\(\int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx\) [259]

Optimal result

Integrand size = 25, antiderivative size = 53 \[ \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx=\frac {\sqrt {d \cos (a+b x)} E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sin (a+b x)}}{b \sqrt {\sin (2 a+2 b x)}} \]

[Out]

-(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticE(cos(a+1/4*Pi+b*x),2^(1/2))*(d*cos(b*x+a))^(1/2)*(c*si
n(b*x+a))^(1/2)/b/sin(2*b*x+2*a)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2652, 2719} \[ \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx=\frac {E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{b \sqrt {\sin (2 a+2 b x)}} \]

[In]

Int[Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]],x]

[Out]

(Sqrt[d*Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(b*Sqrt[Sin[2*a + 2*b*x]])

Rule 2652

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a*Sin[e +
f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]), Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f},
x]

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}\right ) \int \sqrt {\sin (2 a+2 b x)} \, dx}{\sqrt {\sin (2 a+2 b x)}} \\ & = \frac {\sqrt {d \cos (a+b x)} E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sin (a+b x)}}{b \sqrt {\sin (2 a+2 b x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.26 \[ \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx=\frac {2 \sqrt {d \cos (a+b x)} \sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {7}{4},\sin ^2(a+b x)\right ) \sqrt {c \sin (a+b x)} \tan (a+b x)}{3 b} \]

[In]

Integrate[Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]],x]

[Out]

(2*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, Sin[a + b*x]^2]*Sqrt[c*Sin[a +
b*x]]*Tan[a + b*x])/(3*b)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(392\) vs. \(2(73)=146\).

Time = 0.26 (sec) , antiderivative size = 393, normalized size of antiderivative = 7.42

method	result	size
default	\(-\frac {\sqrt {2}\, \left (2 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )-\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )+2 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \left (\cos ^{2}\left (b x +a \right )\right )-\sqrt {2}\, \cos \left (b x +a \right )\right ) \sqrt {d \cos \left (b x +a \right )}\, \sqrt {c \sin \left (b x +a \right )}\, \sec \left (b x +a \right ) \csc \left (b x +a \right )}{2 b}\)	\(393\)

[In]

int((d*cos(b*x+a))^(1/2)*(c*sin(b*x+a))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2/b*2^(1/2)*(2*(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/
2)*EllipticE((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2*2^(1/2))*cos(b*x+a)-(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*
x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticF((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2*2^(1/2))
*cos(b*x+a)+2*(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*E
llipticE((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2*2^(1/2))-(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)
+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticF((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2*2^(1/2))+2^(1/2)*cos(b*
x+a)^2-2^(1/2)*cos(b*x+a))*(d*cos(b*x+a))^(1/2)*(c*sin(b*x+a))^(1/2)*sec(b*x+a)*csc(b*x+a)

Fricas [F]

\[ \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} \,d x } \]

[In]

integrate((d*cos(b*x+a))^(1/2)*(c*sin(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a)), x)

Sympy [F]

\[ \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx=\int \sqrt {c \sin {\left (a + b x \right )}} \sqrt {d \cos {\left (a + b x \right )}}\, dx \]

[In]

integrate((d*cos(b*x+a))**(1/2)*(c*sin(b*x+a))**(1/2),x)

[Out]

Integral(sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x)), x)

Maxima [F]

\[ \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} \,d x } \]

[In]

integrate((d*cos(b*x+a))^(1/2)*(c*sin(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a)), x)

Giac [F]

\[ \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} \,d x } \]

[In]

integrate((d*cos(b*x+a))^(1/2)*(c*sin(b*x+a))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a)), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx=\int \sqrt {d\,\cos \left (a+b\,x\right )}\,\sqrt {c\,\sin \left (a+b\,x\right )} \,d x \]

[In]

int((d*cos(a + b*x))^(1/2)*(c*sin(a + b*x))^(1/2),x)

[Out]

int((d*cos(a + b*x))^(1/2)*(c*sin(a + b*x))^(1/2), x)