\(\int (a+b \cos (c+d x) \sin (c+d x))^{5/2} \, dx\) [573]

Optimal result

Integrand size = 20, antiderivative size = 265 \[ \int (a+b \cos (c+d x) \sin (c+d x))^{5/2} \, dx=-\frac {2 \sqrt {2} a b \cos (2 c+2 d x) \sqrt {2 a+b \sin (2 c+2 d x)}}{15 d}-\frac {b \cos (2 c+2 d x) (2 a+b \sin (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {\left (92 a^2+9 b^2\right ) E\left (c-\frac {\pi }{4}+d x|\frac {2 b}{2 a+b}\right ) \sqrt {2 a+b \sin (2 c+2 d x)}}{60 \sqrt {2} d \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}}-\frac {2 \sqrt {2} a \left (4 a^2-b^2\right ) \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,\frac {2 b}{2 a+b}\right ) \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}}{15 d \sqrt {2 a+b \sin (2 c+2 d x)}} \]

[Out]

-1/40*b*cos(2*d*x+2*c)*(2*a+b*sin(2*d*x+2*c))^(3/2)/d*2^(1/2)-2/15*a*b*cos(2*d*x+2*c)*2^(1/2)*(2*a+b*sin(2*d*x
+2*c))^(1/2)/d-1/120*(92*a^2+9*b^2)*(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*a+b*sin(2*d*x+2*c))^(1/2)/d*2^(1/2)/((2*a+b*sin(2*d*x+2*c))/(2*a+b))^(1/2)+2/15*
a*(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^(1/2)*((2*a+b*sin(2*d*x+2*c))/(2*a+b))^(1/2)/d/(2*a+b*sin(2*d*x+2*c))^(1/2)

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2745, 2735, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int (a+b \cos (c+d x) \sin (c+d x))^{5/2} \, dx=-\frac {2 \sqrt {2} a \left (4 a^2-b^2\right ) \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}} \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},\frac {2 b}{2 a+b}\right )}{15 d \sqrt {2 a+b \sin (2 c+2 d x)}}+\frac {\left (92 a^2+9 b^2\right ) \sqrt {2 a+b \sin (2 c+2 d x)} E\left (c+d x-\frac {\pi }{4}|\frac {2 b}{2 a+b}\right )}{60 \sqrt {2} d \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}}-\frac {b \cos (2 c+2 d x) (2 a+b \sin (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}-\frac {2 \sqrt {2} a b \cos (2 c+2 d x) \sqrt {2 a+b \sin (2 c+2 d x)}}{15 d} \]

[In]

Int[(a + b*Cos[c + d*x]*Sin[c + d*x])^(5/2),x]

[Out]

(-2*Sqrt[2]*a*b*Cos[2*c + 2*d*x]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/(15*d) - (b*Cos[2*c + 2*d*x]*(2*a + b*Sin[2*c
+ 2*d*x])^(3/2))/(20*Sqrt[2]*d) + ((92*a^2 + 9*b^2)*EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[2*a + b*S
in[2*c + 2*d*x]])/(60*Sqrt[2]*d*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)]) - (2*Sqrt[2]*a*(4*a^2 - b^2)*Ellip
ticF[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)])/(15*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]

