Optimal. Leaf size=283 \[ -\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}-\frac {a x \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3}-\frac {a \left (6 a^4+11 a^2 b^2+3 b^4\right ) \tan (c+d x)}{b^4 d \left (a^2+b^2\right )^2}+\frac {a^4 \left (6 a^4+17 a^2 b^2+15 b^4\right ) \log (a+b \tan (c+d x))}{b^5 d \left (a^2+b^2\right )^3}+\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b^3 d \left (a^2+b^2\right )^2} \]
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Rubi [A] time = 0.80, antiderivative size = 283, 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 = {3565, 3645, 3647, 3626, 3617, 31, 3475} \[ -\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {\left (11 a^2 b^2+6 a^4+b^4\right ) \tan ^2(c+d x)}{2 b^3 d \left (a^2+b^2\right )^2}-\frac {a \left (11 a^2 b^2+6 a^4+3 b^4\right ) \tan (c+d x)}{b^4 d \left (a^2+b^2\right )^2}+\frac {a^4 \left (17 a^2 b^2+6 a^4+15 b^4\right ) \log (a+b \tan (c+d x))}{b^5 d \left (a^2+b^2\right )^3}-\frac {b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}-\frac {a x \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3475
Rule 3565
Rule 3617
Rule 3626
Rule 3645
Rule 3647
Rubi steps
\begin {align*} \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=-\frac {a^2 \tan ^4(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {\tan ^3(c+d x) \left (4 a^2-2 a b \tan (c+d x)+2 \left (2 a^2+b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=-\frac {a^2 \tan ^4(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\tan ^2(c+d x) \left (12 a^2 \left (a^2+2 b^2\right )-4 a b^3 \tan (c+d x)+2 \left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^4(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\tan (c+d x) \left (-4 a \left (6 a^4+11 a^2 b^2+b^4\right )+4 b^3 \left (a^2-b^2\right ) \tan (c+d x)-4 a \left (6 a^4+11 a^2 b^2+3 b^4\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{4 b^3 \left (a^2+b^2\right )^2}\\ &=-\frac {a \left (6 a^4+11 a^2 b^2+3 b^4\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^4(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {4 a^2 \left (6 a^4+11 a^2 b^2+3 b^4\right )+8 a b^5 \tan (c+d x)+4 \left (6 a^2-b^2\right ) \left (a^2+b^2\right )^2 \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{4 b^4 \left (a^2+b^2\right )^2}\\ &=-\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {a \left (6 a^4+11 a^2 b^2+3 b^4\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^4(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left (b \left (3 a^2-b^2\right )\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^3}+\frac {\left (a^4 \left (6 a^4+17 a^2 b^2+15 b^4\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 )^3}\\ &=-\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}-\frac {a \left (6 a^4+11 a^2 b^2+3 b^4\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^4(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left (a^4 \left (6 a^4+17 a^2 b^2+15 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^5 \left (a^2+b^2\right )^3 d}\\ &=-\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a^4 \left (6 a^4+17 a^2 b^2+15 b^4\right ) \log (a+b \tan (c+d x))}{b^5 \left (a^2+b^2\right )^3 d}-\frac {a \left (6 a^4+11 a^2 b^2+3 b^4\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^4(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] time = 4.51, size = 243, normalized size = 0.86 \[ \frac {-\frac {a^4 \left (6 a^2+5 b^2\right )}{b^4 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {2 a^4 \left (6 a^4+17 a^2 b^2+15 b^4\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3}+\frac {4 a^3 \left (6 a^4+11 a^2 b^2+4 b^4\right )}{b^4 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {\tan ^4(c+d x)}{(a+b \tan (c+d x))^2}-\frac {4 a \tan ^3(c+d x)}{b (a+b \tan (c+d x))^2}+\frac {i b \log (-\tan (c+d x)+i)}{(a+i b)^3}-\frac {b \log (\tan (c+d x)+i)}{(b+i a)^3}}{2 b d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 628, normalized size = 2.22 \[ \frac {6 \, a^{8} b^{2} + 14 \, a^{6} b^{4} + 3 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{4} - 4 \, {\left (a^{7} b^{3} + 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )^{3} - 2 \, {\left (a^{5} b^{5} - 3 \, a^{3} b^{7}\right )} d x - {\left (18 \, a^{8} b^{2} + 45 \, a^{6} b^{4} + 30 \, a^{4} b^{6} + 8 \, a^{2} b^{8} - b^{10} + 2 \, {\left (a^{3} b^{7} - 3 \, a b^{9}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left (6 \, a^{10} + 17 \, a^{8} b^{2} + 