Optimal. Leaf size=30 \[ \frac{\tan ^4(c+d x) (a \cot (c+d x)+b)^4}{4 b d} \]
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Rubi [A] time = 0.0474499, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3088, 37} \[ \frac{\tan ^4(c+d x) (a \cot (c+d x)+b)^4}{4 b d} \]
Antiderivative was successfully verified.
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Rule 3088
Rule 37
Rubi steps
\begin{align*} \int \sec ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(b+a x)^3}{x^5} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{(b+a \cot (c+d x))^4 \tan ^4(c+d x)}{4 b d}\\ \end{align*}
Mathematica [A] time = 0.169052, size = 57, normalized size = 1.9 \[ \frac{\tan (c+d x) \left (6 a^2 b \tan (c+d x)+4 a^3+4 a b^2 \tan ^2(c+d x)+b^3 \tan ^3(c+d x)\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.121, size = 72, normalized size = 2.4 \begin{align*}{\frac{1}{d} \left ({a}^{3}\tan \left ( dx+c \right ) +{\frac{3\,{a}^{2}b}{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{a{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.18469, size = 117, normalized size = 3.9 \begin{align*} \frac{4 \, a b^{2} \tan \left (d x + c\right )^{3} + 4 \, a^{3} \tan \left (d x + c\right ) + \frac{{\left (2 \, \sin \left (d x + c\right )^{2} - 1\right )} b^{3}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \frac{6 \, a^{2} b}{\sin \left (d x + c\right )^{2} - 1}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.467081, size = 181, normalized size = 6.03 \begin{align*} \frac{b^{3} + 2 \,{\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left (a b^{2} \cos \left (d x + c\right ) +{\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23172, size = 77, normalized size = 2.57 \begin{align*} \frac{b^{3} \tan \left (d x + c\right )^{4} + 4 \, a b^{2} \tan \left (d x + c\right )^{3} + 6 \, a^{2} b \tan \left (d x + c\right )^{2} + 4 \, a^{3} \tan \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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