Optimal. Leaf size=226 \[ \frac{b^2 \left (3 a^2 b^2+6 a^4+b^4\right )}{a^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{b^2 \left (3 a^2+b^2\right )}{2 a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac{b^2}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac{b^2 \left (5 a^4 b^2+4 a^2 b^4+10 a^6+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^4}-\frac{4 a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^4}+\frac{\log (\sin (c+d x))}{a^4 d} \]
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Rubi [A] time = 0.650748, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {3569, 3649, 3651, 3530, 3475} \[ \frac{b^2 \left (3 a^2 b^2+6 a^4+b^4\right )}{a^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{b^2 \left (3 a^2+b^2\right )}{2 a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac{b^2}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac{b^2 \left (5 a^4 b^2+4 a^2 b^4+10 a^6+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^4}-\frac{4 a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^4}+\frac{\log (\sin (c+d x))}{a^4 d} \]
Antiderivative was successfully verified.
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Rule 3569
Rule 3649
Rule 3651
Rule 3530
Rule 3475
Rubi steps
\begin{align*} \int \frac{\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx &=\frac{b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{\int \frac{\cot (c+d x) \left (3 \left (a^2+b^2\right )-3 a b \tan (c+d x)+3 b^2 \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 a \left (a^2+b^2\right )}\\ &=\frac{b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{b^2 \left (3 a^2+b^2\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{\int \frac{\cot (c+d x) \left (6 \left (a^2+b^2\right )^2-12 a^3 b \tan (c+d x)+6 b^2 \left (3 a^2+b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{6 a^2 \left (a^2+b^2\right )^2}\\ &=\frac{b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{b^2 \left (3 a^2+b^2\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\int \frac{\cot (c+d x) \left (6 \left (a^2+b^2\right )^3-6 a^3 b \left (3 a^2-b^2\right ) \tan (c+d x)+6 b^2 \left (6 a^4+3 a^2 b^2+b^4\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{6 a^3 \left (a^2+b^2\right )^3}\\ &=-\frac{4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}+\frac{b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{b^2 \left (3 a^2+b^2\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\int \cot (c+d x) \, dx}{a^4}-\frac{\left (b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right )\right ) \int \frac{b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^4 \left (a^2+b^2\right )^4}\\ &=-\frac{4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}+\frac{\log (\sin (c+d x))}{a^4 d}-\frac{b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^4 d}+\frac{b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{b^2 \left (3 a^2+b^2\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 1.8606, size = 243, normalized size = 1.08 \[ \frac{\frac{2 a b^2 \left (a^2+b^2\right )}{(a+b \tan (c+d x))^3}+\frac{6 \left (3 a^2 b^4+6 a^4 b^2+b^6\right )}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac{3 \left (3 a^2 b^2+b^4\right )}{(a+b \tan (c+d x))^2}+\frac{3 \left (2 \left (a^2+b^2\right )^4 \log (\tan (c+d x))-2 b^2 \left (5 a^4 b^2+4 a^2 b^4+10 a^6+b^6\right ) \log (a+b \tan (c+d x))-a^4 (a-i b)^4 \log (-\tan (c+d x)+i)-a^4 (a+i b)^4 \log (\tan (c+d x)+i)\right )}{a^2 \left (a^2+b^2\right )^2}}{6 a^2 d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.123, size = 460, normalized size = 2. \begin{align*} 3\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){a}^{2}{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){b}^{4}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){a}^{4}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-4\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}b}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+4\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+{\frac{\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{4}}}+{\frac{{b}^{2}}{3\, \left ({a}^{2}+{b}^{2} \right ) ad \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}}+{\frac{3\,{b}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{b}^{4}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2}{a}^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+6\,{\frac{a{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}+3\,{\frac{{b}^{4}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}a \left ( a+b\tan \left ( dx+c \right ) \right ) }}+{\frac{{b}^{6}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}{a}^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-10\,{\frac{{a}^{2}{b}^{2}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-5\,{\frac{\ln \left ( a+b\tan \left ( dx+c \right ) \right ){b}^{4}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-4\,{\frac{{b}^{6}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}{a}^{2}}}-{\frac{{b}^{8}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.59504, size = 599, normalized size = 2.65 \begin{align*} -\frac{\frac{24 \,{\left (a^{3} b - a b^{3}\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 (10 \, a^{6} b^{2} + 5 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{12} + 4 \, a^{10} b^{2} + 6 \, a^{8} b^{4} + 4 \, a^{6} b^{6} + a^{4} b^{8}} + \frac{3 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\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{47 \, a^{6} b^{2} + 34 \, a^{4} b^{4} + 11 \, a^{2} b^{6} + 6 \,{\left (6 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (27 \, a^{5} b^{3} + 16 \, a^{3} b^{5} + 5 \, a b^{7}\right )} \tan \left (d x + c\right )}{a^{12} + 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} + a^{6} b^{6} +{\left (a^{9} b^{3} + 3 \, a^{7} b^{5} + 3 \, a^{5} b^{7} + a^{3} b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \,{\left (a^{10} b^{2} + 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} + a^{4} b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}\right )} \tan \left (d x + c\right )} - \frac{6 \, \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.55259, size = 1716, normalized size = 7.59 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.38516, size = 643, normalized size = 2.85 \begin{align*} -\frac{\frac{24 \,{\left (a^{3} b - a b^{3}\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{3 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\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{6 \,{\left (10 \, a^{6} b^{3} + 5 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{12} b + 4 \, a^{10} b^{3} + 6 \, a^{8} b^{5} + 4 \, a^{6} b^{7} + a^{4} b^{9}} - \frac{110 \, a^{6} b^{5} \tan \left (d x + c\right )^{3} + 55 \, a^{4} b^{7} \tan \left (d x + c\right )^{3} + 44 \, a^{2} b^{9} \tan \left (d x + c\right )^{3} + 11 \, b^{11} \tan \left (d x + c\right )^{3} + 366 \, a^{7} b^{4} \tan \left (d x + c\right )^{2} + 219 \, a^{5} b^{6} \tan \left (d x + c\right )^{2} + 156 \, a^{3} b^{8} \tan \left (d x + c\right )^{2} + 39 \, a b^{10} \tan \left (d x + c\right )^{2} + 411 \, a^{8} b^{3} \tan \left (d x + c\right ) + 294 \, a^{6} b^{5} \tan \left (d x + c\right ) + 195 \, a^{4} b^{7} \tan \left (d x + c\right ) + 48 \, a^{2} b^{9} \tan \left (d x + c\right ) + 157 \, a^{9} b^{2} + 136 \, a^{7} b^{4} + 89 \, a^{5} b^{6} + 22 \, a^{3} b^{8}}{{\left (a^{12} + 4 \, a^{10} b^{2} + 6 \, a^{8} b^{4} + 4 \, a^{6} b^{6} + a^{4} b^{8}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{3}} - \frac{6 \, \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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