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

Optimal. Leaf size=159 \[ \frac{496 \sin (c+d x)}{63 a^5 d}-\frac{5 \sin (c+d x)}{d \left (a^5 \sec (c+d x)+a^5\right )}-\frac{67 \sin (c+d x)}{63 a^3 d (a \sec (c+d x)+a)^2}-\frac{29 \sin (c+d x)}{63 a^2 d (a \sec (c+d x)+a)^3}-\frac{5 x}{a^5}-\frac{5 \sin (c+d x)}{21 a d (a \sec (c+d x)+a)^4}-\frac{\sin (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]

[Out]

(-5*x)/a^5 + (496*Sin[c + d*x])/(63*a^5*d) - Sin[c + d*x]/(9*d*(a + a*Sec[c + d*x])^5) - (5*Sin[c + d*x])/(21*
a*d*(a + a*Sec[c + d*x])^4) - (29*Sin[c + d*x])/(63*a^2*d*(a + a*Sec[c + d*x])^3) - (67*Sin[c + d*x])/(63*a^3*
d*(a + a*Sec[c + d*x])^2) - (5*Sin[c + d*x])/(d*(a^5 + a^5*Sec[c + d*x]))

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Rubi [A]  time = 0.397208, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {3817, 4020, 3787, 2637, 8} \[ \frac{496 \sin (c+d x)}{63 a^5 d}-\frac{5 \sin (c+d x)}{d \left (a^5 \sec (c+d x)+a^5\right )}-\frac{67 \sin (c+d x)}{63 a^3 d (a \sec (c+d x)+a)^2}-\frac{29 \sin (c+d x)}{63 a^2 d (a \sec (c+d x)+a)^3}-\frac{5 x}{a^5}-\frac{5 \sin (c+d x)}{21 a d (a \sec (c+d x)+a)^4}-\frac{\sin (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*x)/a^5 + (496*Sin[c + d*x])/(63*a^5*d) - Sin[c + d*x]/(9*d*(a + a*Sec[c + d*x])^5) - (5*Sin[c + d*x])/(21*
a*d*(a + a*Sec[c + d*x])^4) - (29*Sin[c + d*x])/(63*a^2*d*(a + a*Sec[c + d*x])^3) - (67*Sin[c + d*x])/(63*a^3*
d*(a + a*Sec[c + d*x])^2) - (5*Sin[c + d*x])/(d*(a^5 + a^5*Sec[c + d*x]))

Rule 3817

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[(Cot[
e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*(2*m + 1)), x] + Dist[1/(a^2*(2*m + 1)), Int[(a + b*Csc
[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*(a*(2*m + n + 1) - b*(m + n + 1)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d
, e, f, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m])

Rule 4020

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

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\cos (c+d x)}{(a+a \sec (c+d x))^5} \, dx &=-\frac{\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{\int \frac{\cos (c+d x) (-10 a+5 a \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac{\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac{\int \frac{\cos (c+d x) \left (-85 a^2+60 a^2 \sec (c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac{\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac{29 \sin (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac{\int \frac{\cos (c+d x) \left (-570 a^3+435 a^3 \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac{\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac{29 \sin (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac{67 \sin (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}-\frac{\int \frac{\cos (c+d x) \left (-2715 a^4+2010 a^4 \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{945 a^8}\\ &=-\frac{\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac{29 \sin (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac{67 \sin (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}-\frac{5 \sin (c+d x)}{d \left (a^5+a^5 \sec (c+d x)\right )}-\frac{\int \cos (c+d x) \left (-7440 a^5+4725 a^5 \sec (c+d x)\right ) \, dx}{945 a^{10}}\\ &=-\frac{\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac{29 \sin (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac{67 \sin (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}-\frac{5 \sin (c+d x)}{d \left (a^5+a^5 \sec (c+d x)\right )}-\frac{5 \int 1 \, dx}{a^5}+\frac{496 \int \cos (c+d x) \, dx}{63 a^5}\\ &=-\frac{5 x}{a^5}+\frac{496 \sin (c+d x)}{63 a^5 d}-\frac{\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac{29 \sin (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac{67 \sin (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}-\frac{5 \sin (c+d x)}{d \left (a^5+a^5 \sec (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 0.659211, size = 319, normalized size = 2.01 \[ -\frac{\sec \left (\frac{c}{2}\right ) \sec ^9\left (\frac{1}{2} (c+d x)\right ) \left (143010 \sin \left (c+\frac{d x}{2}\right )-138726 \sin \left (c+\frac{3 d x}{2}\right )+73290 \sin \left (2 c+\frac{3 d x}{2}\right )-70389 \sin \left (2 c+\frac{5 d x}{2}\right )+20475 \sin \left (3 c+\frac{5 d x}{2}\right )-21141 \sin \left (3 c+\frac{7 d x}{2}\right )+1575 \sin \left (4 c+\frac{7 d x}{2}\right )-3091 \sin \left (4 c+\frac{9 d x}{2}\right )-567 \sin \left (5 c+\frac{9 d x}{2}\right )-63 \sin \left (5 c+\frac{11 d x}{2}\right )-63 \sin \left (6 c+\frac{11 d x}{2}\right )+79380 d x \cos \left (c+\frac{d x}{2}\right )+52920 d x \cos \left (c+\frac{3 d x}{2}\right )+52920 d x \cos \left (2 c+\frac{3 d x}{2}\right )+22680 d x \cos \left (2 c+\frac{5 d x}{2}\right )+22680 d x \cos \left (3 c+\frac{5 d x}{2}\right )+5670 d x \cos \left (3 c+\frac{7 d x}{2}\right )+5670 d x \cos \left (4 c+\frac{7 d x}{2}\right )+630 d x \cos \left (4 c+\frac{9 d x}{2}\right )+630 d x \cos \left (5 c+\frac{9 d x}{2}\right )-175014 \sin \left (\frac{d x}{2}\right )+79380 d x \cos \left (\frac{d x}{2}\right )\right )}{64512 a^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sec[c/2]*Sec[(c + d*x)/2]^9*(79380*d*x*Cos[(d*x)/2] + 79380*d*x*Cos[c + (d*x)/2] + 52920*d*x*Cos[c + (3*d*x)
/2] + 52920*d*x*Cos[2*c + (3*d*x)/2] + 22680*d*x*Cos[2*c + (5*d*x)/2] + 22680*d*x*Cos[3*c + (5*d*x)/2] + 5670*
d*x*Cos[3*c + (7*d*x)/2] + 5670*d*x*Cos[4*c + (7*d*x)/2] + 630*d*x*Cos[4*c + (9*d*x)/2] + 630*d*x*Cos[5*c + (9
*d*x)/2] - 175014*Sin[(d*x)/2] + 143010*Sin[c + (d*x)/2] - 138726*Sin[c + (3*d*x)/2] + 73290*Sin[2*c + (3*d*x)
/2] - 70389*Sin[2*c + (5*d*x)/2] + 20475*Sin[3*c + (5*d*x)/2] - 21141*Sin[3*c + (7*d*x)/2] + 1575*Sin[4*c + (7
*d*x)/2] - 3091*Sin[4*c + (9*d*x)/2] - 567*Sin[5*c + (9*d*x)/2] - 63*Sin[5*c + (11*d*x)/2] - 63*Sin[6*c + (11*
d*x)/2]))/(64512*a^5*d)

