3.1.41 \(\int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2} \, dx\) [41]

Optimal. Leaf size=193 \[ \frac {10 \left (11 a^2+2 b^2\right ) e^4 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{231 d \sqrt {e \sin (c+d x)}}-\frac {10 \left (11 a^2+2 b^2\right ) e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 \left (11 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e} \]

[Out]

-2/77*(11*a^2+2*b^2)*e*cos(d*x+c)*(e*sin(d*x+c))^(5/2)/d+26/99*a*b*(e*sin(d*x+c))^(9/2)/d/e+2/11*b*(a+b*cos(d*
x+c))*(e*sin(d*x+c))^(9/2)/d/e-10/231*(11*a^2+2*b^2)*e^4*(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)*EllipticF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*sin(d*x+c)^(1/2)/d/(e*sin(d*x+c))^(1/2)-10/231*(11*a^2+2
*b^2)*e^3*cos(d*x+c)*(e*sin(d*x+c))^(1/2)/d

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Rubi [A]
time = 0.13, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2771, 2748, 2715, 2721, 2720} \begin {gather*} \frac {10 e^4 \left (11 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{231 d \sqrt {e \sin (c+d x)}}-\frac {10 e^3 \left (11 a^2+2 b^2\right ) \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 e \left (11 a^2+2 b^2\right ) \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*(11*a^2 + 2*b^2)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(231*d*Sqrt[e*Sin[c + d*x]]) - (
10*(11*a^2 + 2*b^2)*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(231*d) - (2*(11*a^2 + 2*b^2)*e*Cos[c + d*x]*(e*Sin
[c + d*x])^(5/2))/(77*d) + (26*a*b*(e*Sin[c + d*x])^(9/2))/(99*d*e) + (2*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x
])^(9/2))/(11*d*e)

Rule 2715

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] && Integ
erQ[2*n]

Rule 2720

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

Rule 2721

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

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[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 2771

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])

Rubi steps

\begin {align*} \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2} \, dx &=\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}+\frac {2}{11} \int \left (\frac {11 a^2}{2}+b^2+\frac {13}{2} a b \cos (c+d x)\right ) (e \sin (c+d x))^{7/2} \, dx\\ &=\frac {26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}+\frac {1}{11} \left (11 a^2+2 b^2\right ) \int (e \sin (c+d x))^{7/2} \, dx\\ &=-\frac {2 \left (11 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}+\frac {1}{77} \left (5 \left (11 a^2+2 b^2\right ) e^2\right ) \int (e \sin (c+d x))^{3/2} \, dx\\ &=-\frac {10 \left (11 a^2+2 b^2\right ) e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 \left (11 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}+\frac {1}{231} \left (5 \left (11 a^2+2 b^2\right ) e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx\\ &=-\frac {10 \left (11 a^2+2 b^2\right ) e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 \left (11 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}+\frac {\left (5 \left (11 a^2+2 b^2\right ) e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{231 \sqrt {e \sin (c+d x)}}\\ &=\frac {10 \left (11 a^2+2 b^2\right ) e^4 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{231 d \sqrt {e \sin (c+d x)}}-\frac {10 \left (11 a^2+2 b^2\right ) e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 \left (11 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}\\ \end {align*}

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Mathematica [A]
time = 1.73, size = 157, normalized size = 0.81 \begin {gather*} \frac {\left (\frac {1}{6} \left (924 a b-6 \left (506 a^2+71 b^2\right ) \cos (c+d x)-1232 a b \cos (2 (c+d x))+396 a^2 \cos (3 (c+d x))-117 b^2 \cos (3 (c+d x))+308 a b \cos (4 (c+d x))+63 b^2 \cos (5 (c+d x))\right ) \csc ^3(c+d x)-\frac {40 \left (11 a^2+2 b^2\right ) F\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right )}{\sin ^{\frac {7}{2}}(c+d x)}\right ) (e \sin (c+d x))^{7/2}}{924 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((((924*a*b - 6*(506*a^2 + 71*b^2)*Cos[c + d*x] - 1232*a*b*Cos[2*(c + d*x)] + 396*a^2*Cos[3*(c + d*x)] - 117*b
^2*Cos[3*(c + d*x)] + 308*a*b*Cos[4*(c + d*x)] + 63*b^2*Cos[5*(c + d*x)])*Csc[c + d*x]^3)/6 - (40*(11*a^2 + 2*
b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, 2])/Sin[c + d*x]^(7/2))*(e*Sin[c + d*x])^(7/2))/(924*d)

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Maple [A]
time = 0.14, size = 252, normalized size = 1.31

method	result	size
default	\(\frac {\frac {4 a b \left (e \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9 e}-\frac {e^{4} \left (-42 b^{2} \left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right )-66 a^{2} \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+72 b^{2} \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+55 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2}+10 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2}+176 a^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-10 b^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )\right )}{231 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\)	\(252\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cos(d*x+c))^2*(e*sin(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

(4/9/e*a*b*(e*sin(d*x+c))^(9/2)-1/231*e^4*(-42*b^2*cos(d*x+c)^6*sin(d*x+c)-66*a^2*cos(d*x+c)^4*sin(d*x+c)+72*b
^2*cos(d*x+c)^4*sin(d*x+c)+55*(-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^2+10*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((-si
n(d*x+c)+1)^(1/2),1/2*2^(1/2))*b^2+176*a^2*cos(d*x+c)^2*sin(d*x+c)-10*b^2*cos(d*x+c)^2*sin(d*x+c))/cos(d*x+c)/
(e*sin(d*x+c))^(1/2))/d

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.13, size = 188, normalized size = 0.97 \begin {gather*} \frac {15 \, \sqrt {2} \sqrt {-i} {\left (11 \, a^{2} + 2 \, b^{2}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} \sqrt {i} {\left (11 \, a^{2} + 2 \, b^{2}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (63 \, b^{2} \cos \left (d x + c\right )^{5} e^{\frac {7}{2}} + 154 \, a b \cos \left (d x + c\right )^{4} e^{\frac {7}{2}} - 308 \, a b \cos \left (d x + c\right )^{2} e^{\frac {7}{2}} + 9 \, {\left (11 \, a^{2} - 12 \, b^{2}\right )} \cos \left (d x + c\right )^{3} e^{\frac {7}{2}} + 154 \, a b e^{\frac {7}{2}} - 3 \, {\left (88 \, a^{2} - 5 \, b^{2}\right )} \cos \left (d x + c\right ) e^{\frac {7}{2}}\right )} \sqrt {\sin \left (d x + c\right )}}{693 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/693*(15*sqrt(2)*sqrt(-I)*(11*a^2 + 2*b^2)*e^(7/2)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) +
15*sqrt(2)*sqrt(I)*(11*a^2 + 2*b^2)*e^(7/2)*weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2*(63*
b^2*cos(d*x + c)^5*e^(7/2) + 154*a*b*cos(d*x + c)^4*e^(7/2) - 308*a*b*cos(d*x + c)^2*e^(7/2) + 9*(11*a^2 - 12*
b^2)*cos(d*x + c)^3*e^(7/2) + 154*a*b*e^(7/2) - 3*(88*a^2 - 5*b^2)*cos(d*x + c)*e^(7/2))*sqrt(sin(d*x + c)))/d

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5986 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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