Optimal. Leaf size=58 \[ -\frac{\left (a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac{(a+b \tan (c+d x))^2}{2 d}+\frac{a b \tan (c+d x)}{d}-2 a b x \]
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Rubi [A] time = 0.0427957, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3528, 3525, 3475} \[ -\frac{\left (a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac{(a+b \tan (c+d x))^2}{2 d}+\frac{a b \tan (c+d x)}{d}-2 a b x \]
Antiderivative was successfully verified.
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Rule 3528
Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int \tan (c+d x) (a+b \tan (c+d x))^2 \, dx &=\frac{(a+b \tan (c+d x))^2}{2 d}+\int (-b+a \tan (c+d x)) (a+b \tan (c+d x)) \, dx\\ &=-2 a b x+\frac{a b \tan (c+d x)}{d}+\frac{(a+b \tan (c+d x))^2}{2 d}+\left (a^2-b^2\right ) \int \tan (c+d x) \, dx\\ &=-2 a b x-\frac{\left (a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac{a b \tan (c+d x)}{d}+\frac{(a+b \tan (c+d x))^2}{2 d}\\ \end{align*}
Mathematica [C] time = 0.274971, size = 74, normalized size = 1.28 \[ \frac{4 a b \tan (c+d x)+(a-i b)^2 \log (\tan (c+d x)+i)+(a+i b)^2 \log (-\tan (c+d x)+i)+b^2 \tan ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 83, normalized size = 1.4 \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{2}{b}^{2}}{2\,d}}+2\,{\frac{ab\tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{2}\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){b}^{2}}{2\,d}}-2\,{\frac{ab\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.66487, size = 78, normalized size = 1.34 \begin{align*} \frac{b^{2} \tan \left (d x + c\right )^{2} - 4 \,{\left (d x + c\right )} a b + 4 \, a b \tan \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89436, size = 140, normalized size = 2.41 \begin{align*} -\frac{4 \, a b d x - b^{2} \tan \left (d x + c\right )^{2} - 4 \, a b \tan \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.277232, size = 85, normalized size = 1.47 \begin{align*} \begin{cases} \frac{a^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - 2 a b x + \frac{2 a b \tan{\left (c + d x \right )}}{d} - \frac{b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{b^{2} \tan ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a + b \tan{\left (c \right )}\right )^{2} \tan{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.90075, size = 748, normalized size = 12.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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