Optimal. Leaf size=70 \[ x \left (a^2 C+2 a b B-b^2 C\right )+\frac {a^2 B \log (\sin (c+d x))}{d}-\frac {b (2 a C+b B) \log (\cos (c+d x))}{d}+\frac {b^2 C \tan (c+d x)}{d} \]
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Rubi [A] time = 0.18, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3632, 3606, 3624, 3475} \[ x \left (a^2 C+2 a b B-b^2 C\right )+\frac {a^2 B \log (\sin (c+d x))}{d}-\frac {b (2 a C+b B) \log (\cos (c+d x))}{d}+\frac {b^2 C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3606
Rule 3624
Rule 3632
Rubi steps
\begin {align*} \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx &=\int \cot (c+d x) (a+b \tan (c+d x))^2 (B+C \tan (c+d x)) \, dx\\ &=\frac {b^2 C \tan (c+d x)}{d}+\int \cot (c+d x) \left (a^2 B+\left (2 a b B+\left (a^2-b^2\right ) C\right ) \tan (c+d x)+\left (b^2 B+2 a b C\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\left (2 a b B+a^2 C-b^2 C\right ) x+\frac {b^2 C \tan (c+d x)}{d}+\left (a^2 B\right ) \int \cot (c+d x) \, dx+(b (b B+2 a C)) \int \tan (c+d x) \, dx\\ &=\left (2 a b B+a^2 C-b^2 C\right ) x-\frac {b (b B+2 a C) \log (\cos (c+d x))}{d}+\frac {a^2 B \log (\sin (c+d x))}{d}+\frac {b^2 C \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [C] time = 0.28, size = 91, normalized size = 1.30 \[ -\frac {-2 a^2 B \log (\tan (c+d x))+(a+i b)^2 (B+i C) \log (-\tan (c+d x)+i)+(a-i b)^2 (B-i C) \log (\tan (c+d x)+i)-2 b^2 C \tan (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 92, normalized size = 1.31 \[ \frac {B a^{2} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, C b^{2} \tan \left (d x + c\right ) + 2 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} d x - {\left (2 \, C a b + B b^{2}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 7.51, size = 86, normalized size = 1.23 \[ \frac {2 \, B a^{2} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + 2 \, C b^{2} \tan \left (d x + c\right ) + 2 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} {\left (d x + c\right )} - {\left (B a^{2} - 2 \, C a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 109, normalized size = 1.56 \[ 2 B x a b +a^{2} C x -b^{2} C x +\frac {a^{2} B \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {b^{2} B \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {2 B a b c}{d}+\frac {b^{2} C \tan \left (d x +c \right )}{d}-\frac {2 C a b \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {C \,a^{2} c}{d}-\frac {C \,b^{2} c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 85, normalized size = 1.21 \[ \frac {2 \, B a^{2} \log \left (\tan \left (d x + c\right )\right ) + 2 \, C b^{2} \tan \left (d x + c\right ) + 2 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} {\left (d x + c\right )} - {\left (B a^{2} - 2 \, C a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.85, size = 90, normalized size = 1.29 \[ \frac {B\,a^2\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^2}{2\,d}+\frac {C\,b^2\,\mathrm {tan}\left (c+d\,x\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^2}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.61, size = 136, normalized size = 1.94 \[ \begin {cases} - \frac {B a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B a^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + 2 B a b x + \frac {B b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + C a^{2} x + \frac {C a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - C b^{2} x + \frac {C b^{2} \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\relax (c )}\right )^{2} \left (B \tan {\relax (c )} + C \tan ^{2}{\relax (c )}\right ) \cot ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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