Optimal. Leaf size=44 \[ \frac{a \tan (c+d x)}{d}-a x+\frac{b \tan ^2(c+d x)}{2 d}+\frac{b \log (\cos (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0379417, antiderivative size = 44, 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} \[ \frac{a \tan (c+d x)}{d}-a x+\frac{b \tan ^2(c+d x)}{2 d}+\frac{b \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3528
Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int \tan ^2(c+d x) (a+b \tan (c+d x)) \, dx &=\frac{b \tan ^2(c+d x)}{2 d}+\int \tan (c+d x) (-b+a \tan (c+d x)) \, dx\\ &=-a x+\frac{a \tan (c+d x)}{d}+\frac{b \tan ^2(c+d x)}{2 d}-b \int \tan (c+d x) \, dx\\ &=-a x+\frac{b \log (\cos (c+d x))}{d}+\frac{a \tan (c+d x)}{d}+\frac{b \tan ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.10126, size = 51, normalized size = 1.16 \[ -\frac{a \tan ^{-1}(\tan (c+d x))}{d}+\frac{a \tan (c+d x)}{d}+\frac{b \left (\tan ^2(c+d x)+2 \log (\cos (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 57, normalized size = 1.3 \begin{align*}{\frac{b \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{a\tan \left ( dx+c \right ) }{d}}-{\frac{b\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}}-{\frac{a\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.69948, size = 63, normalized size = 1.43 \begin{align*} \frac{b \tan \left (d x + c\right )^{2} - 2 \,{\left (d x + c\right )} a - b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, a \tan \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.73999, size = 119, normalized size = 2.7 \begin{align*} -\frac{2 \, a d x - b \tan \left (d x + c\right )^{2} - b \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, a \tan \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.25122, size = 56, normalized size = 1.27 \begin{align*} \begin{cases} - a x + \frac{a \tan{\left (c + d x \right )}}{d} - \frac{b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{b \tan ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a + b \tan{\left (c \right )}\right ) \tan ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.66851, size = 441, normalized size = 10.02 \begin{align*} -\frac{2 \, a d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - b \log \left (\frac{4 \,{\left (\tan \left (c\right )^{2} + 1\right )}}{\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1}\right ) \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 4 \, a d x \tan \left (d x\right ) \tan \left (c\right ) - b \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 2 \, b \log \left (\frac{4 \,{\left (\tan \left (c\right )^{2} + 1\right )}}{\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) + 2 \, a \tan \left (d x\right )^{2} \tan \left (c\right ) + 2 \, a \tan \left (d x\right ) \tan \left (c\right )^{2} + 2 \, a d x - b \tan \left (d x\right )^{2} - b \tan \left (c\right )^{2} - b \log \left (\frac{4 \,{\left (\tan \left (c\right )^{2} + 1\right )}}{\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1}\right ) - 2 \, a \tan \left (d x\right ) - 2 \, a \tan \left (c\right ) - b}{2 \,{\left (d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, d \tan \left (d x\right ) \tan \left (c\right ) + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]