3.82 \(\int \frac{\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx\)

Optimal. Leaf size=225 \[ -\frac{7664 \sin (c+d x)}{315 a^5 d}-\frac{28 \sin (c+d x) \cos ^4(c+d x)}{45 a^2 d (a \cos (c+d x)+a)^3}-\frac{577 \sin (c+d x) \cos ^3(c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac{3832 \sin (c+d x) \cos ^2(c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}+\frac{31 \sin (c+d x) \cos (c+d x)}{2 a^5 d}+\frac{31 x}{2 a^5}-\frac{\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{17 \sin (c+d x) \cos ^5(c+d x)}{63 a d (a \cos (c+d x)+a)^4} \]

[Out]

(31*x)/(2*a^5) - (7664*Sin[c + d*x])/(315*a^5*d) + (31*Cos[c + d*x]*Sin[c + d*x])/(2*a^5*d) - (Cos[c + d*x]^6*
Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (17*Cos[c + d*x]^5*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4)
- (28*Cos[c + d*x]^4*Sin[c + d*x])/(45*a^2*d*(a + a*Cos[c + d*x])^3) - (577*Cos[c + d*x]^3*Sin[c + d*x])/(315*
a^3*d*(a + a*Cos[c + d*x])^2) - (3832*Cos[c + d*x]^2*Sin[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))

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Rubi [A]  time = 0.515535, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2765, 2977, 2734} \[ -\frac{7664 \sin (c+d x)}{315 a^5 d}-\frac{28 \sin (c+d x) \cos ^4(c+d x)}{45 a^2 d (a \cos (c+d x)+a)^3}-\frac{577 \sin (c+d x) \cos ^3(c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac{3832 \sin (c+d x) \cos ^2(c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}+\frac{31 \sin (c+d x) \cos (c+d x)}{2 a^5 d}+\frac{31 x}{2 a^5}-\frac{\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{17 \sin (c+d x) \cos ^5(c+d x)}{63 a d (a \cos (c+d x)+a)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(31*x)/(2*a^5) - (7664*Sin[c + d*x])/(315*a^5*d) + (31*Cos[c + d*x]*Sin[c + d*x])/(2*a^5*d) - (Cos[c + d*x]^6*
Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (17*Cos[c + d*x]^5*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4)
- (28*Cos[c + d*x]^4*Sin[c + d*x])/(45*a^2*d*(a + a*Cos[c + d*x])^3) - (577*Cos[c + d*x]^3*Sin[c + d*x])/(315*
a^3*d*(a + a*Cos[c + d*x])^2) - (3832*Cos[c + d*x]^2*Sin[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))

Rule 2765

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[((b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1))/(a*f*(2*m + 1)), x] + Dist[1/
(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)*Simp[b*(c^2*(m + 1) + d^2*(n -
1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e,
f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ
[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2977

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^n)/(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ
[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 2734

