3.6.47 \(\int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2 \, dx\) [547]

Optimal. Leaf size=188 \[ -\frac {26 a b (e \cos (c+d x))^{9/2}}{99 d e}+\frac {10 \left (11 a^2+2 b^2\right ) e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {10 \left (11 a^2+2 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (11 a^2+2 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 188, 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 {\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 {10 e^3 \left (11 a^2+2 b^2\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{231 d}+\frac {2 e \left (11 a^2+2 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac {26 a b (e \cos (c+d x))^{9/2}}{99 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-26*a*b*(e*Cos[c + d*x])^(9/2))/(99*d*e) + (10*(11*a^2 + 2*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2,
2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (10*(11*a^2 + 2*b^2)*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(
11*a^2 + 2*b^2)*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*d) - (2*b*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x
]))/(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 (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2 \, dx &=-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}+\frac {2}{11} \int (e \cos (c+d x))^{7/2} \left (\frac {11 a^2}{2}+b^2+\frac {13}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac {26 a b (e \cos (c+d x))^{9/2}}{99 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}+\frac {1}{11} \left (11 a^2+2 b^2\right ) \int (e \cos (c+d x))^{7/2} \, dx\\ &=-\frac {26 a b (e \cos (c+d x))^{9/2}}{99 d e}+\frac {2 \left (11 a^2+2 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}+\frac {1}{77} \left (5 \left (11 a^2+2 b^2\right ) e^2\right ) \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac {26 a b (e \cos (c+d x))^{9/2}}{99 d e}+\frac {10 \left (11 a^2+2 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (11 a^2+2 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}+\frac {1}{231} \left (5 \left (11 a^2+2 b^2\right ) e^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {26 a b (e \cos (c+d x))^{9/2}}{99 d e}+\frac {10 \left (11 a^2+2 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (11 a^2+2 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}+\frac {\left (5 \left (11 a^2+2 b^2\right ) 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 {26 a b (e \cos (c+d x))^{9/2}}{99 d e}+\frac {10 \left (11 a^2+2 b^2\right ) 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 {10 \left (11 a^2+2 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (11 a^2+2 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.92, size = 160, normalized size = 0.85 \begin {gather*} \frac {(e \cos (c+d x))^{7/2} \left (-154 a b \sqrt {\cos (c+d x)}+40 \left (11 a^2+2 b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\frac {1}{6} \sqrt {\cos (c+d x)} \left (6 \left (572 a^2+41 b^2\right ) \sin (c+d x)-14 b \cos (4 (c+d x)) (22 a+9 b \sin (c+d x))+8 \cos (2 (c+d x)) \left (-154 a b+9 \left (11 a^2-5 b^2\right ) \sin (c+d x)\right )\right )\right )}{924 d \cos ^{\frac {7}{2}}(c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((e*Cos[c + d*x])^(7/2)*(-154*a*b*Sqrt[Cos[c + d*x]] + 40*(11*a^2 + 2*b^2)*EllipticF[(c + d*x)/2, 2] + (Sqrt[C
os[c + d*x]]*(6*(572*a^2 + 41*b^2)*Sin[c + d*x] - 14*b*Cos[4*(c + d*x)]*(22*a + 9*b*Sin[c + d*x]) + 8*Cos[2*(c
+ d*x)]*(-154*a*b + 9*(11*a^2 - 5*b^2)*Sin[c + d*x])))/6))/(924*d*Cos[c + d*x]^(7/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(472\) vs. \(2(192)=384\).
time = 5.16, size = 473, normalized size = 2.52

method	result	size
default	\(-\frac {2 e^{4} \left (-4032 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4928 a b \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10080 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1584 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12320 a b \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9792 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2376 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12320 a b \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4608 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1848 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6160 a b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-924 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+165 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}+30 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}-528 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1540 a b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+30 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+154 a b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{693 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\)	\(473\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/693/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^4*(-4032*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/
2*c)^12-4928*a*b*sin(1/2*d*x+1/2*c)^11+10080*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10+1584*a^2*cos(1/2*d*x
+1/2*c)*sin(1/2*d*x+1/2*c)^8+12320*a*b*sin(1/2*d*x+1/2*c)^9-9792*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-2
376*a^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-12320*a*b*sin(1/2*d*x+1/2*c)^7+4608*b^2*cos(1/2*d*x+1/2*c)*sin
(1/2*d*x+1/2*c)^6+1848*a^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+6160*a*b*sin(1/2*d*x+1/2*c)^5-924*b^2*cos(1
/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+165*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF
(cos(1/2*d*x+1/2*c),2^(1/2))*a^2+30*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(co
s(1/2*d*x+1/2*c),2^(1/2))*b^2-528*a^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-1540*a*b*sin(1/2*d*x+1/2*c)^3+30
*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+154*a*b*sin(1/2*d*x+1/2*c))/d

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.13, size = 165, normalized size = 0.88 \begin {gather*} \frac {-15 i \, \sqrt {2} {\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 i \, \sqrt {2} {\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 (154 \, a b \cos \left (d x + c\right )^{4} e^{\frac {7}{2}} + 3 \, {\left (21 \, b^{2} \cos \left (d x + c\right )^{4} e^{\frac {7}{2}} - 3 \, {\left (11 \, a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} e^{\frac {7}{2}} - 5 \, {\left (11 \, a^{2} + 2 \, b^{2}\right )} e^{\frac {7}{2}}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{693 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/693*(-15*I*sqrt(2)*(11*a^2 + 2*b^2)*e^(7/2)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*I
*sqrt(2)*(11*a^2 + 2*b^2)*e^(7/2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 2*(154*a*b*cos(d
*x + c)^4*e^(7/2) + 3*(21*b^2*cos(d*x + c)^4*e^(7/2) - 3*(11*a^2 + 2*b^2)*cos(d*x + c)^2*e^(7/2) - 5*(11*a^2 +
2*b^2)*e^(7/2))*sin(d*x + c))*sqrt(cos(d*x + c)))/d

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________