Optimal. Leaf size=199 \[ -\frac {2 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{5 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (a^2+3 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{5 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac {2 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{5 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (a^2+3 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{5 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac {2 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \, dx &=\frac {2 (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {2}{5} \int \left (\frac {3 b}{2}+\frac {3}{2} a \cos (c+d x)\right ) \sqrt {a+b \cos (c+d x)} \, dx\\ &=\frac {2 a \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{5 d}+\frac {2 (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {4}{15} \int \frac {3 a b+\frac {3}{4} \left (a^2+3 b^2\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx\\ &=\frac {2 a \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{5 d}+\frac {2 (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}-\frac {\left (a \left (a^2-b^2\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{5 b}+\frac {\left (a^2+3 b^2\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{5 b}\\ &=\frac {2 a \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{5 d}+\frac {2 (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {\left (\left (a^2+3 b^2\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{5 b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{5 b \sqrt {a+b \cos (c+d x)}}\\ &=\frac {2 \left (a^2+3 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{5 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{5 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 a \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{5 d}+\frac {2 (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.78, size = 174, normalized size = 0.87 \[ \frac {b \sin (c+d x) \left (4 a^2+6 a b \cos (c+d x)+b^2 \cos (2 (c+d x))+b^2\right )-2 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+2 \left (a^3+a^2 b+3 a b^2+3 b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{5 b d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.76, size = 663, normalized size = 3.33 \[ -\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (8 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+12 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-16 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+4 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -18 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+10 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{3}+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a \,b^{2}+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{3}-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{2} b +3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a \,b^{2}-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) b^{3}-4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b +6 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}\right )}{5 b \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \cos \left (c+d\,x\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \cos {\left (c + d x \right )}\right )^{\frac {3}{2}} \cos {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________