Optimal. Leaf size=250 \[ \frac{a (A b-a B)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{a^3 A+3 a^2 b B-3 a A b^2-b^3 B}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{a^2 A+2 a b B-A b^2}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{\left (-6 a^2 A b^2+a^4 A+4 a^3 b B-4 a b^3 B+A b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{x \left (4 a^3 A b+6 a^2 b^2 B+a^4 (-B)-4 a A b^3-b^4 B\right )}{\left (a^2+b^2\right )^4} \]
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Rubi [A] time = 0.427556, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {3591, 3529, 3531, 3530} \[ \frac{a (A b-a B)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{a^3 A+3 a^2 b B-3 a A b^2-b^3 B}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{a^2 A+2 a b B-A b^2}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{\left (-6 a^2 A b^2+a^4 A+4 a^3 b B-4 a b^3 B+A b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{x \left (4 a^3 A b+6 a^2 b^2 B+a^4 (-B)-4 a A b^3-b^4 B\right )}{\left (a^2+b^2\right )^4} \]
Antiderivative was successfully verified.
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Rule 3591
Rule 3529
Rule 3531
Rule 3530
Rubi steps
\begin{align*} \int \frac{\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx &=\frac{a (A b-a B)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{\int \frac{A b-a B+(a A+b B) \tan (c+d x)}{(a+b \tan (c+d x))^3} \, dx}{a^2+b^2}\\ &=\frac{a (A b-a B)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a^2 A-A b^2+2 a b B}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{\int \frac{2 a A b-a^2 B+b^2 B+\left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx}{\left (a^2+b^2\right )^2}\\ &=\frac{a (A b-a B)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a^2 A-A b^2+2 a b B}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{a^3 A-3 a A b^2+3 a^2 b B-b^3 B}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\int \frac{3 a^2 A b-A b^3-a^3 B+3 a b^2 B+\left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^3}\\ &=\frac{\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) x}{\left (a^2+b^2\right )^4}+\frac{a (A b-a B)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a^2 A-A b^2+2 a b B}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{a^3 A-3 a A b^2+3 a^2 b B-b^3 B}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac{\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\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 (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) x}{\left (a^2+b^2\right )^4}-\frac{\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac{a (A b-a B)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a^2 A-A b^2+2 a b B}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{a^3 A-3 a A b^2+3 a^2 b B-b^3 B}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 1.13117, size = 248, normalized size = 0.99 \[ \frac{\frac{2 a (A b-a B)}{b \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{6 \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{3 \left (a^2 A+2 a b B-A b^2\right )}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{6 \left (-6 a^2 A b^2+a^4 A+4 a^3 b B-4 a b^3 B+A b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}+\frac{3 (A+i B) \log (-\tan (c+d x)+i)}{(a+i b)^4}+\frac{3 (A-i B) \log (\tan (c+d x)+i)}{(a-i b)^4}}{6 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 702, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.59517, size = 706, normalized size = 2.82 \begin{align*} -\frac{\frac{6 \,{\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B 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 a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\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{3 \,{\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A 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{2 \, B a^{6} - 11 \, A a^{5} b - 20 \, B a^{4} b^{2} + 14 \, A a^{3} b^{3} + 2 \, B a^{2} b^{4} + A a b^{5} - 6 \,{\left (A a^{3} b^{3} + 3 \, B a^{2} b^{4} - 3 \, A a b^{5} - B b^{6}\right )} \tan \left (d x + c\right )^{2} - 3 \,{\left (5 \, A a^{4} b^{2} + 14 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4} - 2 \, B a b^{5} - A b^{6}\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 )}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.10397, size = 1831, normalized size = 7.32 \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(-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 = 1.2896, size = 861, normalized size = 3.44 \begin{align*} -\frac{\frac{6 \,{\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B 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{3 \,{\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A 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 (A a^{4} b + 4 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3} - 4 \, B a b^{4} + A b^{5}\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{11 \, A a^{4} b^{4} \tan \left (d x + c\right )^{3} + 44 \, B a^{3} b^{5} \tan \left (d x + c\right )^{3} - 66 \, A a^{2} b^{6} \tan \left (d x + c\right )^{3} - 44 \, B a b^{7} \tan \left (d x + c\right )^{3} + 11 \, A b^{8} \tan \left (d x + c\right )^{3} + 39 \, A a^{5} b^{3} \tan \left (d x + c\right )^{2} + 150 \, B a^{4} b^{4} \tan \left (d x + c\right )^{2} - 210 \, A a^{3} b^{5} \tan \left (d x + c\right )^{2} - 120 \, B a^{2} b^{6} \tan \left (d x + c\right )^{2} + 15 \, A a b^{7} \tan \left (d x + c\right )^{2} - 6 \, B b^{8} \tan \left (d x + c\right )^{2} + 48 \, A a^{6} b^{2} \tan \left (d x + c\right ) + 174 \, B a^{5} b^{3} \tan \left (d x + c\right ) - 219 \, A a^{4} b^{4} \tan \left (d x + c\right ) - 96 \, B a^{3} b^{5} \tan \left (d x + c\right ) - 6 \, A a^{2} b^{6} \tan \left (d x + c\right ) - 6 \, B a b^{7} \tan \left (d x + c\right ) - 3 \, A b^{8} \tan \left (d x + c\right ) - 2 \, B a^{8} + 22 \, A a^{7} b + 62 \, B a^{6} b^{2} - 69 \, A a^{5} b^{3} - 26 \, B a^{4} b^{4} - 4 \, A a^{3} b^{5} - 2 \, B a^{2} b^{6} - A a b^{7}}{{\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}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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