Integrand size = 23, antiderivative size = 68 \[ \int (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)} \, dx=\frac {2 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}+\frac {2 b (e \sin (c+d x))^{3/2}}{3 d e} \] Output:
-2*a*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*sin(d*x+c))^(1/2)/d/s in(d*x+c)^(1/2)+2/3*b*(e*sin(d*x+c))^(3/2)/d/e
Time = 0.34 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.88 \[ \int (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)} \, dx=\frac {2 \sqrt {e \sin (c+d x)} \left (-3 a E\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right )+b \sin ^{\frac {3}{2}}(c+d x)\right )}{3 d \sqrt {\sin (c+d x)}} \] Input:
Integrate[(a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]],x]
Output:
(2*Sqrt[e*Sin[c + d*x]]*(-3*a*EllipticE[(-2*c + Pi - 2*d*x)/4, 2] + b*Sin[ c + d*x]^(3/2)))/(3*d*Sqrt[Sin[c + d*x]])
Time = 0.33 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 3148, 3042, 3121, 3042, 3119}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {e \sin (c+d x)} (a+b \cos (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )dx\) |
\(\Big \downarrow \) 3148 |
\(\displaystyle a \int \sqrt {e \sin (c+d x)}dx+\frac {2 b (e \sin (c+d x))^{3/2}}{3 d e}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \sqrt {e \sin (c+d x)}dx+\frac {2 b (e \sin (c+d x))^{3/2}}{3 d e}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle \frac {a \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}+\frac {2 b (e \sin (c+d x))^{3/2}}{3 d e}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}+\frac {2 b (e \sin (c+d x))^{3/2}}{3 d e}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}+\frac {2 b (e \sin (c+d x))^{3/2}}{3 d e}\) |
Input:
Int[(a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]],x]
Output:
(2*a*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]]) + (2*b*(e*Sin[c + d*x])^(3/2))/(3*d*e)
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]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Time = 2.14 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.72
method | result | size |
default | \(\frac {\frac {2 b \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3 e}-\frac {a e \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \left (2 \operatorname {EllipticE}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )\right )}{\cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(117\) |
parts | \(-\frac {a e \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \left (2 \operatorname {EllipticE}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )\right )}{\cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {2 b \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d e}\) | \(119\) |
Input:
int((a+cos(d*x+c)*b)*(e*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
(2/3*b/e*(e*sin(d*x+c))^(3/2)-a*e*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1 /2)*sin(d*x+c)^(1/2)*(2*EllipticE((1-sin(d*x+c))^(1/2),1/2*2^(1/2))-Ellipt icF((1-sin(d*x+c))^(1/2),1/2*2^(1/2)))/cos(d*x+c)/(e*sin(d*x+c))^(1/2))/d
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.21 \[ \int (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)} \, dx=\frac {2 \, {\left (\sqrt {e \sin \left (d x + c\right )} b \sin \left (d x + c\right ) + 3 i \, a \sqrt {-\frac {1}{2} i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 3 i \, a \sqrt {\frac {1}{2} i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )\right )}}{3 \, d} \] Input:
integrate((a+b*cos(d*x+c))*(e*sin(d*x+c))^(1/2),x, algorithm="fricas")
Output:
2/3*(sqrt(e*sin(d*x + c))*b*sin(d*x + c) + 3*I*a*sqrt(-1/2*I*e)*weierstras sZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) - 3* I*a*sqrt(1/2*I*e)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c))))/d
\[ \int (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)} \, dx=\int \sqrt {e \sin {\left (c + d x \right )}} \left (a + b \cos {\left (c + d x \right )}\right )\, dx \] Input:
integrate((a+b*cos(d*x+c))*(e*sin(d*x+c))**(1/2),x)
Output:
Integral(sqrt(e*sin(c + d*x))*(a + b*cos(c + d*x)), x)
\[ \int (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )} \sqrt {e \sin \left (d x + c\right )} \,d x } \] Input:
integrate((a+b*cos(d*x+c))*(e*sin(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate((b*cos(d*x + c) + a)*sqrt(e*sin(d*x + c)), x)
\[ \int (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )} \sqrt {e \sin \left (d x + c\right )} \,d x } \] Input:
integrate((a+b*cos(d*x+c))*(e*sin(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate((b*cos(d*x + c) + a)*sqrt(e*sin(d*x + c)), x)
Time = 42.66 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.88 \[ \int (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)} \, dx=\frac {2\,b\,\sin \left (c+d\,x\right )\,\sqrt {e\,\sin \left (c+d\,x\right )}}{3\,d}+\frac {2\,a\,\sqrt {e\,\sin \left (c+d\,x\right )}\,\mathrm {E}\left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\middle |2\right )}{d\,\sqrt {\sin \left (c+d\,x\right )}} \] Input:
int((e*sin(c + d*x))^(1/2)*(a + b*cos(c + d*x)),x)
Output:
(2*b*sin(c + d*x)*(e*sin(c + d*x))^(1/2))/(3*d) + (2*a*(e*sin(c + d*x))^(1 /2)*ellipticE(c/2 - pi/4 + (d*x)/2, 2))/(d*sin(c + d*x)^(1/2))
\[ \int (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)} \, dx=\frac {\sqrt {e}\, \left (2 \sqrt {\sin \left (d x +c \right )}\, \sin \left (d x +c \right ) b +3 \left (\int \sqrt {\sin \left (d x +c \right )}d x \right ) a d \right )}{3 d} \] Input:
int((a+b*cos(d*x+c))*(e*sin(d*x+c))^(1/2),x)
Output:
(sqrt(e)*(2*sqrt(sin(c + d*x))*sin(c + d*x)*b + 3*int(sqrt(sin(c + d*x)),x )*a*d))/(3*d)