Optimal. Leaf size=57 \[ \frac{a^2 \sin (c+d x)}{d}+\frac{(a-b)^2 \sin ^5(c+d x)}{5 d}-\frac{2 a (a-b) \sin ^3(c+d x)}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.059917, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3676, 194} \[ \frac{a^2 \sin (c+d x)}{d}+\frac{(a-b)^2 \sin ^5(c+d x)}{5 d}-\frac{2 a (a-b) \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3676
Rule 194
Rubi steps
\begin{align*} \int \cos ^5(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a-(a-b) x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2-2 a (a-b) x^2+(a-b)^2 x^4\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{a^2 \sin (c+d x)}{d}-\frac{2 a (a-b) \sin ^3(c+d x)}{3 d}+\frac{(a-b)^2 \sin ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.158537, size = 52, normalized size = 0.91 \[ \frac{15 a^2 \sin (c+d x)+3 (a-b)^2 \sin ^5(c+d x)-10 a (a-b) \sin ^3(c+d x)}{15 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.052, size = 89, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ({\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}+2\,ab \left ( -1/5\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+1/15\, \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) +{\frac{{a}^{2}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01503, size = 76, normalized size = 1.33 \begin{align*} \frac{3 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \sin \left (d x + c\right )^{5} - 10 \,{\left (a^{2} - a b\right )} \sin \left (d x + c\right )^{3} + 15 \, a^{2} \sin \left (d x + c\right )}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.47696, size = 169, normalized size = 2.96 \begin{align*} \frac{{\left (3 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (2 \, a^{2} + a b - 3 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \, a^{2} + 4 \, a b + 3 \, b^{2}\right )} \sin \left (d x + c\right )}{15 \, d} \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 [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]