Optimal. Leaf size=169 \[ -\frac{a^2}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{a b}{d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac{b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac{4 a b \left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}-\frac{x \left (-6 a^2 b^2+a^4+b^4\right )}{\left (a^2+b^2\right )^4} \]
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Rubi [A] time = 0.276632, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, 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 3542
Rule 3529
Rule 3531
Rule 3530
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*}
Mathematica [C] time = 6.19019, 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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Maple [A] time = 0.036, size = 311, normalized size = 1.8 \begin{align*} 2\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){a}^{3}b}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-2\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) a{b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{4}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+6\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{2}{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){b}^{4}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-{\frac{{a}^{2}}{3\,b \left ({a}^{2}+{b}^{2} \right ) d \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}}+{\frac{ab}{ \left ({a}^{2}+{b}^{2} \right ) ^{2}d \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{b{a}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-{\frac{{b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-4\,{\frac{b{a}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+4\,{\frac{{b}^{3}a\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.61005, size = 525, normalized size = 3.11 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.00467, size = 1130, normalized size = 6.69 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.57182, size = 508, normalized size = 3.01 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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