frac {a^2 \tan ^4(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (2 a^2+5 b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (a^4+3 a^2 b^2+4 b^4\right ) \tan ^2(c+d x)}{b^3 \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 = 6.53, size = 1281, normalized size = 4.07 \begin {gather*} -\frac {4 i \left (a^{16} b^4-i a^{15} b^5+7 a^{14} b^6-7 i a^{13} b^7+21 a^{12} b^8-21 i a^{11} b^9+36 a^{10} b^{10}-36 i a^9 b^{11}+37 a^8 b^{12}-37 i a^7 b^{13}+21 a^6 b^{14}-21 i a^5 b^{15}+5 a^4 b^{16}-5 i a^3 b^{17}\right ) (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4}{(a-i b)^8 (a+i b)^7 b^9 d (a+b \tan (c+d x))^4}+\frac {4 i \left (a^9+4 a^7 b^2+6 a^5 b^4+5 a^3 b^6\right ) \text {ArcTan}(\tan (c+d x)) \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4}{b^5 \left (a^2+b^2\right )^4 d (a+b \tan (c+d x))^4}+\frac {4 a \log (\cos (c+d x)) \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4}{b^5 d (a+b \tan (c+d x))^4}-\frac {2 \left (a^9+4 a^7 b^2+6 a^5 b^4+5 a^3 b^6\right ) \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right ) \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4}{b^5 \left (a^2+b^2\right )^4 d (a+b \tan (c+d x))^4}+\frac {\sec ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (39 a^{10} b+171 a^8 b^3+276 a^6 b^5+180 a^4 b^7+45 a^2 b^9+9 b^{11}-9 a^7 b^4 (c+d x)+45 a^5 b^6 (c+d x)+45 a^3 b^8 (c+d x)-9 a b^{10} (c+d x)+12 a^{10} b \cos (2 (c+d x))+32 a^8 b^3 \cos (2 (c+d x))-16 a^6 b^5 \cos (2 (c+d x))-72 a^4 b^7 \cos (2 (c+d x))-48 a^2 b^9 \cos (2 (c+d x))-12 b^{11} \cos (2 (c+d x))-12 a^7 b^4 (c+d x) \cos (2 (c+d x))+72 a^5 b^6 (c+d x) \cos (2 (c+d x))-12 a^3 b^8 (c+d x) \cos (2 (c+d x))-27 a^{10} b \cos (4 (c+d x))-115 a^8 b^3 \cos (4 (c+d x))-196 a^6 b^5 \cos (4 (c+d x))-108 a^4 b^7 \cos (4 (c+d x))+3 a^2 b^9 \cos (4 (c+d x))+3 b^{11} \cos (4 (c+d x))-3 a^7 b^4 (c+d x) \cos (4 (c+d x))+27 a^5 b^6 (c+d x) \cos (4 (c+d x))-57 a^3 b^8 (c+d x) \cos (4 (c+d x))+9 a b^{10} (c+d x) \cos (4 (c+d x))+24 a^{11} \sin (2 (c+d x))+158 a^9 b^2 \sin (2 (c+d x))+396 a^7 b^4 \sin (2 (c+d x))+412 a^5 b^6 \sin (2 (c+d x))+168 a^3 b^8 \sin (2 (c+d x))+18 a b^{10} \sin (2 (c+d x))-18 a^6 b^5 (c+d x) \sin (2 (c+d x))+102 a^4 b^7 (c+d x) \sin (2 (c+d x))+18 a^2 b^9 (c+d x) \sin (2 (c+d x))-6 b^{11} (c+d x) \sin (2 (c+d x))+12 a^{11} \sin (4 (c+d x))+35 a^9 b^2 \sin (4 (c+d x))+18 a^7 b^4 \sin (4 (c+d x))-74 a^5 b^6 \sin (4 (c+d x))-78 a^3 b^8 \sin (4 (c+d x))-9 a b^{10} \sin (4 (c+d x))-9 a^6 b^5 (c+d x) \sin (4 (c+d x))+57 a^4 b^7 (c+d x) \sin (4 (c+d x))-27 a^2 b^9 (c+d x) \sin (4 (c+d x))+3 b^{11} (c+d x) \sin (4 (c+d x))\right )}{24 b^4 (-i a+b)^4 (i a+b)^4 d (a+b \tan (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 249, normalized size = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {\tan \left (d x +c \right )}{b^{4}}+\frac {\frac {\left (4 a^{3} b -4 a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {a^{6}}{3 b^{5} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {a^{4} \left (6 a^{4}+17 a^{2} b^{2}+15 b^{4}\right )}{b^{5} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {4 a^{3} \left (a^{6}+4 a^{4} b^{2}+6 a^{2} b^{4}+5 b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{5} \left (a^{2}+b^{2}\right )^{4}}+\frac {a^{5} \left (2 a^{2}+3 b^{2}\right )}{b^{5} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\) | \(249\) |
