3.214 \(\int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=203 \[ \frac{170 a^3 e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{170 a^3 e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d \sqrt{e \cos (c+d x)}}-\frac{34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}-\frac{34 \left (a^3 \sin (c+d x)+a^3\right ) (e \cos (c+d x))^{9/2}}{143 d e}+\frac{34 a^3 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac{2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{9/2}}{13 d e} \]

[Out]

(-34*a^3*(e*Cos[c + d*x])^(9/2))/(99*d*e) + (170*a^3*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*
Sqrt[e*Cos[c + d*x]]) + (170*a^3*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (34*a^3*e*(e*Cos[c + d*x])^(
5/2)*Sin[c + d*x])/(77*d) - (2*a*(e*Cos[c + d*x])^(9/2)*(a + a*Sin[c + d*x])^2)/(13*d*e) - (34*(e*Cos[c + d*x]
)^(9/2)*(a^3 + a^3*Sin[c + d*x]))/(143*d*e)

________________________________________________________________________________________

Rubi [A]  time = 0.211256, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2678, 2669, 2635, 2642, 2641} \[ \frac{170 a^3 e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{170 a^3 e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d \sqrt{e \cos (c+d x)}}-\frac{34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}-\frac{34 \left (a^3 \sin (c+d x)+a^3\right ) (e \cos (c+d x))^{9/2}}{143 d e}+\frac{34 a^3 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac{2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{9/2}}{13 d e} \]

Antiderivative was successfully verified.

[In]

Int[(e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^3,x]

[Out]

(-34*a^3*(e*Cos[c + d*x])^(9/2))/(99*d*e) + (170*a^3*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*
Sqrt[e*Cos[c + d*x]]) + (170*a^3*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (34*a^3*e*(e*Cos[c + d*x])^(
5/2)*Sin[c + d*x])/(77*d) - (2*a*(e*Cos[c + d*x])^(9/2)*(a + a*Sin[c + d*x])^2)/(13*d*e) - (34*(e*Cos[c + d*x]
)^(9/2)*(a^3 + a^3*Sin[c + d*x]))/(143*d*e)

Rule 2678

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[(a*(2*m + p - 1))/(m + p), Int[(
g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0]
&& GtQ[m, 0] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]

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 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2641

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

Rubi steps

\begin{align*} \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^3 \, dx &=-\frac{2 a (e \cos (c+d x))^{9/2} (a+a \sin (c+d x))^2}{13 d e}+\frac{1}{13} (17 a) \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx\\ &=-\frac{2 a (e \cos (c+d x))^{9/2} (a+a \sin (c+d x))^2}{13 d e}-\frac{34 (e \cos (c+d x))^{9/2} \left (a^3+a^3 \sin (c+d x)\right )}{143 d e}+\frac{1}{11} \left (17 a^2\right ) \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx\\ &=-\frac{34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}-\frac{2 a (e \cos (c+d x))^{9/2} (a+a \sin (c+d x))^2}{13 d e}-\frac{34 (e \cos (c+d x))^{9/2} \left (a^3+a^3 \sin (c+d x)\right )}{143 d e}+\frac{1}{11} \left (17 a^3\right ) \int (e \cos (c+d x))^{7/2} \, dx\\ &=-\frac{34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}+\frac{34 a^3 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac{2 a (e \cos (c+d x))^{9/2} (a+a \sin (c+d x))^2}{13 d e}-\frac{34 (e \cos (c+d x))^{9/2} \left (a^3+a^3 \sin (c+d x)\right )}{143 d e}+\frac{1}{77} \left (85 a^3 e^2\right ) \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac{34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}+\frac{170 a^3 e^3 \sqrt{e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac{34 a^3 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac{2 a (e \cos (c+d x))^{9/2} (a+a \sin (c+d x))^2}{13 d e}-\frac{34 (e \cos (c+d x))^{9/2} \left (a^3+a^3 \sin (c+d x)\right )}{143 d e}+\frac{1}{231} \left (85 a^3 e^4\right ) \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx\\ &=-\frac{34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}+\frac{170 a^3 e^3 \sqrt{e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac{34 a^3 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac{2 a (e \cos (c+d x))^{9/2} (a+a \sin (c+d x))^2}{13 d e}-\frac{34 (e \cos (c+d x))^{9/2} \left (a^3+a^3 \sin (c+d x)\right )}{143 d e}+\frac{\left (85 a^3 e^4 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{231 \sqrt{e \cos (c+d x)}}\\ &=-\frac{34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}+\frac{170 a^3 e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d \sqrt{e \cos (c+d x)}}+\frac{170 a^3 e^3 \sqrt{e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac{34 a^3 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac{2 a (e \cos (c+d x))^{9/2} (a+a \sin (c+d x))^2}{13 d e}-\frac{34 (e \cos (c+d x))^{9/2} \left (a^3+a^3 \sin (c+d x)\right )}{143 d e}\\ \end{align*}

