Optimal. Leaf size=76 \[ \frac {\sqrt {2 a+b \sin (2 c+2 d x)} E\left (c+d x-\frac {\pi }{4}|\frac {2 b}{2 a+b}\right )}{\sqrt {2} d \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2666, 2655, 2653} \[ \frac {\sqrt {2 a+b \sin (2 c+2 d x)} E\left (c+d x-\frac {\pi }{4}|\frac {2 b}{2 a+b}\right )}{\sqrt {2} d \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2653
Rule 2655
Rule 2666
Rubi steps
\begin {align*} \int \sqrt {a+b \cos (c+d x) \sin (c+d x)} \, dx &=\int \sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)} \, dx\\ &=\frac {\sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)} \int \sqrt {\frac {a}{a+\frac {b}{2}}+\frac {b \sin (2 c+2 d x)}{2 \left (a+\frac {b}{2}\right )}} \, dx}{\sqrt {\frac {a+\frac {1}{2} b \sin (2 c+2 d x)}{a+\frac {b}{2}}}}\\ &=\frac {E\left (c-\frac {\pi }{4}+d x|\frac {2 b}{2 a+b}\right ) \sqrt {2 a+b \sin (2 c+2 d x)}}{\sqrt {2} d \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 75, normalized size = 0.99 \[ \frac {(2 a+b) \sqrt {\frac {2 a+b \sin (2 (c+d x))}{2 a+b}} E\left (c+d x-\frac {\pi }{4}|\frac {2 b}{2 a+b}\right )}{d \sqrt {4 a+2 b \sin (2 (c+d x))}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 2.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.41, size = 312, normalized size = 4.11 \[ -\frac {\sqrt {\frac {2 a +b \sin \left (2 d x +2 c \right )}{2 a -b}}\, \sqrt {-\frac {\left (\sin \left (2 d x +2 c \right )-1\right ) b}{2 a +b}}\, \sqrt {-\frac {\left (1+\sin \left (2 d x +2 c \right )\right ) b}{2 a -b}}\, \left (2 a -b \right ) \left (2 \EllipticE \left (\sqrt {\frac {2 a +b \sin \left (2 d x +2 c \right )}{2 a -b}}, \sqrt {\frac {2 a -b}{2 a +b}}\right ) a +\EllipticE \left (\sqrt {\frac {2 a +b \sin \left (2 d x +2 c \right )}{2 a -b}}, \sqrt {\frac {2 a -b}{2 a +b}}\right ) b -2 a \EllipticF \left (\sqrt {\frac {2 a +b \sin \left (2 d x +2 c \right )}{2 a -b}}, \sqrt {\frac {2 a -b}{2 a +b}}\right )-\EllipticF \left (\sqrt {\frac {2 a +b \sin \left (2 d x +2 c \right )}{2 a -b}}, \sqrt {\frac {2 a -b}{2 a +b}}\right ) b \right )}{b \cos \left (2 d x +2 c \right ) \sqrt {4 a +2 b \sin \left (2 d x +2 c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {a+b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________