Optimal. Leaf size=46 \[ \frac {b (2 a-b) \tan (c+d x)}{d}+x (a-b)^2+\frac {b^2 \tan ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3661, 390, 203} \[ \frac {b (2 a-b) \tan (c+d x)}{d}+x (a-b)^2+\frac {b^2 \tan ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 390
Rule 3661
Rubi steps
\begin {align*} \int \left (a+b \tan ^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^2}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left ((2 a-b) b+b^2 x^2+\frac {(a-b)^2}{1+x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {(2 a-b) b \tan (c+d x)}{d}+\frac {b^2 \tan ^3(c+d x)}{3 d}+\frac {(a-b)^2 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=(a-b)^2 x+\frac {(2 a-b) b \tan (c+d x)}{d}+\frac {b^2 \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 73, normalized size = 1.59 \[ \frac {\tan (c+d x) \left (b \left (6 a-b \left (3-\tan ^2(c+d x)\right )\right )+\frac {3 (a-b)^2 \tanh ^{-1}\left (\sqrt {-\tan ^2(c+d x)}\right )}{\sqrt {-\tan ^2(c+d x)}}\right )}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 51, normalized size = 1.11 \[ \frac {b^{2} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} d x + 3 \, {\left (2 \, a b - b^{2}\right )} \tan \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.48, size = 359, normalized size = 7.80 \[ \frac {3 \, a^{2} d x \tan \left (d x\right )^{3} \tan \relax (c)^{3} - 6 \, a b d x \tan \left (d x\right )^{3} \tan \relax (c)^{3} + 3 \, b^{2} d x \tan \left (d x\right )^{3} \tan \relax (c)^{3} - 9 \, a^{2} d x \tan \left (d x\right )^{2} \tan \relax (c)^{2} + 18 \, a b d x \tan \left (d x\right )^{2} \tan \relax (c)^{2} - 9 \, b^{2} d x \tan \left (d x\right )^{2} \tan \relax (c)^{2} - 6 \, a b \tan \left (d x\right )^{3} \tan \relax (c)^{2} + 3 \, b^{2} \tan \left (d x\right )^{3} \tan \relax (c)^{2} - 6 \, a b \tan \left (d x\right )^{2} \tan \relax (c)^{3} + 3 \, b^{2} \tan \left (d x\right )^{2} \tan \relax (c)^{3} + 9 \, a^{2} d x \tan \left (d x\right ) \tan \relax (c) - 18 \, a b d x \tan \left (d x\right ) \tan \relax (c) + 9 \, b^{2} d x \tan \left (d x\right ) \tan \relax (c) - b^{2} \tan \left (d x\right )^{3} + 12 \, a b \tan \left (d x\right )^{2} \tan \relax (c) - 9 \, b^{2} \tan \left (d x\right )^{2} \tan \relax (c) + 12 \, a b \tan \left (d x\right ) \tan \relax (c)^{2} - 9 \, b^{2} \tan \left (d x\right ) \tan \relax (c)^{2} - b^{2} \tan \relax (c)^{3} - 3 \, a^{2} d x + 6 \, a b d x - 3 \, b^{2} d x - 6 \, a b \tan \left (d x\right ) + 3 \, b^{2} \tan \left (d x\right ) - 6 \, a b \tan \relax (c) + 3 \, b^{2} \tan \relax (c)}{3 \, {\left (d \tan \left (d x\right )^{3} \tan \relax (c)^{3} - 3 \, d \tan \left (d x\right )^{2} \tan \relax (c)^{2} + 3 \, d \tan \left (d x\right ) \tan \relax (c) - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 87, normalized size = 1.89 \[ \frac {b^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {2 a b \tan \left (d x +c \right )}{d}-\frac {b^{2} \tan \left (d x +c \right )}{d}+\frac {\arctan \left (\tan \left (d x +c \right )\right ) a^{2}}{d}-\frac {2 \arctan \left (\tan \left (d x +c \right )\right ) a b}{d}+\frac {\arctan \left (\tan \left (d x +c \right )\right ) b^{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 58, normalized size = 1.26 \[ a^{2} x - \frac {2 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a b}{d} + \frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} b^{2}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.42, size = 76, normalized size = 1.65 \[ \frac {\mathrm {tan}\left (c+d\,x\right )\,\left (2\,a\,b-b^2\right )}{d}+\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,{\left (a-b\right )}^2}{a^2-2\,a\,b+b^2}\right )\,{\left (a-b\right )}^2}{d}+\frac {b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 68, normalized size = 1.48 \[ \begin {cases} a^{2} x - 2 a b x + \frac {2 a b \tan {\left (c + d x \right )}}{d} + b^{2} x + \frac {b^{2} \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {b^{2} \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan ^{2}{\relax (c )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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