3.2.82 \(\int \frac {\cos ^{10}(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx\) [182]

Optimal. Leaf size=95 \[ -\frac {64 a^3 \cos ^{11}(c+d x)}{2145 d (a+a \sin (c+d x))^{11/2}}-\frac {16 a^2 \cos ^{11}(c+d x)}{195 d (a+a \sin (c+d x))^{9/2}}-\frac {2 a \cos ^{11}(c+d x)}{15 d (a+a \sin (c+d x))^{7/2}} \]

[Out]

-64/2145*a^3*cos(d*x+c)^11/d/(a+a*sin(d*x+c))^(11/2)-16/195*a^2*cos(d*x+c)^11/d/(a+a*sin(d*x+c))^(9/2)-2/15*a*
cos(d*x+c)^11/d/(a+a*sin(d*x+c))^(7/2)

________________________________________________________________________________________

Rubi [A]
time = 0.12, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2753, 2752} \begin {gather*} -\frac {64 a^3 \cos ^{11}(c+d x)}{2145 d (a \sin (c+d x)+a)^{11/2}}-\frac {16 a^2 \cos ^{11}(c+d x)}{195 d (a \sin (c+d x)+a)^{9/2}}-\frac {2 a \cos ^{11}(c+d x)}{15 d (a \sin (c+d x)+a)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^10/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(-64*a^3*Cos[c + d*x]^11)/(2145*d*(a + a*Sin[c + d*x])^(11/2)) - (16*a^2*Cos[c + d*x]^11)/(195*d*(a + a*Sin[c
+ d*x])^(9/2)) - (2*a*Cos[c + d*x]^11)/(15*d*(a + a*Sin[c + d*x])^(7/2))

Rule 2752

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

Rule 2753

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] && IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]

Rubi steps

\begin {align*} \int \frac {\cos ^{10}(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=-\frac {2 a \cos ^{11}(c+d x)}{15 d (a+a \sin (c+d x))^{7/2}}+\frac {1}{15} (8 a) \int \frac {\cos ^{10}(c+d x)}{(a+a \sin (c+d x))^{7/2}} \, dx\\ &=-\frac {16 a^2 \cos ^{11}(c+d x)}{195 d (a+a \sin (c+d x))^{9/2}}-\frac {2 a \cos ^{11}(c+d x)}{15 d (a+a \sin (c+d x))^{7/2}}+\frac {1}{195} \left (32 a^2\right ) \int \frac {\cos ^{10}(c+d x)}{(a+a \sin (c+d x))^{9/2}} \, dx\\ &=-\frac {64 a^3 \cos ^{11}(c+d x)}{2145 d (a+a \sin (c+d x))^{11/2}}-\frac {16 a^2 \cos ^{11}(c+d x)}{195 d (a+a \sin (c+d x))^{9/2}}-\frac {2 a \cos ^{11}(c+d x)}{15 d (a+a \sin (c+d x))^{7/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.41, size = 59, normalized size = 0.62 \begin {gather*} -\frac {2 \cos ^{11}(c+d x) \left (263+374 \sin (c+d x)+143 \sin ^2(c+d x)\right )}{2145 d (1+\sin (c+d x))^3 (a (1+\sin (c+d x)))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^10/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(-2*Cos[c + d*x]^11*(263 + 374*Sin[c + d*x] + 143*Sin[c + d*x]^2))/(2145*d*(1 + Sin[c + d*x])^3*(a*(1 + Sin[c
+ d*x]))^(5/2))

________________________________________________________________________________________

Maple [A]
time = 0.43, size = 67, normalized size = 0.71

method	result	size
default	\(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) \left (\sin \left (d x +c \right )-1\right )^{6} \left (143 \left (\sin ^{2}\left (d x +c \right )\right )+374 \sin \left (d x +c \right )+263\right )}{2145 a^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^10/(a+a*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/2145/a^2*(1+sin(d*x+c))*(sin(d*x+c)-1)^6*(143*sin(d*x+c)^2+374*sin(d*x+c)+263)/cos(d*x+c)/(a+a*sin(d*x+c))^
(1/2)/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(cos(d*x+c)^10/(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^10/(a*sin(d*x + c) + a)^(5/2), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (83) = 166\).
time = 0.36, size = 201, normalized size = 2.12 \begin {gather*} -\frac {2 \, {\left (143 \, \cos \left (d x + c\right )^{8} - 341 \, \cos \left (d x + c\right )^{7} - 736 \, \cos \left (d x + c\right )^{6} + 28 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{4} + 64 \, \cos \left (d x + c\right )^{3} - 128 \, \cos \left (d x + c\right )^{2} + {\left (143 \, \cos \left (d x + c\right )^{7} + 484 \, \cos \left (d x + c\right )^{6} - 252 \, \cos \left (d x + c\right )^{5} - 280 \, \cos \left (d x + c\right )^{4} - 320 \, \cos \left (d x + c\right )^{3} - 384 \, \cos \left (d x + c\right )^{2} - 512 \, \cos \left (d x + c\right ) - 1024\right )} \sin \left (d x + c\right ) + 512 \, \cos \left (d x + c\right ) + 1024\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{2145 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^10/(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

-2/2145*(143*cos(d*x + c)^8 - 341*cos(d*x + c)^7 - 736*cos(d*x + c)^6 + 28*cos(d*x + c)^5 - 40*cos(d*x + c)^4
+ 64*cos(d*x + c)^3 - 128*cos(d*x + c)^2 + (143*cos(d*x + c)^7 + 484*cos(d*x + c)^6 - 252*cos(d*x + c)^5 - 280
*cos(d*x + c)^4 - 320*cos(d*x + c)^3 - 384*cos(d*x + c)^2 - 512*cos(d*x + c) - 1024)*sin(d*x + c) + 512*cos(d*
x + c) + 1024)*sqrt(a*sin(d*x + c) + a)/(a^3*d*cos(d*x + c) + a^3*d*sin(d*x + c) + a^3*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(cos(d*x+c)**10/(a+a*sin(d*x+c))**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4962 deep

________________________________________________________________________________________

Giac [A]
time = 3.05, size = 90, normalized size = 0.95 \begin {gather*} \frac {256 \, {\left (143 \, \sqrt {2} \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 330 \, \sqrt {2} \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 195 \, \sqrt {2} \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11}\right )}}{2145 \, a^{3} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^10/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

256/2145*(143*sqrt(2)*sqrt(a)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^15 - 330*sqrt(2)*sqrt(a)*sin(-1/4*pi + 1/2*d*x +
1/2*c)^13 + 195*sqrt(2)*sqrt(a)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^11)/(a^3*d*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)))

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^{10}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^10/(a + a*sin(c + d*x))^(5/2),x)

[Out]

int(cos(c + d*x)^10/(a + a*sin(c + d*x))^(5/2), x)

________________________________________________________________________________________