Optimal. Leaf size=39 \[ x \left (a^2-b^2\right )-\frac {2 a b \log (\cos (c+d x))}{d}+\frac {b^2 \tan (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3477, 3475} \[ x \left (a^2-b^2\right )-\frac {2 a b \log (\cos (c+d x))}{d}+\frac {b^2 \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3475
Rule 3477
Rubi steps
\begin {align*} \int (a+b \tan (c+d x))^2 \, dx &=\left (a^2-b^2\right ) x+\frac {b^2 \tan (c+d x)}{d}+(2 a b) \int \tan (c+d x) \, dx\\ &=\left (a^2-b^2\right ) x-\frac {2 a b \log (\cos (c+d x))}{d}+\frac {b^2 \tan (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.11, size = 69, normalized size = 1.77 \[ \frac {2 b^2 \tan (c+d x)-i \left ((a+i b)^2 \log (-\tan (c+d x)+i)-(a-i b)^2 \log (\tan (c+d x)+i)\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 44, normalized size = 1.13 \[ \frac {{\left (a^{2} - b^{2}\right )} d x - a b \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + b^{2} \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.96, size = 201, normalized size = 5.15 \[ \frac {a^{2} d x \tan \left (d x\right ) \tan \relax (c) - b^{2} d x \tan \left (d x\right ) \tan \relax (c) - a b \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \relax (c)^{2} - 2 \, \tan \left (d x\right )^{3} \tan \relax (c) + \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \relax (c) + 1\right )}}{\tan \relax (c)^{2} + 1}\right ) \tan \left (d x\right ) \tan \relax (c) - a^{2} d x + b^{2} d x + a b \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \relax (c)^{2} - 2 \, \tan \left (d x\right )^{3} \tan \relax (c) + \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \relax (c) + 1\right )}}{\tan \relax (c)^{2} + 1}\right ) - b^{2} \tan \left (d x\right ) - b^{2} \tan \relax (c)}{d \tan \left (d x\right ) \tan \relax (c) - d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 61, normalized size = 1.56 \[ \frac {\arctan \left (\tan \left (d x +c \right )\right ) a^{2}}{d}-\frac {\arctan \left (\tan \left (d x +c \right )\right ) b^{2}}{d}+\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {b^{2} \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.85, size = 41, normalized size = 1.05 \[ a^{2} x - \frac {{\left (d x + c - \tan \left (d x + c\right )\right )} b^{2}}{d} + \frac {2 \, a b \log \left (\sec \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.09, size = 136, normalized size = 3.49 \[ \frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (c+d\,x\right )}{a^2-b^2}-\frac {b^2\,\mathrm {tan}\left (c+d\,x\right )}{a^2-b^2}\right )}{d}-\frac {b^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (c+d\,x\right )}{a^2-b^2}-\frac {b^2\,\mathrm {tan}\left (c+d\,x\right )}{a^2-b^2}\right )}{d}+\frac {b^2\,\mathrm {tan}\left (c+d\,x\right )}{d}+\frac {a\,b\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.18, size = 48, normalized size = 1.23 \[ \begin {cases} a^{2} x + \frac {a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - b^{2} x + \frac {b^{2} \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\relax (c )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________