default | \(\frac {\frac {\tan \left (d x +c \right )}{b^{4}}+\frac {\frac {\left (4 a^{3} b -4 a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {a^{6}}{3 b^{5} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {a^{4} \left (6 a^{4}+17 a^{2} b^{2}+15 b^{4}\right )}{b^{5} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {4 a^{3} \left (a^{6}+4 a^{4} b^{2}+6 a^{2} b^{4}+5 b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{5} \left (a^{2}+b^{2}\right )^{4}}+\frac {a^{5} \left (2 a^{2}+3 b^{2}\right )}{b^{5} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\) | \(249\) |
norman | \(\frac {\frac {\tan ^{4}\left (d x +c \right )}{b d}-\frac {a^{3} \left (22 a^{7}+65 a^{5} b^{2}+64 a^{3} b^{4}+9 a \,b^{6}\right )}{3 d \,b^{5} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) a^{3} x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {b^{3} \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) x \left (\tan ^{3}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {a \left (12 a^{7}+35 a^{5} b^{2}+33 a^{3} b^{4}+6 a \,b^{6}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d \,b^{3} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {a^{2} \left (18 a^{7}+53 a^{5} b^{2}+51 a^{3} b^{4}+8 a \,b^{6}\right ) \tan \left (d x +c \right )}{d \,b^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {3 b \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) a^{2} x \tan \left (d x +c \right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {3 b^{2} \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) a x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {2 a b \left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {4 a^{3} \left (a^{6}+4 a^{4} b^{2}+6 a^{2} b^{4}+5 b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right ) d \,b^{5}}\) | \(619\) |
risch | \(\text {Expression too large to display}\) | \(1370\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 444, normalized size = 1.41 \begin {gather*} -\frac {\frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\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 {12 \, {\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 5 \, a^{3} b^{6}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} b^{5} + 4 \, a^{6} b^{7} + 6 \, a^{4} b^{9} + 4 \, a^{2} b^{11} + b^{13}} - \frac {6 \, {\left (a^{3} b - a b^{3}\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 {13 \, a^{10} + 38 \, a^{8} b^{2} + 37 \, a^{6} b^{4} + 3 \, {\left (6 \, a^{8} b^{2} + 17 \, a^{6} b^{4} + 15 \, a^{4} b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (10 \, a^{9} b + 29 \, a^{7} b^{3} + 27 \, a^{5} b^{5}\right )} \tan \left (d x + c\right )}{a^{9} b^{5} + 3 \, a^{7} b^{7} + 3 \, a^{5} b^{9} + a^{3} b^{11} + {\left (a^{6} b^{8} + 3 \, a^{4} b^{10} + 3 \, a^{2} b^{12} + b^{14}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{7} + 3 \, a^{5} b^{9} + 3 \, a^{3} b^{11} + a b^{13}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b^{6} + 3 \, a^{6} b^{8} + 3 \, a^{4} b^{10} + a^{2} b^{12}\right )} \tan \left (d x + c\right )} - \frac {3 \, \tan \left (d x + c\right )}{b^{4}}}{3 \, 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. 886 vs.
\(2 (311) = 622\).
