Optimal. Leaf size=331 \[ \frac{a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{a \left (a^2 b B-3 a^3 C-7 a b^2 C+5 b^3 B\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{\left (-6 a^2 b^2 C+a^3 b B-3 a^4 C+3 a b^3 B-b^4 C\right ) \tan (c+d x)}{b^3 d \left (a^2+b^2\right )^2}+\frac{a^2 \left (3 a^2 b^3 B-9 a^3 b^2 C+a^4 b B-3 a^5 C-10 a b^4 C+6 b^5 B\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^3}+\frac{\left (3 a^2 b B+a^3 (-C)+3 a b^2 C-b^3 B\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}+\frac{x \left (3 a^2 b C+a^3 B-3 a b^2 B-b^3 C\right )}{\left (a^2+b^2\right )^3} \]
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Rubi [A] time = 0.860573, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3632, 3605, 3645, 3647, 3626, 3617, 31, 3475} \[ \frac{a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{a \left (a^2 b B-3 a^3 C-7 a b^2 C+5 b^3 B\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{\left (-6 a^2 b^2 C+a^3 b B-3 a^4 C+3 a b^3 B-b^4 C\right ) \tan (c+d x)}{b^3 d \left (a^2+b^2\right )^2}+\frac{a^2 \left (3 a^2 b^3 B-9 a^3 b^2 C+a^4 b B-3 a^5 C-10 a b^4 C+6 b^5 B\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^3}+\frac{\left (3 a^2 b B+a^3 (-C)+3 a b^2 C-b^3 B\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}+\frac{x \left (3 a^2 b C+a^3 B-3 a b^2 B-b^3 C\right )}{\left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 3632
Rule 3605
Rule 3645
Rule 3647
Rule 3626
Rule 3617
Rule 31
Rule 3475
Rubi steps
\begin{align*} \int \frac{\tan ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx &=\int \frac{\tan ^4(c+d x) (B+C \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\\ &=\frac{a (b B-a C) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{\int \frac{\tan ^2(c+d x) \left (-3 a (b B-a C)+2 b (b B-a C) \tan (c+d x)-\left (a b B-3 a^2 C-2 b^2 C\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=\frac{a (b B-a C) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a \left (a^2 b B+5 b^3 B-3 a^3 C-7 a b^2 C\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{\int \frac{\tan (c+d x) \left (-2 a \left (a^2 b B+5 b^3 B-3 a^3 C-7 a b^2 C\right )-2 b^2 \left (a^2 B-b^2 B+2 a b C\right ) \tan (c+d x)-2 \left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac{\left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac{a (b B-a C) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a \left (a^2 b B+5 b^3 B-3 a^3 C-7 a b^2 C\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{\int \frac{2 a \left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right )-2 b^3 \left (2 a b B-a^2 C+b^2 C\right ) \tan (c+d x)+2 \left (a^2+b^2\right )^2 (b B-3 a C) \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{2 b^3 \left (a^2+b^2\right )^2}\\ &=\frac{\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) x}{\left (a^2+b^2\right )^3}-\frac{\left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac{a (b B-a C) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a \left (a^2 b B+5 b^3 B-3 a^3 C-7 a b^2 C\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^3}+\frac{\left (a^2 \left (a^4 b B+3 a^2 b^3 B+6 b^5 B-3 a^5 C-9 a^3 b^2 C-10 a b^4 C\right )\right ) \int \frac{1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^3 \left (a^2+b^2\right )^3}\\ &=\frac{\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) x}{\left (a^2+b^2\right )^3}+\frac{\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}-\frac{\left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac{a (b B-a C) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a \left (a^2 b B+5 b^3 B-3 a^3 C-7 a b^2 C\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{\left (a^2 \left (a^4 b B+3 a^2 b^3 B+6 b^5 B-3 a^5 C-9 a^3 b^2 C-10 a b^4 C\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^4 \left (a^2+b^2\right )^3 d}\\ &=\frac{\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) x}{\left (a^2+b^2\right )^3}+\frac{\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac{a^2 \left (a^4 b B+3 a^2 b^3 B+6 b^5 B-3 a^5 C-9 a^3 b^2 C-10 a b^4 C\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3 d}-\frac{\left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac{a (b