\(\int (a+b \cos (c+d x) \sin (c+d x))^{3/2} \, dx\) [574]

Optimal result

Integrand size = 20, antiderivative size = 212 \[ \int (a+b \cos (c+d x) \sin (c+d x))^{3/2} \, dx=-\frac {b \cos (2 c+2 d x) \sqrt {2 a+b \sin (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {2 \sqrt {2} a E\left (c-\frac {\pi }{4}+d x|\frac {2 b}{2 a+b}\right ) \sqrt {2 a+b \sin (2 c+2 d x)}}{3 d \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}}-\frac {\left (4 a^2-b^2\right ) \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,\frac {2 b}{2 a+b}\right ) \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}}{6 \sqrt {2} d \sqrt {2 a+b \sin (2 c+2 d x)}} \]

[Out]

-1/12*b*cos(2*d*x+2*c)*(2*a+b*sin(2*d*x+2*c))^(1/2)/d*2^(1/2)-2/3*a*(sin(c+1/4*Pi+d*x)^2)^(1/2)/sin(c+1/4*Pi+d
*x)*EllipticE(cos(c+1/4*Pi+d*x),2^(1/2)*(b/(2*a+b))^(1/2))*2^(1/2)*(2*a+b*sin(2*d*x+2*c))^(1/2)/d/((2*a+b*sin(
2*d*x+2*c))/(2*a+b))^(1/2)+1/12*(4*a^2-b^2)*(sin(c+1/4*Pi+d*x)^2)^(1/2)/sin(c+1/4*Pi+d*x)*EllipticF(cos(c+1/4*
Pi+d*x),2^(1/2)*(b/(2*a+b))^(1/2))*((2*a+b*sin(2*d*x+2*c))/(2*a+b))^(1/2)/d*2^(1/2)/(2*a+b*sin(2*d*x+2*c))^(1/
2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2745, 2735, 2831, 2742, 2740, 2734, 2732} \[ \int (a+b \cos (c+d x) \sin (c+d x))^{3/2} \, dx=-\frac {\left (4 a^2-b^2\right ) \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}} \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},\frac {2 b}{2 a+b}\right )}{6 \sqrt {2} d \sqrt {2 a+b \sin (2 c+2 d x)}}-\frac {b \cos (2 c+2 d x) \sqrt {2 a+b \sin (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {2 \sqrt {2} a \sqrt {2 a+b \sin (2 c+2 d x)} E\left (c+d x-\frac {\pi }{4}|\frac {2 b}{2 a+b}\right )}{3 d \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}} \]

[In]

Int[(a + b*Cos[c + d*x]*Sin[c + d*x])^(3/2),x]

[Out]

-1/6*(b*Cos[2*c + 2*d*x]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/(Sqrt[2]*d) + (2*Sqrt[2]*a*EllipticE[c - Pi/4 + d*x,
(2*b)/(2*a + b)]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/(3*d*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)]) - ((4*a^2 -
b^2)*EllipticF[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)])/(6*Sqrt[2]*d*Sqrt[
2*a + b*Sin[2*c + 2*d*x]])

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2735

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((a + b*Sin[c + d*x])^(n
- 1)/(d*n)), x] + Dist[1/n, Int[(a + b*Sin[c + d*x])^(n - 2)*Simp[a^2*n + b^2*(n - 1) + a*b*(2*n - 1)*Sin[c +
d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2745

Int[((a_) + cos[(c_.) + (d_.)*(x_)]*(b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[(a + b*(Sin[2*c + 2*
d*x]/2))^n, x] /; FreeQ[{a, b, c, d, n}, x]