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((2*a*c
+ b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin{align*} \int \frac{\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx &=-\frac{\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{\int \frac{\cos ^5(c+d x) (6 a-11 a \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac{\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{\int \frac{\cos ^4(c+d x) \left (85 a^2-111 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac{\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{28 \cos ^4(c+d x) \sin (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac{\int \frac{\cos ^3(c+d x) \left (784 a^3-947 a^3 \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac{\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{28 \cos ^4(c+d x) \sin (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac{577 \cos ^3(c+d x) \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac{\int \frac{\cos ^2(c+d x) \left (5193 a^4-6303 a^4 \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{945 a^8}\\ &=-\frac{\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{28 \cos ^4(c+d x) \sin (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac{577 \cos ^3(c+d x) \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac{3832 \cos ^2(c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}-\frac{\int \cos (c+d x) \left (22992 a^5-29295 a^5 \cos (c+d x)\right ) \, dx}{945 a^{10}}\\ &=\frac{31 x}{2 a^5}-\frac{7664 \sin (c+d x)}{315 a^5 d}+\frac{31 \cos (c+d x) \sin (c+d x)}{2 a^5 d}-\frac{\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{28 \cos ^4(c+d x) \sin (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac{577 \cos ^3(c+d x) \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac{3832 \cos ^2(c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.724873, size = 345, normalized size = 1.53 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^9\left (\frac{1}{2} (c+d x)\right ) \left (7194600 \sin \left (c+\frac{d x}{2}\right )-7472241 \sin \left (c+\frac{3 d x}{2}\right )+3432975 \sin \left (2 c+\frac{3 d x}{2}\right )-3871989 \sin \left (2 c+\frac{5 d x}{2}\right )+801675 \sin \left (3 c+\frac{5 d x}{2}\right )-1186056 \sin \left (3 c+\frac{7 d x}{2}\right )-17640 \sin \left (4 c+\frac{7 d x}{2}\right )-175184 \sin \left (4 c+\frac{9 d x}{2}\right )-45360 \sin \left (5 c+\frac{9 d x}{2}\right )-3465 \sin \left (5 c+\frac{11 d x}{2}\right )-3465 \sin \left (6 c+\frac{11 d x}{2}\right )+315 \sin \left (6 c+\frac{13 d x}{2}\right )+315 \sin \left (7 c+\frac{13 d x}{2}\right )+4921560 d x \cos \left (c+\frac{d x}{2}\right )+3281040 d x \cos \left (c+\frac{3 d x}{2}\right )+3281040 d x \cos \left (2 c+\frac{3 d x}{2}\right )+1406160 d x \cos \left (2 c+\frac{5 d x}{2}\right )+1406160 d x \cos \left (3 c+\frac{5 d x}{2}\right )+351540 d x \cos \left (3 c+\frac{7 d x}{2}\right )+351540 d x \cos \left (4 c+\frac{7 d x}{2}\right )+39060 d x \cos \left (4 c+\frac{9 d x}{2}\right )+39060 d x \cos \left (5 c+\frac{9 d x}{2}\right )-9163224 \sin \left (\frac{d x}{2}\right )+4921560 d x \cos \left (\frac{d x}{2}\right )\right )}{1290240 a^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]^9*(4921560*d*x*Cos[(d*x)/2] + 4921560*d*x*Cos[c + (d*x)/2] + 3281040*d*x*Cos[c + (3
*d*x)/2] + 3281040*d*x*Cos[2*c + (3*d*x)/2] + 1406160*d*x*Cos[2*c + (5*d*x)/2] + 1406160*d*x*Cos[3*c + (5*d*x)
/2] + 351540*d*x*Cos[3*c + (7*d*x)/2] + 351540*d*x*Cos[4*c + (7*d*x)/2] + 39060*d*x*Cos[4*c + (9*d*x)/2] + 390
60*d*x*Cos[5*c + (9*d*x)/2] - 9163224*Sin[(d*x)/2] + 7194600*Sin[c + (d*x)/2] - 7472241*Sin[c + (3*d*x)/2] + 3
432975*Sin[2*c + (3*d*x)/2] - 3871989*Sin[2*c + (5*d*x)/2] + 801675*Sin[3*c + (5*d*x)/2] - 1186056*Sin[3*c + (
7*d*x)/2] - 17640*Sin[4*c + (7*d*x)/2] - 175184*Sin[4*c + (9*d*x)/2] - 45360*Sin[5*c + (9*d*x)/2] - 3465*Sin[5
*c + (11*d*x)/2] - 3465*Sin[6*c + (11*d*x)/2] + 315*Sin[6*c + (13*d*x)/2] + 315*Sin[7*c + (13*d*x)/2]))/(12902
40*a^5*d)

