3.567 \(\int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^4 \, dx\)

Optimal. Leaf size=210 \[ -\frac{22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{315 d e}-\frac{2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}+\frac{2 \left (36 a^2 b^2+15 a^4+4 b^4\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}-\frac{10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e} \]

[Out]

(-22*a*b*(17*a^2 + 18*b^2)*(e*Cos[c + d*x])^(3/2))/(315*d*e) + (2*(15*a^4 + 36*a^2*b^2 + 4*b^4)*Sqrt[e*Cos[c +
d*x]]*EllipticE[(c + d*x)/2, 2])/(15*d*Sqrt[Cos[c + d*x]]) - (2*b*(41*a^2 + 14*b^2)*(e*Cos[c + d*x])^(3/2)*(a
+ b*Sin[c + d*x]))/(105*d*e) - (10*a*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2)/(21*d*e) - (2*b*(e*Cos[
c + d*x])^(3/2)*(a + b*Sin[c + d*x])^3)/(9*d*e)

________________________________________________________________________________________

Rubi [A]  time = 0.441776, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2692, 2862, 2669, 2640, 2639} \[ -\frac{22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{315 d e}-\frac{2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}+\frac{2 \left (36 a^2 b^2+15 a^4+4 b^4\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}-\frac{10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^4,x]

[Out]

(-22*a*b*(17*a^2 + 18*b^2)*(e*Cos[c + d*x])^(3/2))/(315*d*e) + (2*(15*a^4 + 36*a^2*b^2 + 4*b^4)*Sqrt[e*Cos[c +
d*x]]*EllipticE[(c + d*x)/2, 2])/(15*d*Sqrt[Cos[c + d*x]]) - (2*b*(41*a^2 + 14*b^2)*(e*Cos[c + d*x])^(3/2)*(a
+ b*Sin[c + d*x]))/(105*d*e) - (10*a*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2)/(21*d*e) - (2*b*(e*Cos[
c + d*x])^(3/2)*(a + b*Sin[c + d*x])^3)/(9*d*e)

Rule 2692

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

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 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 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]

Rubi steps

\begin{align*} \int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^4 \, dx &=-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}+\frac{2}{9} \int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2 \left (\frac{9 a^2}{2}+3 b^2+\frac{15}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac{10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}+\frac{4}{63} \int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x)) \left (\frac{3}{4} a \left (21 a^2+34 b^2\right )+\frac{3}{4} b \left (41 a^2+14 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac{2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}-\frac{10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}+\frac{8}{315} \int \sqrt{e \cos (c+d x)} \left (\frac{21}{8} \left (15 a^4+36 a^2 b^2+4 b^4\right )+\frac{33}{8} a b \left (17 a^2+18 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac{22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{315 d e}-\frac{2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}-\frac{10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}+\frac{1}{15} \left (15 a^4+36 a^2 b^2+4 b^4\right ) \int \sqrt{e \cos (c+d x)} \, dx\\ &=-\frac{22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{315 d e}-\frac{2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}-\frac{10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}+\frac{\left (\left (15 a^4+36 a^2 b^2+4 b^4\right ) \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{15 \sqrt{\cos (c+d x)}}\\ &=-\frac{22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{315 d e}+\frac{2 \left (15 a^4+36 a^2 b^2+4 b^4\right ) \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d \sqrt{\cos (c+d x)}}-\frac{2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}-\frac{10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\\ \end{align*}

Mathematica [A]  time = 1.0896, size = 137, normalized size = 0.65 \[ \frac{\sqrt{e \cos (c+d x)} \left (84 \left (36 a^2 b^2+15 a^4+4 b^4\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )-b \cos ^{\frac{3}{2}}(c+d x) \left (21 b \left (72 a^2+13 b^2\right ) \sin (c+d x)+5 \left (336 a^3+264 a b^2-7 b^3 \sin (3 (c+d x))\right )-360 a b^2 \cos (2 (c+d x))\right )\right )}{630 d \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^4,x]

[Out]

(Sqrt[e*Cos[c + d*x]]*(84*(15*a^4 + 36*a^2*b^2 + 4*b^4)*EllipticE[(c + d*x)/2, 2] - b*Cos[c + d*x]^(3/2)*(-360
*a*b^2*Cos[2*(c + d*x)] + 21*b*(72*a^2 + 13*b^2)*Sin[c + d*x] + 5*(336*a^3 + 264*a*b^2 - 7*b^3*Sin[3*(c + d*x)
]))))/(630*d*Sqrt[Cos[c + d*x]])

________________________________________________________________________________________

Maple [B]  time = 1.846, size = 525, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(d*x+c))^4*(e*cos(d*x+c))^(1/2),x)

[Out]

2/315/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e*(1120*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)
^10+2880*a*b^3*sin(1/2*d*x+1/2*c)^9-2240*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-3024*a^2*b^2*cos(1/2*d*x+
1/2*c)*sin(1/2*d*x+1/2*c)^6-5760*a*b^3*sin(1/2*d*x+1/2*c)^7+1064*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-1
680*a^3*b*sin(1/2*d*x+1/2*c)^5+3024*a^2*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+2640*a*b^3*sin(1/2*d*x+1/2
*c)^5+56*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+315*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-
1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^4+756*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)
^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2*b^2+84*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1
)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^4+1680*a^3*b*sin(1/2*d*x+1/2*c)^3-756*a^2*b^2*cos(1/2*d*x+1/2*
c)*sin(1/2*d*x+1/2*c)^2+240*a*b^3*sin(1/2*d*x+1/2*c)^3-84*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-420*a^3*
b*sin(1/2*d*x+1/2*c)-240*a*b^3*sin(1/2*d*x+1/2*c))/d

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e \cos \left (d x + c\right )}{\left (b \sin \left (d x + c\right ) + a\right )}^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c))^4*(e*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(e*cos(d*x + c))*(b*sin(d*x + c) + a)^4, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{4} \cos \left (d x + c\right )^{4} + a^{4} + 6 \, a^{2} b^{2} + b^{4} - 2 \,{\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} - 4 \,{\left (a b^{3} \cos \left (d x + c\right )^{2} - a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt{e \cos \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c))^4*(e*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral((b^4*cos(d*x + c)^4 + a^4 + 6*a^2*b^2 + b^4 - 2*(3*a^2*b^2 + b^4)*cos(d*x + c)^2 - 4*(a*b^3*cos(d*x +
c)^2 - a^3*b - a*b^3)*sin(d*x + c))*sqrt(e*cos(d*x + c)), x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c))**4*(e*cos(d*x+c))**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e \cos \left (d x + c\right )}{\left (b \sin \left (d x + c\right ) + a\right )}^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c))^4*(e*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(e*cos(d*x + c))*(b*sin(d*x + c) + a)^4, x)