3.54 \(\int \frac {(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=165 \[ -\frac {2 b \left (3 a^2+4 b^2\right ) (e \sin (c+d x))^{3/2}}{3 d e^3}-\frac {2 a \left (a^2+6 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}}-\frac {2 a b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}{d e^3}-\frac {2 (a \cos (c+d x)+b) (a+b \cos (c+d x))^2}{d e \sqrt {e \sin (c+d x)}} \]

[Out]

-2/3*b*(3*a^2+4*b^2)*(e*sin(d*x+c))^(3/2)/d/e^3-2*a*b*(a+b*cos(d*x+c))*(e*sin(d*x+c))^(3/2)/d/e^3-2*(b+a*cos(d
*x+c))*(a+b*cos(d*x+c))^2/d/e/(e*sin(d*x+c))^(1/2)+2*a*(a^2+6*b^2)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2
*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*sin(d*x+c))^(1/2)/d/e^2/sin(d*x+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.25, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2691, 2862, 2669, 2640, 2639} \[ -\frac {2 b \left (3 a^2+4 b^2\right ) (e \sin (c+d x))^{3/2}}{3 d e^3}-\frac {2 a \left (a^2+6 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}}-\frac {2 a b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}{d e^3}-\frac {2 (a \cos (c+d x)+b) (a+b \cos (c+d x))^2}{d e \sqrt {e \sin (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2639

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

Rule 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2669

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] + Dist[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])

Rule 2691

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[((g*C
os[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(b + a*Sin[e + f*x]))/(f*g*(p + 1)), x] + Dist[1/(g^2*(p + 1
)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*Sin
[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (IntegersQ[
2*m, 2*p] || IntegerQ[m])

Rule 2862

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> -Simp[(d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(f*g*(m + p + 1)), x]
+ Dist[1/(m + p + 1), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Simp[a*c*(m + p + 1) + b*d*m + (a*d*
m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && Gt
Q[m, 0] &&  !LtQ[p, -1] && IntegerQ[2*m] &&  !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x])

Rubi steps

\begin {align*} \int \frac {(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{3/2}} \, dx &=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))^2}{d e \sqrt {e \sin (c+d x)}}-\frac {2 \int (a+b \cos (c+d x)) \left (\frac {a^2}{2}+2 b^2+\frac {5}{2} a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)} \, dx}{e^2}\\ &=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))^2}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}{d e^3}-\frac {4 \int \left (\frac {5}{4} a \left (a^2+6 b^2\right )+\frac {5}{4} b \left (3 a^2+4 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)} \, dx}{5 e^2}\\ &=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))^2}{d e \sqrt {e \sin (c+d x)}}-\frac {2 b \left (3 a^2+4 b^2\right ) (e \sin (c+d x))^{3/2}}{3 d e^3}-\frac {2 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}{d e^3}-\frac {\left (a \left (a^2+6 b^2\right )\right ) \int \sqrt {e \sin (c+d x)} \, dx}{e^2}\\ &=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))^2}{d e \sqrt {e \sin (c+d x)}}-\frac {2 b \left (3 a^2+4 b^2\right ) (e \sin (c+d x))^{3/2}}{3 d e^3}-\frac {2 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}{d e^3}-\frac {\left (a \left (a^2+6 b^2\right ) \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{e^2 \sqrt {\sin (c+d x)}}\\ &=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))^2}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a \left (a^2+6 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}}-\frac {2 b \left (3 a^2+4 b^2\right ) (e \sin (c+d x))^{3/2}}{3 d e^3}-\frac {2 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}{d e^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.32, size = 101, normalized size = 0.61 \[ -\frac {2 \left (3 a \left (a^2+3 b^2\right ) \cos (c+d x)-3 a \left (a^2+6 b^2\right ) \sqrt {\sin (c+d x)} E\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )+9 a^2 b+b^3 \sin ^2(c+d x)+3 b^3\right )}{3 d e \sqrt {e \sin (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(9*a^2*b + 3*b^3 + 3*a*(a^2 + 3*b^2)*Cos[c + d*x] - 3*a*(a^2 + 6*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, 2]*
Sqrt[Sin[c + d*x]] + b^3*Sin[c + d*x]^2))/(3*d*e*Sqrt[e*Sin[c + d*x]])

________________________________________________________________________________________

fricas [F]  time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}\right )} \sqrt {e \sin \left (d x + c\right )}}{e^{2} \cos \left (d x + c\right )^{2} - e^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.32, size = 313, normalized size = 1.90 \[ \frac {6 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticE \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{3}+36 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticE \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a \,b^{2}-3 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{3}-18 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a \,b^{2}+2 b^{3} \left (\cos ^{3}\left (d x +c \right )\right )-6 a^{3} \left (\cos ^{2}\left (d x +c \right )\right )-18 b^{2} a \left (\cos ^{2}\left (d x +c \right )\right )-18 a^{2} b \cos \left (d x +c \right )-8 b^{3} \cos \left (d x +c \right )}{3 e \sqrt {e \sin \left (d x +c \right )}\, \cos \left (d x +c \right ) d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/e/(e*sin(d*x+c))^(1/2)/cos(d*x+c)*(6*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*Ellipti
cE((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))*a^3+36*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*Ell
ipticE((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))*a*b^2-3*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2
)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))*a^3-18*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^
(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))*a*b^2+2*b^3*cos(d*x+c)^3-6*a^3*cos(d*x+c)^2-18*b^2*a*cos(d*
x+c)^2-18*a^2*b*cos(d*x+c)-8*b^3*cos(d*x+c))/d

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \cos {\left (c + d x \right )}\right )^{3}}{\left (e \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________