Rule 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a+\frac {1}{2} b \sin (2 c+2 d x)\right )^{3/2} \, dx \\ & = -\frac {b \cos (2 c+2 d x) \sqrt {2 a+b \sin (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {2}{3} \int \frac {\frac {1}{8} \left (12 a^2+b^2\right )+a b \sin (2 c+2 d x)}{\sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)}} \, dx \\ & = -\frac {b \cos (2 c+2 d x) \sqrt {2 a+b \sin (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {1}{3} (4 a) \int \sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)} \, dx+\frac {1}{12} \left (-4 a^2+b^2\right ) \int \frac {1}{\sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)}} \, dx \\ & = -\frac {b \cos (2 c+2 d x) \sqrt {2 a+b \sin (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {\left (4 a \sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)}\right ) \int \sqrt {\frac {a}{a+\frac {b}{2}}+\frac {b \sin (2 c+2 d x)}{2 \left (a+\frac {b}{2}\right )}} \, dx}{3 \sqrt {\frac {a+\frac {1}{2} b \sin (2 c+2 d x)}{a+\frac {b}{2}}}}+\frac {\left (\left (-4 a^2+b^2\right ) \sqrt {\frac {a+\frac {1}{2} b \sin (2 c+2 d x)}{a+\frac {b}{2}}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+\frac {b}{2}}+\frac {b \sin (2 c+2 d x)}{2 \left (a+\frac {b}{2}\right )}}} \, dx}{12 \sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)}} \\ & = -\frac {b \cos (2 c+2 d x) \sqrt {2 a+b \sin (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {2 \sqrt {2} a 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)}}{3 d \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}}-\frac {\left (4 a^2-b^2\right ) \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,\frac {2 b}{2 a+b}\right ) \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}}{6 \sqrt {2} d \sqrt {2 a+b \sin (2 c+2 d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.79 \[ \int (a+b \cos (c+d x) \sin (c+d x))^{3/2} \, dx=\frac {-b \cos (2 (c+d x)) (2 a+b \sin (2 (c+d x)))+8 a (2 a+b) E\left (c-\frac {\pi }{4}+d x|\frac {2 b}{2 a+b}\right ) \sqrt {\frac {2 a+b \sin (2 (c+d x))}{2 a+b}}-\left (4 a^2-b^2\right ) \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,\frac {2 b}{2 a+b}\right ) \sqrt {\frac {2 a+b \sin (2 (c+d x))}{2 a+b}}}{6 d \sqrt {4 a+2 b \sin (2 (c+d x))}} \]

[In]

Integrate[(a + b*Cos[c + d*x]*Sin[c + d*x])^(3/2),x]

[Out]

(-(b*Cos[2*(c + d*x)]*(2*a + b*Sin[2*(c + d*x)])) + 8*a*(2*a + b)*EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*S
qrt[(2*a + b*Sin[2*(c + d*x)])/(2*a + b)] - (4*a^2 - b^2)*EllipticF[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a
+ b*Sin[2*(c + d*x)])/(2*a + b)])/(6*d*Sqrt[4*a + 2*b*Sin[2*(c + d*x)]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(843\) vs. \(2(246)=492\).

Time = 1.57 (sec) , antiderivative size = 844, normalized size of antiderivative = 3.98

method	result	size
default	\(\frac {24 a^{3} \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}}\, \operatorname {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 )+4 \operatorname {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 ) \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}}\, a^{2} b -6 \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}}\, \operatorname {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^{2} a -\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}}\, \operatorname {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^{3}-32 \operatorname {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 ) \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}}\, a^{3}+8 \operatorname {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 ) \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}}\, a \,b^{2}+\sin \left (2 d x +2 c \right )^{3} b^{3}+2 \sin \left (2 d x +2 c \right )^{2} a \,b^{2}-\sin \left (2 d x +2 c \right ) b^{3}-2 a \,b^{2}}{6 b \cos \left (2 d x +2 c \right ) \sqrt {4 a +2 b \sin \left (2 d x +2 c \right )}\, d}\)	\(844\)