15 \, a^{6} b^{4} + {\left (6 \, a^{8} b^{2} + 17 \, a^{6} b^{4} + 15 \, a^{4} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{9} b + 17 \, a^{7} b^{3} + 15 \, a^{5} b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (6 \, a^{10} + 17 \, a^{8} b^{2} + 15 \, a^{6} b^{4} + 3 \, a^{4} b^{6} - a^{2} b^{8} + {\left (6 \, a^{8} b^{2} + 17 \, a^{6} b^{4} + 15 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{9} b + 17 \, a^{7} b^{3} + 15 \, a^{5} b^{5} + 3 \, a^{3} b^{7} - a b^{9}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (6 \, a^{9} b + 11 \, a^{7} b^{3} - a b^{9} + 2 \, {\left (a^{4} b^{6} - 3 \, a^{2} b^{8}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{7} + 3 \, a^{4} b^{9} + 3 \, a^{2} b^{11} + b^{13}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{6} + 3 \, a^{5} b^{8} + 3 \, a^{3} b^{10} + a b^{12}\right )} d \tan \left (d x + c\right ) + {\left (a^{8} b^{5} + 3 \, a^{6} b^{7} + 3 \, a^{4} b^{9} + a^{2} b^{11}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 24.83, size = 345, normalized size = 1.22 \[ -\frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (6 \, a^{8} + 17 \, a^{6} b^{2} + 15 \, a^{4} b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}} + \frac {18 \, a^{8} b^{2} \tan \left (d x + c\right )^{2} + 51 \, a^{6} b^{4} \tan \left (d x + c\right )^{2} + 45 \, a^{4} b^{6} \tan \left (d x + c\right )^{2} + 28 \, a^{9} b \tan \left (d x + c\right ) + 82 \, a^{7} b^{3} \tan \left (d x + c\right ) + 78 \, a^{5} b^{5} \tan \left (d x + c\right ) + 11 \, a^{10} + 33 \, a^{8} b^{2} + 34 \, a^{6} b^{4}}{{\left (a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}} - \frac {b^{3} \tan \left (d x + c\right )^{2} - 6 \, a b^{2} \tan \left (d x + c\right )}{b^{6}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 328, normalized size = 1.16 \[ \frac {\tan ^{2}\left (d x +c \right )}{2 b^{3} d}-\frac {3 a \tan \left (d x +c \right )}{b^{4} d}-\frac {a^{6}}{2 d \,b^{5} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {6 a^{8} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,b^{5} \left (a^{2}+b^{2}\right )^{3}}+\frac {17 a^{6} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,b^{3} \left (a^{2}+b^{2}\right )^{3}}+\frac {15 a^{4} \ln \left (a +b \tan \left (d x +c \right )\right )}{d b \left (a^{2}+b^{2}\right )^{3}}+\frac {4 a^{7}}{d \,b^{5} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {6 a^{5}}{d \,b^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b}{2 d \left (a^{2}+b^{2}\right )^{3}}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{3}}{2 d \left (a^{2}+b^{2}\right )^{3}}-\frac {\arctan \left (\tan \left (d x +c \right )\right ) a^{3}}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 \arctan \left (\tan \left (d x +c \right )\right ) a \,b^{2}}{d \left (a^{2}+b^{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 308, normalized size = 1.09 \[ -\frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (6 \, a^{8} + 17 \, a^{6} b^{2} + 15 \, a^{4} b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {7 \, a^{8} + 11 \, a^{6} b^{2} + 4 \, {\left (2 \, a^{7} b + 3 \, a^{5} b^{3}\right )} \tan \left (d x + c\right )}{a^{6} b^{5} + 2 \, a^{4} b^{7} + a^{2} b^{9} + {\left (a^{4} b^{7} + 2 \, a^{2} b^{9} + b^{11}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{6} + 2 \, a^{3} b^{8} + a b^{10}\right )} \tan \left (d x + c\right )} - \frac {b \tan \left (d x + c\right )^{2} - 6 \, a \tan \left (d x + c\right )}{b^{4}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.35, size = 284, normalized size = 1.00 \[ \frac {\frac {2\,\mathrm {tan}\left (c+d\,x\right )\,\left (2\,a^7+3\,a^5\,b^2\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {7\,a^8+11\,a^6\,b^2}{2\,b\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2\,b^4+2\,a\,b^5\,\mathrm {tan}\left (c+d\,x\right )+b^6\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}+\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {b}{{\left (a^2+b^2\right )}^2}-\frac {1}{b^3}+\frac {6\,a^2}{b^5}-\frac {4\,a^2\,b}{{\left (a^2+b^2\right )}^3}\right )}{d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,b^3\,d}-\frac {3\,a\,\mathrm {tan}\left (c+d\,x\right )}{b^4\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )} \]
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 \[ \text {Exception raised: AttributeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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