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Maple [A]  time = 0.07, size = 145, normalized size = 0.9 \begin{align*}{\frac{1}{144\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}-{\frac{1}{14\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{3}{8\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{3}{2\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{129}{16\,d{a}^{5}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\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 ) }}-10\,{\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)/(a+a*sec(d*x+c))^5,x)

[Out]

1/144/d/a^5*tan(1/2*d*x+1/2*c)^9-1/14/d/a^5*tan(1/2*d*x+1/2*c)^7+3/8/d/a^5*tan(1/2*d*x+1/2*c)^5-3/2/d/a^5*tan(
1/2*d*x+1/2*c)^3+129/16/d/a^5*tan(1/2*d*x+1/2*c)+2/d/a^5*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)-10/d/a^5*
arctan(tan(1/2*d*x+1/2*c))

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Maxima [A]  time = 1.52837, size = 240, normalized size = 1.51 \begin{align*} \frac{\frac{2016 \, \sin \left (d x + c\right )}{{\left (a^{5} + \frac{a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}} + \frac{\frac{8127 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{1512 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{378 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{72 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{7 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac{10080 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{1008 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1008*(2016*sin(d*x + c)/((a^5 + a^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) + (8127*sin(d*x
+ c)/(cos(d*x + c) + 1) - 1512*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 378*sin(d*x + c)^5/(cos(d*x + c) + 1)^5
- 72*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 7*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/a^5 - 10080*arctan(sin(d*x +
c)/(cos(d*x + c) + 1))/a^5)/d

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Fricas [A]  time = 1.71159, size = 541, normalized size = 3.4 \begin{align*} -\frac{315 \, d x \cos \left (d x + c\right )^{5} + 1575 \, d x \cos \left (d x + c\right )^{4} + 3150 \, d x \cos \left (d x + c\right )^{3} + 3150 \, d x \cos \left (d x + c\right )^{2} + 1575 \, d x \cos \left (d x + c\right ) + 315 \, d x -{\left (63 \, \cos \left (d x + c\right )^{5} + 946 \, \cos \left (d x + c\right )^{4} + 2840 \, \cos \left (d x + c\right )^{3} + 3633 \, \cos \left (d x + c\right )^{2} + 2165 \, \cos \left (d x + c\right ) + 496\right )} \sin \left (d x + c\right )}{63 \,{\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)/(a+a*sec(d*x+c))^5,x, algorithm="fricas")

[Out]

-1/63*(315*d*x*cos(d*x + c)^5 + 1575*d*x*cos(d*x + c)^4 + 3150*d*x*cos(d*x + c)^3 + 3150*d*x*cos(d*x + c)^2 +
1575*d*x*cos(d*x + c) + 315*d*x - (63*cos(d*x + c)^5 + 946*cos(d*x + c)^4 + 2840*cos(d*x + c)^3 + 3633*cos(d*x
+ c)^2 + 2165*cos(d*x + c) + 496)*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)/(a+a*sec(d*x+c))**5,x)

[Out]

Timed out

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Giac [A]  time = 1.56401, size = 174, normalized size = 1.09 \begin{align*} -\frac{\frac{5040 \,{\left (d x + c\right )}}{a^{5}} - \frac{2016 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )} a^{5}} - \frac{7 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 72 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 378 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 1512 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 8127 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{45}}}{1008 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1008*(5040*(d*x + c)/a^5 - 2016*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*a^5) - (7*a^40*tan(1/2*d
*x + 1/2*c)^9 - 72*a^40*tan(1/2*d*x + 1/2*c)^7 + 378*a^40*tan(1/2*d*x + 1/2*c)^5 - 1512*a^40*tan(1/2*d*x + 1/2
*c)^3 + 8127*a^40*tan(1/2*d*x + 1/2*c))/a^45)/d