B-a C) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a \left (a^2 b B+5 b^3 B-3 a^3 C-7 a b^2 C\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 6.67839, size = 1146, normalized size = 3.46 \[ \frac{(a C-b B) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) (B+C \tan (c+d x)) a^4}{2 (a-i b)^2 (a+i b)^2 b^2 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \left (2 C \sin (c+d x) a^5-b B \sin (c+d x) a^4+5 b^2 C \sin (c+d x) a^3-4 b^3 B \sin (c+d x) a^2\right ) (B+C \tan (c+d x))}{(a-i b)^2 (a+i b)^2 b^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{C \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \tan (c+d x) (B+C \tan (c+d x))}{b^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{\left (B a^3+3 b C a^2-3 b^2 B a-b^3 C\right ) (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{(a-i b)^3 (a+i b)^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{\left (6 a^2 B b^{13}+6 i a^3 B b^{12}-10 a^3 C b^{12}+15 a^4 B b^{11}-10 i a^4 C b^{11}+15 i a^5 B b^{10}-29 a^5 C b^{10}+13 a^6 B b^9-29 i a^6 C b^9+13 i a^7 B b^8-31 a^7 C b^8+5 a^8 B b^7-31 i a^8 C b^7+5 i a^9 B b^6-15 a^9 C b^6+a^{10} B b^5-15 i a^{10} C b^5+i a^{11} B b^4-3 a^{11} C b^4-3 i a^{12} C b^3\right ) (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{(a-i b)^6 (a+i b)^5 b^7 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}-\frac{i \left (-3 C a^7+b B a^6-9 b^2 C a^5+3 b^3 B a^4-10 b^4 C a^3+6 b^5 B a^2\right ) \tan ^{-1}(\tan (c+d x)) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{(3 a C-b B) \log (\cos (c+d x)) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{b^4 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{\left (-3 C a^7+b B a^6-9 b^2 C a^5+3 b^3 B a^4-10 b^4 C a^3+6 b^5 B a^2\right ) \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{2 b^4 \left (a^2+b^2\right )^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 619, normalized size = 1.9 \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 [A] time = 1.7538, size = 525, normalized size = 1.59 \begin{align*} \frac{\frac{2 \,{\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{2 \,{\left (3 \, C a^{7} - B a^{6} b + 9 \, C a^{5} b^{2} - 3 \, B a^{4} b^{3} + 10 \, C a^{3} b^{4} - 6 \, B a^{2} b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}} + \frac{{\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{5 \, C a^{7} - 3 \, B a^{6} b + 9 \, C a^{5} b^{2} - 7 \, B a^{4} b^{3} + 2 \,{\left (3 \, C a^{6} b - 2 \, B a^{5} b^{2} + 5 \, C a^{4} b^{3} - 4 \, B a^{3} b^{4}\right )} \tan \left (d x + c\right )}{a^{6} b^{4} + 2 \, a^{4} b^{6} + a^{2} b^{8} +{\left (a^{4} b^{6} + 2 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{5} b^{5} + 2 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )} + \frac{2 \, C \tan \left (d x + c\right )}{b^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85253, size = 1895, normalized size = 5.73 \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 = 2.0753, size = 682, normalized size = 2.06 \begin{align*} \frac{\frac{2 \,{\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{{\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{2 \,{\left (3 \, C a^{7} - B a^{6} b + 9 \, C a^{5} b^{2} - 3 \, B a^{4} b^{3} + 10 \, C a^{3} b^{4} - 6 \, B a^{2} b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}} + \frac{2 \, C \tan \left (d x + c\right )}{b^{3}} + \frac{9 \, C a^{7} b^{2} \tan \left (d x + c\right )^{2} - 3 \, B a^{6} b^{3} \tan \left (d x + c\right )^{2} + 27 \, C a^{5} b^{4} \tan \left (d x + c\right )^{2} - 9 \, B a^{4} b^{5} \tan \left (d x + c\right )^{2} + 30 \, C a^{3} b^{6} \tan \left (d x + c\right )^{2} - 18 \, B a^{2} b^{7} \tan \left (d x + c\right )^{2} + 12 \, C a^{8} b \tan \left (d x + c\right ) - 2 \, B a^{7} b^{2} \tan \left (d x + c\right ) + 38 \, C a^{6} b^{3} \tan \left (d x + c\right ) - 6 \, B a^{5} b^{4} \tan \left (d x + c\right ) + 50 \, C a^{4} b^{5} \tan \left (d x + c\right ) - 28 \, B a^{3} b^{6} \tan \left (d x + c\right ) + 4 \, C a^{9} + 13 \, C a^{7} b^{2} + B a^{6} b^{3} + 21 \, C a^{5} b^{4} - 11 \, B a^{4} b^{5}}{{\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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