Rule 2832

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

Rubi steps \begin{align*} \text {integral}& = \int \left (a+\frac {1}{2} b \sin (2 c+2 d x)\right )^{5/2} \, dx \\ & = -\frac {b \cos (2 c+2 d x) (2 a+b \sin (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {2}{5} \int \sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)} \left (\frac {1}{8} \left (20 a^2+3 b^2\right )+2 a b \sin (2 c+2 d x)\right ) \, dx \\ & = -\frac {2 \sqrt {2} a b \cos (2 c+2 d x) \sqrt {2 a+b \sin (2 c+2 d x)}}{15 d}-\frac {b \cos (2 c+2 d x) (2 a+b \sin (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {4}{15} \int \frac {\frac {1}{16} a \left (60 a^2+17 b^2\right )+\frac {1}{32} b \left (92 a^2+9 b^2\right ) \sin (2 c+2 d x)}{\sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)}} \, dx \\ & = -\frac {2 \sqrt {2} a b \cos (2 c+2 d x) \sqrt {2 a+b \sin (2 c+2 d x)}}{15 d}-\frac {b \cos (2 c+2 d x) (2 a+b \sin (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}-\frac {1}{15} \left (2 a \left (4 a^2-b^2\right )\right ) \int \frac {1}{\sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)}} \, dx+\frac {1}{60} \left (92 a^2+9 b^2\right ) \int \sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)} \, dx \\ & = -\frac {2 \sqrt {2} a b \cos (2 c+2 d x) \sqrt {2 a+b \sin (2 c+2 d x)}}{15 d}-\frac {b \cos (2 c+2 d x) (2 a+b \sin (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {\left (\left (92 a^2+9 b^2\right ) \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}{60 \sqrt {\frac {a+\frac {1}{2} b \sin (2 c+2 d x)}{a+\frac {b}{2}}}}-\frac {\left (2 a \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}{15 \sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)}} \\ & = -\frac {2 \sqrt {2} a b \cos (2 c+2 d x) \sqrt {2 a+b \sin (2 c+2 d x)}}{15 d}-\frac {b \cos (2 c+2 d x) (2 a+b \sin (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {\left (92 a^2+9 b^2\right ) 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)}}{60 \sqrt {2} d \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}}-\frac {2 \sqrt {2} a \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}}}{15 d \sqrt {2 a+b \sin (2 c+2 d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.05 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.76 \[ \int (a+b \cos (c+d x) \sin (c+d x))^{5/2} \, dx=\frac {2 \left (184 a^3+92 a^2 b+18 a b^2+9 b^3\right ) E\left (c-\frac {\pi }{4}+d x|\frac {2 b}{2 a+b}\right ) \sqrt {\frac {2 a+b \sin (2 (c+d x))}{2 a+b}}-32 a \left (4 a^2-b^2\right ) \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,\frac {2 b}{2 a+b}\right ) \sqrt {\frac {2 a+b \sin (2 (c+d x))}{2 a+b}}-b \left (88 a^2 \cos (2 (c+d x))+b (28 a+3 b \sin (2 (c+d x))) \sin (4 (c+d x))\right )}{120 d \sqrt {4 a+2 b \sin (2 (c+d x))}} \]

[In]

Integrate[(a + b*Cos[c + d*x]*Sin[c + d*x])^(5/2),x]

[Out]

(2*(184*a^3 + 92*a^2*b + 18*a*b^2 + 9*b^3)*EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*(c +
d*x)])/(2*a + b)] - 32*a*(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)] - b*(88*a^2*Cos[2*(c + d*x)] + b*(28*a + 3*b*Sin[2*(c + d*x)])*Sin[4*(c + d*x)]))/(120*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. \(1137\) vs. \(2(293)=586\).

Time = 2.58 (sec) , antiderivative size = 1138, normalized size of antiderivative = 4.29

method	result	size
default	\(\text {Expression too large to display}\)	\(1138\)

[In]

int((a+cos(d*x+c)*sin(d*x+c)*b)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/60*(240*a^4*((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))+64*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^3*b-24*a^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)*EllipticF(((2*a+b*sin(2*d*x
+2*c))/(2*a-b))^(1/2),((2*a-b)/(2*a+b))^(1/2))*b^2-16*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^3-9*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)*
b^4-368*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^4+56*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^2*b^2+9*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)*b^4+3*b^4*sin(2*d*x+2*c)^4+28*a*b^3*sin(2*d*x+2*c)^3+44*a^2*b^2*sin(
2*d*x+2*c)^2-3*b^4*sin(2*d*x+2*c)^2-28*a*b^3*sin(2*d*x+2*c)-44*a^2*b^2)/b/cos(2*d*x+2*c)/(4*a+2*b*sin(2*d*x+2*
c))^(1/2)/d

Fricas [F]

\[ \int (a+b \cos (c+d x) \sin (c+d x))^{5/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((a+b*cos(d*x+c)*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

integral(-(b^2*cos(d*x + c)^4 - b^2*cos(d*x + c)^2 - 2*a*b*cos(d*x + c)*sin(d*x + c) - a^2)*sqrt(b*cos(d*x + c
)*sin(d*x + c) + a), x)

Sympy [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x) \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate((a+b*cos(d*x+c)*sin(d*x+c))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int (a+b \cos (c+d x) \sin (c+d x))^{5/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((a+b*cos(d*x+c)*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*cos(d*x + c)*sin(d*x + c) + a)^(5/2), x)

Giac [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x) \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate((a+b*cos(d*x+c)*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x) \sin (c+d x))^{5/2} \, dx=\int {\left (a+b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]

[In]

int((a + b*cos(c + d*x)*sin(c + d*x))^(5/2),x)

[Out]

int((a + b*cos(c + d*x)*sin(c + d*x))^(5/2), x)