Mathematica [C]  time = 0.0742641, size = 66, normalized size = 0.33 \[ -\frac{64 \sqrt [4]{2} a^3 (e \cos (c+d x))^{9/2} \, _2F_1\left (-\frac{17}{4},\frac{9}{4};\frac{13}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{9 d e (\sin (c+d x)+1)^{9/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^3,x]

[Out]

(-64*2^(1/4)*a^3*(e*Cos[c + d*x])^(9/2)*Hypergeometric2F1[-17/4, 9/4, 13/4, (1 - Sin[c + d*x])/2])/(9*d*e*(1 +
Sin[c + d*x])^(9/4))

________________________________________________________________________________________

Maple [A]  time = 0.595, size = 321, normalized size = 1.6 \begin{align*} -{\frac{2\,{a}^{3}{e}^{4}}{9009\,d} \left ( 88704\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{15}-157248\,\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{12}-310464\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{13}+393120\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}\cos \left ( 1/2\,dx+c/2 \right ) +337568\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{11}-361296\,\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}-67760\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{9}+148824\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) -126280\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}-12012\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +101948\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+3315\,\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -5694\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) -30338\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+3311\,\sin \left ( 1/2\,dx+c/2 \right ) \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(d*x+c))^(7/2)*(a+a*sin(d*x+c))^3,x)

[Out]

-2/9009/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*a^3*e^4*(88704*sin(1/2*d*x+1/2*c)^15-157248*cos
(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12-310464*sin(1/2*d*x+1/2*c)^13+393120*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/
2*c)+337568*sin(1/2*d*x+1/2*c)^11-361296*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-67760*sin(1/2*d*x+1/2*c)^9+14
8824*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-126280*sin(1/2*d*x+1/2*c)^7-12012*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*
x+1/2*c)+101948*sin(1/2*d*x+1/2*c)^5+3315*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*Ellipt
icF(cos(1/2*d*x+1/2*c),2^(1/2))-5694*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-30338*sin(1/2*d*x+1/2*c)^3+3311*s
in(1/2*d*x+1/2*c))/d

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \cos \left (d x + c\right )\right )^{\frac{7}{2}}{\left (a \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(7/2)*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

integrate((e*cos(d*x + c))^(7/2)*(a*sin(d*x + c) + a)^3, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (3 \, a^{3} e^{3} \cos \left (d x + c\right )^{5} - 4 \, a^{3} e^{3} \cos \left (d x + c\right )^{3} +{\left (a^{3} e^{3} \cos \left (d x + c\right )^{5} - 4 \, a^{3} e^{3} \cos \left (d x + c\right )^{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((e*cos(d*x+c))^(7/2)*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

integral(-(3*a^3*e^3*cos(d*x + c)^5 - 4*a^3*e^3*cos(d*x + c)^3 + (a^3*e^3*cos(d*x + c)^5 - 4*a^3*e^3*cos(d*x +
c)^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((e*cos(d*x+c))**(7/2)*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \cos \left (d x + c\right )\right )^{\frac{7}{2}}{\left (a \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(7/2)*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((e*cos(d*x + c))^(7/2)*(a*sin(d*x + c) + a)^3, x)