Optimal. Leaf size=119 \[ -\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{9 a^3}{4 d (a-a \sin (c+d x))}-\frac{2 a^2 \sin (c+d x)}{d}-\frac{31 a^2 \log (1-\sin (c+d x))}{8 d}-\frac{a^2 \log (\sin (c+d x)+1)}{8 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0818173, antiderivative size = 119, 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 = {2707, 88} \[ -\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{9 a^3}{4 d (a-a \sin (c+d x))}-\frac{2 a^2 \sin (c+d x)}{d}-\frac{31 a^2 \log (1-\sin (c+d x))}{8 d}-\frac{a^2 \log (\sin (c+d x)+1)}{8 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2707
Rule 88
Rubi steps
\begin{align*} \int (a+a \sin (c+d x))^2 \tan ^5(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^5}{(a-x)^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-2 a+\frac{a^4}{2 (a-x)^3}-\frac{9 a^3}{4 (a-x)^2}+\frac{31 a^2}{8 (a-x)}-x-\frac{a^2}{8 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{31 a^2 \log (1-\sin (c+d x))}{8 d}-\frac{a^2 \log (1+\sin (c+d x))}{8 d}-\frac{2 a^2 \sin (c+d x)}{d}-\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{9 a^3}{4 d (a-a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.244541, size = 75, normalized size = 0.63 \[ -\frac{a^2 \left (4 \sin ^2(c+d x)+16 \sin (c+d x)-\frac{18}{\sin (c+d x)-1}-\frac{2}{(\sin (c+d x)-1)^2}+31 \log (1-\sin (c+d x))+\log (\sin (c+d x)+1)\right )}{8 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.085, size = 261, normalized size = 2.2 \begin{align*}{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{2\,d}}-{\frac{3\,{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{3\,{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-4\,{\frac{{a}^{2}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{4\,d}}-{\frac{5\,{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d}}-{\frac{15\,{a}^{2}\sin \left ( dx+c \right ) }{4\,d}}+{\frac{15\,{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{4\,d}}+{\frac{{a}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{{a}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.12912, size = 130, normalized size = 1.09 \begin{align*} -\frac{4 \, a^{2} \sin \left (d x + c\right )^{2} + a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) + 31 \, a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) + 16 \, a^{2} \sin \left (d x + c\right ) - \frac{2 \,{\left (9 \, a^{2} \sin \left (d x + c\right ) - 8 \, a^{2}\right )}}{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.55703, size = 412, normalized size = 3.46 \begin{align*} \frac{4 \, a^{2} \cos \left (d x + c\right )^{4} + 22 \, a^{2} \cos \left (d x + c\right )^{2} - 12 \, a^{2} -{\left (a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 31 \,{\left (a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (4 \, a^{2} \cos \left (d x + c\right )^{2} - 5 \, a^{2}\right )} \sin \left (d x + c\right )}{8 \,{\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \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]