\(\int \frac {\cos ^7(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx\) [185]

Optimal result

Integrand size = 23, antiderivative size = 97 \[ \int \frac {\cos ^7(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {16 (a+a \sin (c+d x))^{3/2}}{3 a^4 d}-\frac {24 (a+a \sin (c+d x))^{5/2}}{5 a^5 d}+\frac {12 (a+a \sin (c+d x))^{7/2}}{7 a^6 d}-\frac {2 (a+a \sin (c+d x))^{9/2}}{9 a^7 d} \]

[Out]

16/3*(a+a*sin(d*x+c))^(3/2)/a^4/d-24/5*(a+a*sin(d*x+c))^(5/2)/a^5/d+12/7*(a+a*sin(d*x+c))^(7/2)/a^6/d-2/9*(a+a
*sin(d*x+c))^(9/2)/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} \[ \int \frac {\cos ^7(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 (a \sin (c+d x)+a)^{9/2}}{9 a^7 d}+\frac {12 (a \sin (c+d x)+a)^{7/2}}{7 a^6 d}-\frac {24 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d}+\frac {16 (a \sin (c+d x)+a)^{3/2}}{3 a^4 d} \]

[In]

Int[Cos[c + d*x]^7/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(16*(a + a*Sin[c + d*x])^(3/2))/(3*a^4*d) - (24*(a + a*Sin[c + d*x])^(5/2))/(5*a^5*d) + (12*(a + a*Sin[c + d*x
])^(7/2))/(7*a^6*d) - (2*(a + a*Sin[c + d*x])^(9/2))/(9*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 \sqrt {a+x} \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \left (8 a^3 \sqrt {a+x}-12 a^2 (a+x)^{3/2}+6 a (a+x)^{5/2}-(a+x)^{7/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {16 (a+a \sin (c+d x))^{3/2}}{3 a^4 d}-\frac {24 (a+a \sin (c+d x))^{5/2}}{5 a^5 d}+\frac {12 (a+a \sin (c+d x))^{7/2}}{7 a^6 d}-\frac {2 (a+a \sin (c+d x))^{9/2}}{9 a^7 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.56 \[ \int \frac {\cos ^7(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 (a (1+\sin (c+d x)))^{3/2} \left (-319+321 \sin (c+d x)-165 \sin ^2(c+d x)+35 \sin ^3(c+d x)\right )}{315 a^4 d} \]

[In]

Integrate[Cos[c + d*x]^7/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(-2*(a*(1 + Sin[c + d*x]))^(3/2)*(-319 + 321*Sin[c + d*x] - 165*Sin[c + d*x]^2 + 35*Sin[c + d*x]^3))/(315*a^4*
d)

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75

method	result	size
derivativedivides	\(-\frac {2 \left (\frac {\left (a +a \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9}-\frac {6 a \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {12 a^{2} \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {8 a^{3} \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3}\right )}{d \,a^{7}}\)	\(73\)
default	\(-\frac {2 \left (\frac {\left (a +a \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9}-\frac {6 a \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {12 a^{2} \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {8 a^{3} \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3}\right )}{d \,a^{7}}\)	\(73\)

[In]

int(cos(d*x+c)^7/(a+a*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/d/a^7*(1/9*(a+a*sin(d*x+c))^(9/2)-6/7*a*(a+a*sin(d*x+c))^(7/2)+12/5*a^2*(a+a*sin(d*x+c))^(5/2)-8/3*a^3*(a+a
*sin(d*x+c))^(3/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.64 \[ \int \frac {\cos ^7(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 \, {\left (35 \, \cos \left (d x + c\right )^{4} - 226 \, \cos \left (d x + c\right )^{2} + 2 \, {\left (65 \, \cos \left (d x + c\right )^{2} - 64\right )} \sin \left (d x + c\right ) - 128\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{315 \, a^{3} d} \]

[In]

integrate(cos(d*x+c)^7/(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

-2/315*(35*cos(d*x + c)^4 - 226*cos(d*x + c)^2 + 2*(65*cos(d*x + c)^2 - 64)*sin(d*x + c) - 128)*sqrt(a*sin(d*x
+ c) + a)/(a^3*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^7(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**7/(a+a*sin(d*x+c))**(5/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.74 \[ \int \frac {\cos ^7(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 \, {\left (35 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 270 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 756 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 840 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3}\right )}}{315 \, a^{7} d} \]

[In]

integrate(cos(d*x+c)^7/(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

-2/315*(35*(a*sin(d*x + c) + a)^(9/2) - 270*(a*sin(d*x + c) + a)^(7/2)*a + 756*(a*sin(d*x + c) + a)^(5/2)*a^2
- 840*(a*sin(d*x + c) + a)^(3/2)*a^3)/(a^7*d)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.15 \[ \int \frac {\cos ^7(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {32 \, {\left (35 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 135 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 189 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}\right )}}{315 \, a^{3} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \]

[In]

integrate(cos(d*x+c)^7/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

-32/315*(35*sqrt(2)*sqrt(a)*cos(-1/4*pi + 1/2*d*x + 1/2*c)^9 - 135*sqrt(2)*sqrt(a)*cos(-1/4*pi + 1/2*d*x + 1/2
*c)^7 + 189*sqrt(2)*sqrt(a)*cos(-1/4*pi + 1/2*d*x + 1/2*c)^5 - 105*sqrt(2)*sqrt(a)*cos(-1/4*pi + 1/2*d*x + 1/2
*c)^3)/(a^3*d*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)))

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^7(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^7}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]

[In]

int(cos(c + d*x)^7/(a + a*sin(c + d*x))^(5/2),x)

[Out]

int(cos(c + d*x)^7/(a + a*sin(c + d*x))^(5/2), x)