3.565 \(\int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^4 \, dx\)

Optimal. Leaf size=258 \[ \frac{2 e^2 \left (52 a^2 b^2+39 a^4+4 b^4\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{65 d \sqrt{\cos (c+d x)}}-\frac{10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{3003 d e}-\frac{2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}+\frac{2 e \left (52 a^2 b^2+39 a^4+4 b^4\right ) \sin (c+d x) (e \cos (c+d x))^{3/2}}{195 d}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}-\frac{38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e} \]

[Out]

(-10*a*b*(115*a^2 + 94*b^2)*(e*Cos[c + d*x])^(7/2))/(3003*d*e) + (2*(39*a^4 + 52*a^2*b^2 + 4*b^4)*e^2*Sqrt[e*C
os[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(65*d*Sqrt[Cos[c + d*x]]) + (2*(39*a^4 + 52*a^2*b^2 + 4*b^4)*e*(e*Cos[
c + d*x])^(3/2)*Sin[c + d*x])/(195*d) - (2*b*(73*a^2 + 22*b^2)*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x]))/(4
29*d*e) - (38*a*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^2)/(143*d*e) - (2*b*(e*Cos[c + d*x])^(7/2)*(a +
b*Sin[c + d*x])^3)/(13*d*e)

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Rubi [A]  time = 0.508205, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2692, 2862, 2669, 2635, 2640, 2639} \[ \frac{2 e^2 \left (52 a^2 b^2+39 a^4+4 b^4\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{65 d \sqrt{\cos (c+d x)}}-\frac{10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{3003 d e}-\frac{2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}+\frac{2 e \left (52 a^2 b^2+39 a^4+4 b^4\right ) \sin (c+d x) (e \cos (c+d x))^{3/2}}{195 d}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}-\frac{38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e} \]

Antiderivative was successfully verified.

[In]

Int[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^4,x]

[Out]

(-10*a*b*(115*a^2 + 94*b^2)*(e*Cos[c + d*x])^(7/2))/(3003*d*e) + (2*(39*a^4 + 52*a^2*b^2 + 4*b^4)*e^2*Sqrt[e*C
os[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(65*d*Sqrt[Cos[c + d*x]]) + (2*(39*a^4 + 52*a^2*b^2 + 4*b^4)*e*(e*Cos[
c + d*x])^(3/2)*Sin[c + d*x])/(195*d) - (2*b*(73*a^2 + 22*b^2)*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x]))/(4
29*d*e) - (38*a*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^2)/(143*d*e) - (2*b*(e*Cos[c + d*x])^(7/2)*(a +
b*Sin[c + d*x])^3)/(13*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 2635

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

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 (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^4 \, dx &=-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}+\frac{2}{13} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \left (\frac{13 a^2}{2}+3 b^2+\frac{19}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac{38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}+\frac{4}{143} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x)) \left (\frac{1}{4} a \left (143 a^2+142 b^2\right )+\frac{3}{4} b \left (73 a^2+22 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac{2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}-\frac{38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}+\frac{8 \int (e \cos (c+d x))^{5/2} \left (\frac{33}{8} \left (39 a^4+52 a^2 b^2+4 b^4\right )+\frac{15}{8} a b \left (115 a^2+94 b^2\right ) \sin (c+d x)\right ) \, dx}{1287}\\ &=-\frac{10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{3003 d e}-\frac{2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}-\frac{38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}+\frac{1}{39} \left (39 a^4+52 a^2 b^2+4 b^4\right ) \int (e \cos (c+d x))^{5/2} \, dx\\ &=-\frac{10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{3003 d e}+\frac{2 \left (39 a^4+52 a^2 b^2+4 b^4\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{195 d}-\frac{2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}-\frac{38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}+\frac{1}{65} \left (\left (39 a^4+52 a^2 b^2+4 b^4\right ) e^2\right ) \int \sqrt{e \cos (c+d x)} \, dx\\ &=-\frac{10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{3003 d e}+\frac{2 \left (39 a^4+52 a^2 b^2+4 b^4\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{195 d}-\frac{2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}-\frac{38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}+\frac{\left (\left (39 a^4+52 a^2 b^2+4 b^4\right ) e^2 \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{65 \sqrt{\cos (c+d x)}}\\ &=-\frac{10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{3003 d e}+\frac{2 \left (39 a^4+52 a^2 b^2+4 b^4\right ) e^2 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{65 d \sqrt{\cos (c+d x)}}+\frac{2 \left (39 a^4+52 a^2 b^2+4 b^4\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{195 d}-\frac{2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}-\frac{38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\\ \end{align*}