[In]

int((a+cos(d*x+c)*sin(d*x+c)*b)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/6*(24*a^3*((2*a+b*sin(2*d*x+2*c))/(2*a-b))^(1/2)*(-(sin(2*d*x+2*c)-1)*b/(2*a+b))^(1/2)*(-(1+sin(2*d*x+2*c))*
b/(2*a-b))^(1/2)*EllipticF(((2*a+b*sin(2*d*x+2*c))/(2*a-b))^(1/2),((2*a-b)/(2*a+b))^(1/2))+4*EllipticF(((2*a+b
*sin(2*d*x+2*c))/(2*a-b))^(1/2),((2*a-b)/(2*a+b))^(1/2))*((2*a+b*sin(2*d*x+2*c))/(2*a-b))^(1/2)*(-(sin(2*d*x+2
*c)-1)*b/(2*a+b))^(1/2)*(-(1+sin(2*d*x+2*c))*b/(2*a-b))^(1/2)*a^2*b-6*((2*a+b*sin(2*d*x+2*c))/(2*a-b))^(1/2)*(
-(sin(2*d*x+2*c)-1)*b/(2*a+b))^(1/2)*(-(1+sin(2*d*x+2*c))*b/(2*a-b))^(1/2)*EllipticF(((2*a+b*sin(2*d*x+2*c))/(
2*a-b))^(1/2),((2*a-b)/(2*a+b))^(1/2))*b^2*a-((2*a+b*sin(2*d*x+2*c))/(2*a-b))^(1/2)*(-(sin(2*d*x+2*c)-1)*b/(2*
a+b))^(1/2)*(-(1+sin(2*d*x+2*c))*b/(2*a-b))^(1/2)*EllipticF(((2*a+b*sin(2*d*x+2*c))/(2*a-b))^(1/2),((2*a-b)/(2
*a+b))^(1/2))*b^3-32*EllipticE(((2*a+b*sin(2*d*x+2*c))/(2*a-b))^(1/2),((2*a-b)/(2*a+b))^(1/2))*((2*a+b*sin(2*d
*x+2*c))/(2*a-b))^(1/2)*(-(sin(2*d*x+2*c)-1)*b/(2*a+b))^(1/2)*(-(1+sin(2*d*x+2*c))*b/(2*a-b))^(1/2)*a^3+8*Elli
pticE(((2*a+b*sin(2*d*x+2*c))/(2*a-b))^(1/2),((2*a-b)/(2*a+b))^(1/2))*((2*a+b*sin(2*d*x+2*c))/(2*a-b))^(1/2)*(
-(sin(2*d*x+2*c)-1)*b/(2*a+b))^(1/2)*(-(1+sin(2*d*x+2*c))*b/(2*a-b))^(1/2)*a*b^2+sin(2*d*x+2*c)^3*b^3+2*sin(2*
d*x+2*c)^2*a*b^2-sin(2*d*x+2*c)*b^3-2*a*b^2)/b/cos(2*d*x+2*c)/(4*a+2*b*sin(2*d*x+2*c))^(1/2)/d

Fricas [F]

\[ \int (a+b \cos (c+d x) \sin (c+d x))^{3/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

integrate((a+b*cos(d*x+c)*sin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

integral((b*cos(d*x + c)*sin(d*x + c) + a)^(3/2), x)

Sympy [F]

\[ \int (a+b \cos (c+d x) \sin (c+d x))^{3/2} \, dx=\int \left (a + b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((a+b*cos(d*x+c)*sin(d*x+c))**(3/2),x)

[Out]

Integral((a + b*sin(c + d*x)*cos(c + d*x))**(3/2), x)

Maxima [F]

\[ \int (a+b \cos (c+d x) \sin (c+d x))^{3/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

integrate((a+b*cos(d*x+c)*sin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*cos(d*x + c)*sin(d*x + c) + a)^(3/2), x)

Giac [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x) \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]

[In]

integrate((a+b*cos(d*x+c)*sin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x) \sin (c+d x))^{3/2} \, dx=\int {\left (a+b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]

[In]

int((a + b*cos(c + d*x)*sin(c + d*x))^(3/2),x)

[Out]

int((a + b*cos(c + d*x)*sin(c + d*x))^(3/2), x)