\(\int \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\) [101]

Optimal result

Integrand size = 23, antiderivative size = 97 \[ \int \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {16 (a+a \sin (c+d x))^{9/2}}{9 a^4 d}-\frac {24 (a+a \sin (c+d x))^{11/2}}{11 a^5 d}+\frac {12 (a+a \sin (c+d x))^{13/2}}{13 a^6 d}-\frac {2 (a+a \sin (c+d x))^{15/2}}{15 a^7 d} \]

[Out]

16/9*(a+a*sin(d*x+c))^(9/2)/a^4/d-24/11*(a+a*sin(d*x+c))^(11/2)/a^5/d+12/13*(a+a*sin(d*x+c))^(13/2)/a^6/d-2/15
*(a+a*sin(d*x+c))^(15/2)/a^7/d

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2746, 45} \[ \int \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 (a \sin (c+d x)+a)^{15/2}}{15 a^7 d}+\frac {12 (a \sin (c+d x)+a)^{13/2}}{13 a^6 d}-\frac {24 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}+\frac {16 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d} \]

[In]

Int[Cos[c + d*x]^7*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(16*(a + a*Sin[c + d*x])^(9/2))/(9*a^4*d) - (24*(a + a*Sin[c + d*x])^(11/2))/(11*a^5*d) + (12*(a + a*Sin[c + d
*x])^(13/2))/(13*a^6*d) - (2*(a + a*Sin[c + d*x])^(15/2))/(15*a^7*d)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2746

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a-x)^3 (a+x)^{7/2} \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \left (8 a^3 (a+x)^{7/2}-12 a^2 (a+x)^{9/2}+6 a (a+x)^{11/2}-(a+x)^{13/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {16 (a+a \sin (c+d x))^{9/2}}{9 a^4 d}-\frac {24 (a+a \sin (c+d x))^{11/2}}{11 a^5 d}+\frac {12 (a+a \sin (c+d x))^{13/2}}{13 a^6 d}-\frac {2 (a+a \sin (c+d x))^{15/2}}{15 a^7 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.63 \[ \int \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 (1+\sin (c+d x))^4 \sqrt {a (1+\sin (c+d x))} \left (-1241+2367 \sin (c+d x)-1683 \sin ^2(c+d x)+429 \sin ^3(c+d x)\right )}{6435 d} \]

[In]

Integrate[Cos[c + d*x]^7*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-2*(1 + Sin[c + d*x])^4*Sqrt[a*(1 + Sin[c + d*x])]*(-1241 + 2367*Sin[c + d*x] - 1683*Sin[c + d*x]^2 + 429*Sin
[c + d*x]^3))/(6435*d)

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75

method	result	size
derivativedivides	\(-\frac {2 \left (\frac {\left (a +a \sin \left (d x +c \right )\right )^{\frac {15}{2}}}{15}-\frac {6 a \left (a +a \sin \left (d x +c \right )\right )^{\frac {13}{2}}}{13}+\frac {12 a^{2} \left (a +a \sin \left (d x +c \right )\right )^{\frac {11}{2}}}{11}-\frac {8 a^{3} \left (a +a \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9}\right )}{d \,a^{7}}\)	\(73\)
default	\(-\frac {2 \left (\frac {\left (a +a \sin \left (d x +c \right )\right )^{\frac {15}{2}}}{15}-\frac {6 a \left (a +a \sin \left (d x +c \right )\right )^{\frac {13}{2}}}{13}+\frac {12 a^{2} \left (a +a \sin \left (d x +c \right )\right )^{\frac {11}{2}}}{11}-\frac {8 a^{3} \left (a +a \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9}\right )}{d \,a^{7}}\)	\(73\)

[In]

int(cos(d*x+c)^7*(a+a*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/d/a^7*(1/15*(a+a*sin(d*x+c))^(15/2)-6/13*a*(a+a*sin(d*x+c))^(13/2)+12/11*a^2*(a+a*sin(d*x+c))^(11/2)-8/9*a^
3*(a+a*sin(d*x+c))^(9/2))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91 \[ \int \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{6} + 56 \, \cos \left (d x + c\right )^{4} + 128 \, \cos \left (d x + c\right )^{2} + {\left (429 \, \cos \left (d x + c\right )^{6} + 504 \, \cos \left (d x + c\right )^{4} + 640 \, \cos \left (d x + c\right )^{2} + 1024\right )} \sin \left (d x + c\right ) + 1024\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{6435 \, d} \]

[In]

integrate(cos(d*x+c)^7*(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

2/6435*(33*cos(d*x + c)^6 + 56*cos(d*x + c)^4 + 128*cos(d*x + c)^2 + (429*cos(d*x + c)^6 + 504*cos(d*x + c)^4
+ 640*cos(d*x + c)^2 + 1024)*sin(d*x + c) + 1024)*sqrt(a*sin(d*x + c) + a)/d

Sympy [F(-1)]

Timed out. \[ \int \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**7*(a+a*sin(d*x+c))**(1/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.74 \[ \int \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \, {\left (429 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {15}{2}} - 2970 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {13}{2}} a + 7020 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a^{2} - 5720 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a^{3}\right )}}{6435 \, a^{7} d} \]

[In]

integrate(cos(d*x+c)^7*(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

-2/6435*(429*(a*sin(d*x + c) + a)^(15/2) - 2970*(a*sin(d*x + c) + a)^(13/2)*a + 7020*(a*sin(d*x + c) + a)^(11/
2)*a^2 - 5720*(a*sin(d*x + c) + a)^(9/2)*a^3)/(a^7*d)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.32 \[ \int \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {256 \, \sqrt {2} {\left (429 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 1485 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 1755 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 715 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sqrt {a}}{6435 \, d} \]

[In]

integrate(cos(d*x+c)^7*(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

-256/6435*sqrt(2)*(429*cos(-1/4*pi + 1/2*d*x + 1/2*c)^15*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) - 1485*cos(-1/4*p
i + 1/2*d*x + 1/2*c)^13*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) + 1755*cos(-1/4*pi + 1/2*d*x + 1/2*c)^11*sgn(cos(-
1/4*pi + 1/2*d*x + 1/2*c)) - 715*cos(-1/4*pi + 1/2*d*x + 1/2*c)^9*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)))*sqrt(a)
/d

Mupad [F(-1)]

Timed out. \[ \int \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int {\cos \left (c+d\,x\right )}^7\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \]

[In]

int(cos(c + d*x)^7*(a + a*sin(c + d*x))^(1/2),x)

[Out]

int(cos(c + d*x)^7*(a + a*sin(c + d*x))^(1/2), x)