Optimal. Leaf size=154 \[ \frac{4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{9/2}}{9 b^5 d}-\frac{8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac{2 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{5/2}}{5 b^5 d}+\frac{2 (a+b \sin (c+d x))^{13/2}}{13 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{11/2}}{11 b^5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.122571, antiderivative size = 154, 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 = {2668, 697} \[ \frac{4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{9/2}}{9 b^5 d}-\frac{8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac{2 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{5/2}}{5 b^5 d}+\frac{2 (a+b \sin (c+d x))^{13/2}}{13 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{11/2}}{11 b^5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2668
Rule 697
Rubi steps
\begin{align*} \int \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^{3/2} \left (b^2-x^2\right )^2 \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\left (a^2-b^2\right )^2 (a+x)^{3/2}-4 \left (a^3-a b^2\right ) (a+x)^{5/2}+2 \left (3 a^2-b^2\right ) (a+x)^{7/2}-4 a (a+x)^{9/2}+(a+x)^{11/2}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{2 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac{8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac{4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{9/2}}{9 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{11/2}}{11 b^5 d}+\frac{2 (a+b \sin (c+d x))^{13/2}}{13 b^5 d}\\ \end{align*}
Mathematica [A] time = 0.70725, size = 131, normalized size = 0.85 \[ \frac{2 \left (\frac{2}{9} \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{9/2}+\frac{1}{5} \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{5/2}+\frac{1}{13} (a+b \sin (c+d x))^{13/2}-\frac{4}{11} a (a+b \sin (c+d x))^{11/2}-\frac{4}{7} a (a-b) (a+b) (a+b \sin (c+d x))^{7/2}\right )}{b^5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.462, size = 126, normalized size = 0.8 \begin{align*}{\frac{6930\,{b}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{4}+5040\,a{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -3360\,{a}^{2}{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}+6160\,{b}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{2}-1920\,{a}^{3}b\sin \left ( dx+c \right ) +6400\,a{b}^{3}\sin \left ( dx+c \right ) +768\,{a}^{4}-1216\,{a}^{2}{b}^{2}+4928\,{b}^{4}}{45045\,{b}^{5}d} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.950118, size = 157, normalized size = 1.02 \begin{align*} \frac{2 \,{\left (3465 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{13}{2}} - 16380 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}} a + 10010 \,{\left (3 \, a^{2} - b^{2}\right )}{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} - 25740 \,{\left (a^{3} - a b^{2}\right )}{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} + 9009 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}\right )}}{45045 \, b^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 3.30025, size = 462, normalized size = 3. \begin{align*} -\frac{2 \,{\left (3465 \, b^{6} \cos \left (d x + c\right )^{6} - 384 \, a^{6} + 2144 \, a^{4} b^{2} - 8256 \, a^{2} b^{4} - 2464 \, b^{6} - 35 \,{\left (3 \, a^{2} b^{4} + 11 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 8 \,{\left (18 \, a^{4} b^{2} - 81 \, a^{2} b^{4} - 77 \, b^{6}\right )} \cos \left (d x + c\right )^{2} - 2 \,{\left (2205 \, a b^{5} \cos \left (d x + c\right )^{4} - 96 \, a^{5} b + 512 \, a^{3} b^{3} + 4064 \, a b^{5} + 20 \,{\left (3 \, a^{3} b^{3} + 137 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt{b \sin \left (d x + c\right ) + a}}{45045 \, b^{5} 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]