Optimal. Leaf size=278 \[ -\frac{b \left (13 a^4 b^2+12 a^2 b^4+a^6+4 b^6\right )}{a^4 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac{b \left (4 a^2 b^2+a^4+2 b^4\right )}{a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{b \left (3 a^2+4 b^2\right )}{3 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{4 b^3 \left (6 a^4 b^2+4 a^2 b^4+5 a^6+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 d \left (a^2+b^2\right )^4}-\frac{x \left (-6 a^2 b^2+a^4+b^4\right )}{\left (a^2+b^2\right )^4}-\frac{4 b \log (\sin (c+d x))}{a^5 d}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^3} \]
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Rubi [A] time = 0.853126, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3569, 3649, 3651, 3530, 3475} \[ -\frac{b \left (13 a^4 b^2+12 a^2 b^4+a^6+4 b^6\right )}{a^4 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac{b \left (4 a^2 b^2+a^4+2 b^4\right )}{a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{b \left (3 a^2+4 b^2\right )}{3 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{4 b^3 \left (6 a^4 b^2+4 a^2 b^4+5 a^6+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 d \left (a^2+b^2\right )^4}-\frac{x \left (-6 a^2 b^2+a^4+b^4\right )}{\left (a^2+b^2\right )^4}-\frac{4 b \log (\sin (c+d x))}{a^5 d}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^3} \]
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 ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx &=-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac{\int \frac{\cot (c+d x) \left (4 b+a \tan (c+d x)+4 b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^4} \, dx}{a}\\ &=-\frac{b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac{\int \frac{\cot (c+d x) \left (12 b \left (a^2+b^2\right )+3 a^3 \tan (c+d x)+3 b \left (3 a^2+4 b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 a^2 \left (a^2+b^2\right )}\\ &=-\frac{b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac{b \left (a^4+4 a^2 b^2+2 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{\int \frac{\cot (c+d x) \left (24 b \left (a^2+b^2\right )^2+6 a^3 \left (a^2-b^2\right ) \tan (c+d x)+12 b \left (a^4+4 a^2 b^2+2 b^4\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{6 a^3 \left (a^2+b^2\right )^2}\\ &=-\frac{b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac{b \left (a^4+4 a^2 b^2+2 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac{\int \frac{\cot (c+d x) \left (24 b \left (a^2+b^2\right )^3+6 a^5 \left (a^2-3 b^2\right ) \tan (c+d x)+6 b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{6 a^4 \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{b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac{b \left (a^4+4 a^2 b^2+2 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac{(4 b) \int \cot (c+d x) \, dx}{a^5}+\frac{\left (4 b^3 \left (5 a^6+6 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^5 \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 b \log (\sin (c+d x))}{a^5 d}+\frac{4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 \left (a^2+b^2\right )^4 d}-\frac{b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac{b \left (a^4+4 a^2 b^2+2 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 3.29757, size = 241, normalized size = 0.87 \[ -\frac{-\frac{b^6}{3 a^5 \left (a^2+b^2\right ) (a \cot (c+d x)+b)^3}+\frac{b^5 \left (3 a^2+2 b^2\right )}{a^5 \left (a^2+b^2\right )^2 (a \cot (c+d x)+b)^2}-\frac{b^4 \left (17 a^2 b^2+15 a^4+6 b^4\right )}{a^5 \left (a^2+b^2\right )^3 (a \cot (c+d x)+b)}-\frac{4 b^3 \left (6 a^4 b^2+4 a^2 b^4+5 a^6+b^6\right ) \log (a \cot (c+d x)+b)}{a^5 \left (a^2+b^2\right )^4}+\frac{\cot (c+d x)}{a^4}+\frac{i \log (-\cot (c+d x)+i)}{2 (a-i b)^4}-\frac{i \log (\cot (c+d x)+i)}{2 (a+i b)^4}}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.113, size = 478, normalized size = 1.7 \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{1}{d{a}^{4}\tan \left ( dx+c \right ) }}-4\,{\frac{b\ln \left ( \tan \left ( dx+c \right ) \right ) }{{a}^{5}d}}-{\frac{{b}^{3}}{3\,d \left ({a}^{2}+{b}^{2} \right ){a}^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}}-10\,{\frac{{b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-9\,{\frac{{b}^{5}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}{a}^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-3\,{\frac{{b}^{7}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}{a}^{4} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-2\,{\frac{{b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}a \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{5}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}{a}^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+20\,{\frac{{b}^{3}a\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+24\,{\frac{{b}^{5}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}a}}+16\,{\frac{{b}^{7}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}{a}^{3}}}+4\,{\frac{{b}^{9}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60198, size = 697, normalized size = 2.51 \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 (5 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{13} + 4 \, a^{11} b^{2} + 6 \, a^{9} b^{4} + 4 \, a^{7} b^{6} + a^{5} 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{3 \, a^{9} + 9 \, a^{7} b^{2} + 9 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + 3 \,{\left (a^{6} b^{3} + 13 \, a^{4} b^{5} + 12 \, a^{2} b^{7} + 4 \, b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \,{\left (3 \, a^{7} b^{2} + 31 \, a^{5} b^{4} + 30 \, a^{3} b^{6} + 10 \, a b^{8}\right )} \tan \left (d x + c\right )^{2} +{\left (9 \, a^{8} b + 64 \, a^{6} b^{3} + 65 \, a^{4} b^{5} + 22 \, a^{2} b^{7}\right )} \tan \left (d x + c\right )}{{\left (a^{10} b^{3} + 3 \, a^{8} b^{5} + 3 \, a^{6} b^{7} + a^{4} b^{9}\right )} \tan \left (d x + c\right )^{4} + 3 \,{\left (a^{11} b^{2} + 3 \, a^{9} b^{4} + 3 \, a^{7} b^{6} + a^{5} b^{8}\right )} \tan \left (d x + c\right )^{3} + 3 \,{\left (a^{12} b + 3 \, a^{10} b^{3} + 3 \, a^{8} b^{5} + a^{6} b^{7}\right )} \tan \left (d x + c\right )^{2} +{\left (a^{13} + 3 \, a^{11} b^{2} + 3 \, a^{9} b^{4} + a^{7} b^{6}\right )} \tan \left (d x + c\right )} + \frac{12 \, b \log \left (\tan \left (d x + c\right )\right )}{a^{5}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.7209, size = 2024, normalized size = 7.28 \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 [A] time = 1.36564, size = 678, normalized size = 2.44 \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 (5 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{13} b + 4 \, a^{11} b^{3} + 6 \, a^{9} b^{5} + 4 \, a^{7} b^{7} + a^{5} b^{9}} + \frac{110 \, a^{6} b^{6} \tan \left (d x + c\right )^{3} + 132 \, a^{4} b^{8} \tan \left (d x + c\right )^{3} + 88 \, a^{2} b^{10} \tan \left (d x + c\right )^{3} + 22 \, b^{12} \tan \left (d x + c\right )^{3} + 360 \, a^{7} b^{5} \tan \left (d x + c\right )^{2} + 453 \, a^{5} b^{7} \tan \left (d x + c\right )^{2} + 300 \, a^{3} b^{9} \tan \left (d x + c\right )^{2} + 75 \, a b^{11} \tan \left (d x + c\right )^{2} + 396 \, a^{8} b^{4} \tan \left (d x + c\right ) + 525 \, a^{6} b^{6} \tan \left (d x + c\right ) + 348 \, a^{4} b^{8} \tan \left (d x + c\right ) + 87 \, a^{2} b^{10} \tan \left (d x + c\right ) + 147 \, a^{9} b^{3} + 207 \, a^{7} b^{5} + 139 \, a^{5} b^{7} + 35 \, a^{3} b^{9}}{{\left (a^{13} + 4 \, a^{11} b^{2} + 6 \, a^{9} b^{4} + 4 \, a^{7} b^{6} + a^{5} b^{8}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{3}} + \frac{12 \, b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{5}} - \frac{3 \,{\left (4 \, b \tan \left (d x + c\right ) - a\right )}}{a^{5} \tan \left (d x + c\right )}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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