Optimal. Leaf size=208 \[ -\frac {a^2 \tan ^2(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {4 a b \left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac {x \left (a^4-6 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4}-\frac {a^2 \left (2 a^4+7 a^2 b^2+17 b^4\right )}{3 b^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac {a^3 \left (a^2+4 b^2\right )}{3 b^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2} \]
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Rubi [A] time = 0.41, antiderivative size = 208, 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 = {3565, 3635, 3628, 3531, 3530} \[ -\frac {a^2 \tan ^2(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {a^3 \left (a^2+4 b^2\right )}{3 b^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2 b^2+2 a^4+17 b^4\right )}{3 b^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac {4 a b \left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac {x \left (-6 a^2 b^2+a^4+b^4\right )}{\left (a^2+b^2\right )^4} \]
Antiderivative was successfully verified.
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Rule 3530
Rule 3531
Rule 3565
Rule 3628
Rule 3635
Rubi steps
\begin {align*} \int \frac {\tan ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx &=-\frac {a^2 \tan ^2(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {\int \frac {\tan (c+d x) \left (2 a^2-3 a b \tan (c+d x)+\left (2 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 ^2(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a^3 \left (a^2+4 b^2\right )}{3 b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\int \frac {2 a^2 \left (a^2+4 b^2\right )-6 a b^3 \tan (c+d x)+\left (a^2+b^2\right ) \left (2 a^2+3 b^2\right ) \tan ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx}{3 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac {a^2 \tan ^2(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a^3 \left (a^2+4 b^2\right )}{3 b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {a^2 \left (2 a^4+7 a^2 b^2+17 b^4\right )}{3 b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \frac {3 a b^2 \left (a^2-3 b^2\right )-3 b^3 \left (3 a^2-b^2\right ) \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{3 b^2 \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 {a^2 \tan ^2(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a^3 \left (a^2+4 b^2\right )}{3 b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {a^2 \left (2 a^4+7 a^2 b^2+17 b^4\right )}{3 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 \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\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 (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {a^2 \tan ^2(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a^3 \left (a^2+4 b^2\right )}{3 b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {a^2 \left (2 a^4+7 a^2 b^2+17 b^4\right )}{3 b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] time = 2.53, size = 236, normalized size = 1.13 \[ -\frac {\frac {6 a b^3}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {6 b^3 \left (b^2-3 a^2\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {24 a b^3 (a-b) (a+b) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}+\frac {2 a^4}{b \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {3 i b^2 \log (-\tan (c+d x)+i)}{(a+i b)^4}-\frac {3 i b^2 \log (\tan (c+d x)+i)}{(b+i a)^4}+\frac {6 b \tan ^2(c+d x)}{(a+b \tan (c+d x))^3}+\frac {6 a \tan (c+d x)}{(a+b \tan (c+d x))^3}}{6 b^2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 510, normalized size = 2.45 \[ \frac {9 \, a^{6} b - 13 \, a^{4} b^{3} + {\left (a^{7} + 3 \, a^{5} b^{2} + 24 \, a^{3} b^{4} + 3 \, {\left (a^{4} b^{3} - 6 \, a^{2} b^{5} + b^{7}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} - 6 \, a^{5} b^{2} + a^{3} b^{4}\right )} d x - 3 \, {\left (a^{6} b - 15 \, a^{4} b^{3} + 6 \, a^{2} b^{5} - 3 \, {\left (a^{5} b^{2} - 6 \, a^{3} b^{4} + a b^{6}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 6 \, {\left (a^{6} b - a^{4} b^{3} + {\left (a^{3} b^{4} - a b^{6}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{5} b^{2} - a^{3} b^{4}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (a^{7} - 11 \, a^{5} b^{2} + 10 \, a^{3} b^{4} - 3 \, {\left (a^{6} b - 6 \, a^{4} b^{3} + a^{2} b^{5}\right )} d x\right )} \tan \left (d x + c\right )}{3 \, {\left ({\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b + 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 5.67, size = 409, normalized size = 1.97 \[ \frac {\frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {12 \, {\left (a^{3} b^{2} - a b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} - \frac {22 \, a^{3} b^{7} \tan \left (d x + c\right )^{3} - 22 \, a b^{9} \tan \left (d x + c\right )^{3} + 3 \, a^{8} b^{2} \tan \left (d x + c\right )^{2} + 12 \, a^{6} b^{4} \tan \left (d x + c\right )^{2} + 93 \, a^{4} b^{6} \tan \left (d x + c\right )^{2} - 48 \, a^{2} b^{8} \tan \left (d x + c\right )^{2} + 3 \, a^{9} b \tan \left (d x + c\right ) + 12 \, a^{7} b^{3} \tan \left (d x + c\right ) + 105 \, a^{5} b^{5} \tan \left (d x + c\right ) - 36 \, a^{3} b^{7} \tan \left (d x + c\right ) + a^{10} + 3 \, a^{8} b^{2} + 37 \, a^{6} b^{4} - 9 \, a^{4} b^{6}}{{\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 380, normalized size = 1.83 \[ -\frac {a^{4}}{3 d \,b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {a^{6}}{d \,b^{3} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {3 a^{4}}{d b \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {6 a^{2} b}{d \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {4 b \,a^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {4 a \,b^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {a^{5}}{d \,b^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {2 a^{3}}{d b \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\arctan \left (\tan \left (d x +c \right )\right ) a^{4}}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {6 \arctan \left (\tan \left (d x +c \right )\right ) a^{2} b^{2}}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {\arctan \left (\tan \left (d x +c \right )\right ) b^{4}}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {2 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{3} b}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {2 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a \,b^{3}}{d \left (a^{2}+b^{2}\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.68, size = 410, normalized size = 1.97 \[ \frac {\frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {12 \, {\left (a^{3} b - a b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {a^{8} + 2 \, a^{6} b^{2} + 13 \, a^{4} b^{4} + 3 \, {\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 6 \, a^{2} b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 10 \, a^{3} b^{5}\right )} \tan \left (d x + c\right )}{a^{9} b^{3} + 3 \, a^{7} b^{5} + 3 \, a^{5} b^{7} + a^{3} b^{9} + {\left (a^{6} b^{6} + 3 \, a^{4} b^{8} + 3 \, a^{2} b^{10} + b^{12}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{5} + 3 \, a^{5} b^{7} + 3 \, a^{3} b^{9} + a b^{11}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b^{4} + 3 \, a^{6} b^{6} + 3 \, a^{4} b^{8} + a^{2} b^{10}\right )} \tan \left (d x + c\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.64, size = 359, normalized size = 1.73 \[ \frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {4\,a\,b}{{\left (a^2+b^2\right )}^3}-\frac {8\,a\,b^3}{{\left (a^2+b^2\right )}^4}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}-\frac {\frac {a^2\,\left (a^6+2\,a^4\,b^2+13\,a^2\,b^4\right )}{3\,b^3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a^6+3\,a^4\,b^2+6\,a^2\,b^4\right )}{b\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\left (a^6+3\,a^4\,b^2+10\,a^2\,b^4\right )}{b^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4-a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm {i}+b^4\right )} \]
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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