Optimal. Leaf size=169 \[ -\frac {a^2}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {a b}{d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {4 a b \left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}-\frac {x \left (a^4-6 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4} \]
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Rubi [A] time = 0.28, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3542, 3529, 3531, 3530} \[ -\frac {a^2}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {a b}{d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {4 a b \left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}-\frac {x \left (-6 a^2 b^2+a^4+b^4\right )}{\left (a^2+b^2\right )^4} \]
Antiderivative was successfully verified.
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Rule 3529
Rule 3530
Rule 3531
Rule 3542
Rubi steps
\begin {align*} \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx &=-\frac {a^2}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {\int \frac {-a+b \tan (c+d x)}{(a+b \tan (c+d x))^3} \, dx}{a^2+b^2}\\ &=-\frac {a^2}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a b}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\int \frac {-a^2+b^2+2 a b \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx}{\left (a^2+b^2\right )^2}\\ &=-\frac {a^2}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a b}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \frac {-a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\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}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a b}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2-b^2\right )}{\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}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a b}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2-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] time = 6.22, size = 324, normalized size = 1.92 \[ \frac {b^2 \tan ^3(c+d x)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {\frac {3 a \tan (c+d x)}{d (a+b \tan (c+d x))^2}-\frac {3 a \left (\frac {2 b}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {4 a b \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^2}-2 a \left (\frac {4 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {b}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {2 b \left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}+\frac {\log (-\tan (c+d x)+i)}{(-b+i a)^3}-\frac {\log (\tan (c+d x)+i)}{(b+i a)^3}\right )+\frac {i \log (-\tan (c+d x)+i)}{(a+i b)^2}-\frac {i \log (\tan (c+d x)+i)}{(a-i b)^2}\right )}{2 d}}{3 a \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 531, normalized size = 3.14 \[ -\frac {6 \, a^{6} b - 15 \, a^{4} b^{3} + a^{2} b^{5} - {\left (a^{5} b^{2} - 15 \, a^{3} b^{4} + 6 \, a b^{6} - 3 \, {\left (a^{4} b^{3} - 6 \, a^{2} b^{5} + b^{7}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} - 6 \, a^{5} b^{2} + a^{3} b^{4}\right )} d x - 3 \, {\left (a^{6} b - 12 \, a^{4} b^{3} + 8 \, a^{2} b^{5} - b^{7} - 3 \, {\left (a^{5} b^{2} - 6 \, a^{3} b^{4} + a b^{6}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 6 \, {\left (a^{6} b - a^{4} b^{3} + {\left (a^{3} b^{4} - a b^{6}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{5} b^{2} - a^{3} b^{4}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (a^{7} - 8 \, a^{5} b^{2} + 12 \, a^{3} b^{4} - a b^{6} - 3 \, {\left (a^{6} b - 6 \, a^{4} b^{3} + a^{2} b^{5}\right )} d x\right )} \tan \left (d x + c\right )}{3 \, {\left ({\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b + 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 5.07, size = 376, normalized size = 2.22 \[ -\frac {\frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {12 \, {\left (a^{3} b^{2} - a b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} - \frac {22 \, a^{3} b^{5} \tan \left (d x + c\right )^{3} - 22 \, a b^{7} \tan \left (d x + c\right )^{3} + 75 \, a^{4} b^{4} \tan \left (d x + c\right )^{2} - 60 \, a^{2} b^{6} \tan \left (d x + c\right )^{2} - 3 \, b^{8} \tan \left (d x + c\right )^{2} + 87 \, a^{5} b^{3} \tan \left (d x + c\right ) - 48 \, a^{3} b^{5} \tan \left (d x + c\right ) - 3 \, a b^{7} \tan \left (d x + c\right ) - a^{8} + 31 \, a^{6} b^{2} - 13 \, a^{4} b^{4} - a^{2} b^{6}}{{\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 311, normalized size = 1.84 \[ -\frac {a^{2}}{3 b \left (a^{2}+b^{2}\right ) d \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {a b}{\left (a^{2}+b^{2}\right )^{2} d \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {3 a^{2} b}{d \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{3}}{d \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {4 b \,a^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {4 a \,b^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {\arctan \left (\tan \left (d x +c \right )\right ) a^{4}}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {6 \arctan \left (\tan \left (d x +c \right )\right ) a^{2} b^{2}}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {\arctan \left (\tan \left (d x +c \right )\right ) b^{4}}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {2 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{3} b}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {2 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a \,b^{3}}{d \left (a^{2}+b^{2}\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.83, size = 389, normalized size = 2.30 \[ -\frac {\frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {12 \, {\left (a^{3} b - a b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {a^{6} - 10 \, a^{4} b^{2} + a^{2} b^{4} - 3 \, {\left (3 \, a^{2} b^{4} - b^{6}\right )} \tan \left (d x + c\right )^{2} - 3 \, {\left (7 \, a^{3} b^{3} - a b^{5}\right )} \tan \left (d x + c\right )}{a^{9} b + 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} + a^{3} b^{7} + {\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{3} + 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b^{2} + 3 \, a^{6} b^{4} + 3 \, a^{4} b^{6} + a^{2} b^{8}\right )} \tan \left (d x + c\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.53, size = 326, normalized size = 1.93 \[ -\frac {\frac {a^6-10\,a^4\,b^2+a^2\,b^4}{3\,b\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (b^5-3\,a^2\,b^3\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (a\,b^4-7\,a^3\,b^2\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}-\frac {4\,a\,b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2-b^2\right )}{d\,{\left (a^2+b^2\right )}^4}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4-a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm {i}+b^4\right )} \]
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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