time = 1.15, size = 886, normalized size = 2.81 \begin {gather*} -\frac {6 \, a^{10} b^{2} + 21 \, a^{8} b^{4} + 37 \, a^{6} b^{6} - 3 \, {\left (a^{8} b^{4} + 4 \, a^{6} b^{6} + 6 \, a^{4} b^{8} + 4 \, a^{2} b^{10} + b^{12}\right )} \tan \left (d x + c\right )^{4} - {\left (22 \, a^{9} b^{3} + 81 \, a^{7} b^{5} + 108 \, a^{5} b^{7} + 36 \, a^{3} b^{9} + 9 \, a b^{11} - 3 \, {\left (a^{4} b^{8} - 6 \, a^{2} b^{10} + b^{12}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{5} - 6 \, a^{5} b^{7} + a^{3} b^{9}\right )} d x - 3 \, {\left (10 \, a^{10} b^{2} + 34 \, a^{8} b^{4} + 40 \, a^{6} b^{6} - 3 \, a^{4} b^{8} + 3 \, a^{2} b^{10} - 3 \, {\left (a^{5} b^{7} - 6 \, a^{3} b^{9} + a b^{11}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 6 \, {\left (a^{12} + 4 \, a^{10} b^{2} + 6 \, a^{8} b^{4} + 5 \, a^{6} b^{6} + {\left (a^{9} b^{3} + 4 \, a^{7} b^{5} + 6 \, a^{5} b^{7} + 5 \, a^{3} b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{10} b^{2} + 4 \, a^{8} b^{4} + 6 \, a^{6} b^{6} + 5 \, a^{4} b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{11} b + 4 \, a^{9} b^{3} + 6 \, a^{7} b^{5} + 5 \, a^{5} b^{7}\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 ) - 6 \, {\left (a^{12} + 4 \, a^{10} b^{2} + 6 \, a^{8} b^{4} + 4 \, a^{6} b^{6} + a^{4} b^{8} + {\left (a^{9} b^{3} + 4 \, a^{7} b^{5} + 6 \, a^{5} b^{7} + 4 \, a^{3} b^{9} + a b^{11}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{10} b^{2} + 4 \, a^{8} b^{4} + 6 \, a^{6} b^{6} + 4 \, a^{4} b^{8} + a^{2} b^{10}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{11} b + 4 \, a^{9} b^{3} + 6 \, a^{7} b^{5} + 4 \, a^{5} b^{7} + a^{3} b^{9}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (4 \, a^{11} b + 10 \, a^{9} b^{3} + 4 \, a^{7} b^{5} - 23 \, a^{5} b^{7} + a^{3} b^{9} - 3 \, {\left (a^{6} b^{6} - 6 \, a^{4} b^{8} + a^{2} b^{10}\right )} d x\right )} \tan \left (d x + c\right )}{3 \, {\left ({\left (a^{8} b^{8} + 4 \, a^{6} b^{10} + 6 \, a^{4} b^{12} + 4 \, a^{2} b^{14} + b^{16}\right )} d \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{7} + 4 \, a^{7} b^{9} + 6 \, a^{5} b^{11} + 4 \, a^{3} b^{13} + a b^{15}\right )} d \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b^{6} + 4 \, a^{8} b^{8} + 6 \, a^{6} b^{10} + 4 \, a^{4} b^{12} + a^{2} b^{14}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} b^{5} + 4 \, a^{9} b^{7} + 6 \, a^{7} b^{9} + 4 \, a^{5} b^{11} + a^{3} b^{13}\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 [A]
time = 2.69, size = 472, normalized size = 1.50 \begin {gather*} -\frac {\frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\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^{3} b - a b^{3}\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 {12 \, {\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 5 \, a^{3} b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b^{5} + 4 \, a^{6} b^{7} + 6 \, a^{4} b^{9} + 4 \, a^{2} b^{11} + b^{13}} - \frac {22 \, a^{9} b^{3} \tan \left (d x + c\right )^{3} + 88 \, a^{7} b^{5} \tan \left (d x + c\right )^{3} + 132 \, a^{5} b^{7} \tan \left (d x + c\right )^{3} + 110 \, a^{3} b^{9} \tan \left (d x + c\right )^{3} + 48 \, a^{10} b^{2} \tan \left (d x + c\right )^{2} + 195 \, a^{8} b^{4} \tan \left (d x + c\right )^{2} + 300 \, a^{6} b^{6} \tan \left (d x + c\right )^{2} + 285 \, a^{4} b^{8} \tan \left (d x + c\right )^{2} + 36 \, a^{11} b \tan \left (d x + c\right ) + 147 \, a^{9} b^{3} \tan \left (d x + c\right ) + 228 \, a^{7} b^{5} \tan \left (d x + c\right ) + 249 \, a^{5} b^{7} \tan \left (d x + c\right ) + 9 \, a^{12} + 37 \, a^{10} b^{2} + 57 \, a^{8} b^{4} + 73 \, a^{6} b^{6}}{{\left (a^{8} b^{5} + 4 \, a^{6} b^{7} + 6 \, a^{4} b^{9} + 4 \, a^{2} b^{11} + b^{13}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}} - \frac {3 \, \tan \left (d x + c\right )}{b^{4}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.89, size = 387, normalized size = 1.23 \begin {gather*} \frac {\mathrm {tan}\left (c+d\,x\right )}{b^4\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{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 )}-\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (10\,a^9+29\,a^7\,b^2+27\,a^5\,b^4\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {13\,a^{10}+38\,a^8\,b^2+37\,a^6\,b^4}{3\,b\,\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 (6\,a^8\,b+17\,a^6\,b^3+15\,a^4\,b^5\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}}{d\,\left (a^3\,b^4+3\,a^2\,b^5\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^6\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^7\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {4\,a^3\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^6+4\,a^4\,b^2+6\,a^2\,b^4+5\,b^6\right )}{b^5\,d\,{\left (a^2+b^2\right )}^4}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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