Optimal. Leaf size=46 \[ \frac{(a+b) \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \tan ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.0405622, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3675, 373} \[ \frac{(a+b) \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \tan ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3675
Rule 373
Rubi steps
\begin{align*} \int \sec ^4(c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \left (a+b x^2\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a+(a+b) x^2+b x^4\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{a \tan (c+d x)}{d}+\frac{(a+b) \tan ^3(c+d x)}{3 d}+\frac{b \tan ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.138067, size = 53, normalized size = 1.15 \[ \frac{\tan (c+d x) \left (5 a \tan ^2(c+d x)+15 a+3 b \sec ^4(c+d x)-b \sec ^2(c+d x)-2 b\right )}{15 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 66, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ( b \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{15\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) -a \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00245, size = 53, normalized size = 1.15 \begin{align*} \frac{3 \, b \tan \left (d x + c\right )^{5} + 5 \,{\left (a + b\right )} \tan \left (d x + c\right )^{3} + 15 \, a \tan \left (d x + c\right )}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31229, size = 135, normalized size = 2.93 \begin{align*} \frac{{\left (2 \,{\left (5 \, a - b\right )} \cos \left (d x + c\right )^{4} +{\left (5 \, a - b\right )} \cos \left (d x + c\right )^{2} + 3 \, b\right )} \sin \left (d x + c\right )}{15 \, d \cos \left (d x + c\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan ^{2}{\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.53058, size = 65, normalized size = 1.41 \begin{align*} \frac{3 \, b \tan \left (d x + c\right )^{5} + 5 \, a \tan \left (d x + c\right )^{3} + 5 \, b \tan \left (d x + c\right )^{3} + 15 \, a \tan \left (d x + c\right )}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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