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Maple [A]  time = 0.05, size = 179, normalized size = 0.8 \begin{align*} -{\frac{1}{144\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}+{\frac{5}{56\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{3}{5\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{25}{8\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{351}{16\,d{a}^{5}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-11\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}}{d{a}^{5} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}-9\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{5} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}+31\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144/d/a^5*tan(1/2*d*x+1/2*c)^9+5/56/d/a^5*tan(1/2*d*x+1/2*c)^7-3/5/d/a^5*tan(1/2*d*x+1/2*c)^5+25/8/d/a^5*ta
n(1/2*d*x+1/2*c)^3-351/16/d/a^5*tan(1/2*d*x+1/2*c)-11/d/a^5/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3-9/
d/a^5/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)+31/d/a^5*arctan(tan(1/2*d*x+1/2*c))

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Maxima [A]  time = 1.72517, size = 302, normalized size = 1.34 \begin{align*} -\frac{\frac{5040 \,{\left (\frac{9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{11 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{5} + \frac{2 \, a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac{\frac{110565 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{15750 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3024 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{450 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac{156240 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{5040 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5040*(5040*(9*sin(d*x + c)/(cos(d*x + c) + 1) + 11*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^5 + 2*a^5*sin(d*
x + c)^2/(cos(d*x + c) + 1)^2 + a^5*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (110565*sin(d*x + c)/(cos(d*x + c)
+ 1) - 15750*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3024*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 450*sin(d*x + c)
^7/(cos(d*x + c) + 1)^7 + 35*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/a^5 - 156240*arctan(sin(d*x + c)/(cos(d*x +
c) + 1))/a^5)/d

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Fricas [A]  time = 1.73784, size = 591, normalized size = 2.63 \begin{align*} \frac{9765 \, d x \cos \left (d x + c\right )^{5} + 48825 \, d x \cos \left (d x + c\right )^{4} + 97650 \, d x \cos \left (d x + c\right )^{3} + 97650 \, d x \cos \left (d x + c\right )^{2} + 48825 \, d x \cos \left (d x + c\right ) + 9765 \, d x +{\left (315 \, \cos \left (d x + c\right )^{6} - 1575 \, \cos \left (d x + c\right )^{5} - 28828 \, \cos \left (d x + c\right )^{4} - 87440 \, \cos \left (d x + c\right )^{3} - 112119 \, \cos \left (d x + c\right )^{2} - 66875 \, \cos \left (d x + c\right ) - 15328\right )} \sin \left (d x + c\right )}{630 \,{\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/630*(9765*d*x*cos(d*x + c)^5 + 48825*d*x*cos(d*x + c)^4 + 97650*d*x*cos(d*x + c)^3 + 97650*d*x*cos(d*x + c)^
2 + 48825*d*x*cos(d*x + c) + 9765*d*x + (315*cos(d*x + c)^6 - 1575*cos(d*x + c)^5 - 28828*cos(d*x + c)^4 - 874
40*cos(d*x + c)^3 - 112119*cos(d*x + c)^2 - 66875*cos(d*x + c) - 15328)*sin(d*x + c))/(a^5*d*cos(d*x + c)^5 +
5*a^5*d*cos(d*x + c)^4 + 10*a^5*d*cos(d*x + c)^3 + 10*a^5*d*cos(d*x + c)^2 + 5*a^5*d*cos(d*x + c) + a^5*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.42491, size = 196, normalized size = 0.87 \begin{align*} \frac{\frac{78120 \,{\left (d x + c\right )}}{a^{5}} - \frac{5040 \,{\left (11 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2} a^{5}} - \frac{35 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 450 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 3024 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 15750 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 110565 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{45}}}{5040 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5040*(78120*(d*x + c)/a^5 - 5040*(11*tan(1/2*d*x + 1/2*c)^3 + 9*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)
^2 + 1)^2*a^5) - (35*a^40*tan(1/2*d*x + 1/2*c)^9 - 450*a^40*tan(1/2*d*x + 1/2*c)^7 + 3024*a^40*tan(1/2*d*x + 1
/2*c)^5 - 15750*a^40*tan(1/2*d*x + 1/2*c)^3 + 110565*a^40*tan(1/2*d*x + 1/2*c))/a^45)/d