Optimal. Leaf size=30 \[ \frac{\tan ^5(c+d x) (a \cot (c+d x)+b)^5}{5 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0476237, 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} \[ \frac{\tan ^5(c+d x) (a \cot (c+d x)+b)^5}{5 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3088
Rule 37
Rubi steps
\begin{align*} \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(b+a x)^4}{x^6} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{(b+a \cot (c+d x))^5 \tan ^5(c+d x)}{5 b d}\\ \end{align*}
Mathematica [B] time = 0.308616, size = 73, normalized size = 2.43 \[ \frac{\tan (c+d x) \left (10 a^2 b^2 \tan ^2(c+d x)+10 a^3 b \tan (c+d x)+5 a^4+5 a b^3 \tan ^3(c+d x)+b^4 \tan ^4(c+d x)\right )}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.146, size = 96, normalized size = 3.2 \begin{align*}{\frac{1}{d} \left ({a}^{4}\tan \left ( dx+c \right ) +2\,{\frac{{a}^{3}b}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+2\,{\frac{{a}^{2}{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{a{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{b}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.23157, size = 139, normalized size = 4.63 \begin{align*} \frac{b^{4} \tan \left (d x + c\right )^{5} + 10 \, a^{2} b^{2} \tan \left (d x + c\right )^{3} + 5 \, a^{4} \tan \left (d x + c\right ) + \frac{5 \,{\left (2 \, \sin \left (d x + c\right )^{2} - 1\right )} a b^{3}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \frac{10 \, a^{3} b}{\sin \left (d x + c\right )^{2} - 1}}{5 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.484743, size = 250, normalized size = 8.33 \begin{align*} \frac{5 \, a b^{3} \cos \left (d x + c\right ) + 10 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left ({\left (5 \, a^{4} - 10 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + b^{4} + 2 \,{\left (5 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{5 \, d \cos \left (d x + c\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.2078, size = 99, normalized size = 3.3 \begin{align*} \frac{b^{4} \tan \left (d x + c\right )^{5} + 5 \, a b^{3} \tan \left (d x + c\right )^{4} + 10 \, a^{2} b^{2} \tan \left (d x + c\right )^{3} + 10 \, a^{3} b \tan \left (d x + c\right )^{2} + 5 \, a^{4} \tan \left (d x + c\right )}{5 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]