Mathematica [A]  time = 2.10344, size = 209, normalized size = 0.81 \[ \frac{(e \cos (c+d x))^{5/2} \left (2 \left (52 a^2 b^2+39 a^4+4 b^4\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+65 \sqrt{\cos (c+d x)} \left (-\frac{1}{78} b^2 \left (13 a^2+b^2\right ) \sin (4 (c+d x))+\frac{\left (-208 a^2 b^2+624 a^4-61 b^4\right ) \sin (2 (c+d x))}{3120}-\frac{1}{77} a b \left (66 a^2+31 b^2\right ) \cos (c+d x)-\frac{1}{154} a b \left (44 a^2+9 b^2\right ) \cos (3 (c+d x))+\frac{1}{22} a b^3 \cos (5 (c+d x))+\frac{1}{208} b^4 \sin (6 (c+d x))\right )\right )}{65 d \cos ^{\frac{5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^4,x]

[Out]

((e*Cos[c + d*x])^(5/2)*(2*(39*a^4 + 52*a^2*b^2 + 4*b^4)*EllipticE[(c + d*x)/2, 2] + 65*Sqrt[Cos[c + d*x]]*(-(
a*b*(66*a^2 + 31*b^2)*Cos[c + d*x])/77 - (a*b*(44*a^2 + 9*b^2)*Cos[3*(c + d*x)])/154 + (a*b^3*Cos[5*(c + d*x)]
)/22 + ((624*a^4 - 208*a^2*b^2 - 61*b^4)*Sin[2*(c + d*x)])/3120 - (b^2*(13*a^2 + b^2)*Sin[4*(c + d*x)])/78 + (
b^4*Sin[6*(c + d*x)])/208)))/(65*d*Cos[c + d*x]^(5/2))

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Maple [B]  time = 2.417, size = 776, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^4,x)

[Out]

2/15015/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^3*(12012*(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+616*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x
+1/2*c)^4+147840*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^14-320320*a^2*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/
2*c)^10+640640*a^2*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-448448*a^2*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1
/2*c)^6+128128*a^2*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4-12012*a^2*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/
2*c)^2+9009*(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+924*(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^4-443520*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+492800*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10-24
6400*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+24024*a^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+48664*b^4*c
os(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-24024*a^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+6006*a^4*cos(1/2*d*x+
1/2*c)*sin(1/2*d*x+1/2*c)^2-924*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+349440*a*b^3*sin(1/2*d*x+1/2*c)^13
-1048320*a*b^3*sin(1/2*d*x+1/2*c)^11-137280*a^3*b*sin(1/2*d*x+1/2*c)^9+1173120*a*b^3*sin(1/2*d*x+1/2*c)^9+2745
60*a^3*b*sin(1/2*d*x+1/2*c)^7-599040*a*b^3*sin(1/2*d*x+1/2*c)^7-205920*a^3*b*sin(1/2*d*x+1/2*c)^5+121680*a*b^3
*sin(1/2*d*x+1/2*c)^5+68640*a^3*b*sin(1/2*d*x+1/2*c)^3+3120*a*b^3*sin(1/2*d*x+1/2*c)^3-8580*a^3*b*sin(1/2*d*x+
1/2*c)-3120*a*b^3*sin(1/2*d*x+1/2*c))/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \cos \left (d x + c\right )\right )^{\frac{5}{2}}{\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((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

integrate((e*cos(d*x + c))^(5/2)*(b*sin(d*x + c) + a)^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{4} e^{2} \cos \left (d x + c\right )^{6} - 2 \,{\left (3 \, a^{2} b^{2} + b^{4}\right )} e^{2} \cos \left (d x + c\right )^{4} +{\left (a^{4} + 6 \, a^{2} b^{2} + b^{4}\right )} e^{2} \cos \left (d x + c\right )^{2} - 4 \,{\left (a b^{3} e^{2} \cos \left (d x + c\right )^{4} -{\left (a^{3} b + a b^{3}\right )} e^{2} \cos \left (d x + c\right )^{2}\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((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

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

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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((e*cos(d*x+c))**(5/2)*(a+b*sin(d*x+c))**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \cos \left (d x + c\right )\right )^{\frac{5}{2}}{\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((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((e*cos(d*x + c))^(5/2)*(b*sin(d*x + c) + a)^4, x)