3.87 \(\int \frac{1}{(a+a \sec (c+d x))^5} \, dx\)

Optimal. Leaf size=144 \[ -\frac{488 \tan (c+d x)}{315 d \left (a^5 \sec (c+d x)+a^5\right )}-\frac{173 \tan (c+d x)}{315 a^3 d (a \sec (c+d x)+a)^2}-\frac{34 \tan (c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}+\frac{x}{a^5}-\frac{13 \tan (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]

[Out]

x/a^5 - Tan[c + d*x]/(9*d*(a + a*Sec[c + d*x])^5) - (13*Tan[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) - (34*Ta
n[c + d*x])/(105*a^2*d*(a + a*Sec[c + d*x])^3) - (173*Tan[c + d*x])/(315*a^3*d*(a + a*Sec[c + d*x])^2) - (488*
Tan[c + d*x])/(315*d*(a^5 + a^5*Sec[c + d*x]))

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Rubi [A]  time = 0.207306, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3777, 3922, 3919, 3794} \[ -\frac{488 \tan (c+d x)}{315 d \left (a^5 \sec (c+d x)+a^5\right )}-\frac{173 \tan (c+d x)}{315 a^3 d (a \sec (c+d x)+a)^2}-\frac{34 \tan (c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}+\frac{x}{a^5}-\frac{13 \tan (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^(-5),x]

[Out]

x/a^5 - Tan[c + d*x]/(9*d*(a + a*Sec[c + d*x])^5) - (13*Tan[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) - (34*Ta
n[c + d*x])/(105*a^2*d*(a + a*Sec[c + d*x])^3) - (173*Tan[c + d*x])/(315*a^3*d*(a + a*Sec[c + d*x])^2) - (488*
Tan[c + d*x])/(315*d*(a^5 + a^5*Sec[c + d*x]))

Rule 3777

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Simp[(Cot[c + d*x]*(a + b*Csc[c + d*x])^n)/(d*(
2*n + 1)), x] + Dist[1/(a^2*(2*n + 1)), Int[(a + b*Csc[c + d*x])^(n + 1)*(a*(2*n + 1) - b*(n + 1)*Csc[c + d*x]
), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]

Rule 3922

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

Rule 3919

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

Rule 3794

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[Cot[e + f*x]/(f*(b + a*
Csc[e + f*x])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{1}{(a+a \sec (c+d x))^5} \, dx &=-\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{\int \frac{-9 a+4 a \sec (c+d x)}{(a+a \sec (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{13 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac{\int \frac{63 a^2-39 a^2 \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{13 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac{34 \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}-\frac{\int \frac{-315 a^3+204 a^3 \sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{13 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac{34 \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}-\frac{173 \tan (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}+\frac{\int \frac{945 a^4-519 a^4 \sec (c+d x)}{a+a \sec (c+d x)} \, dx}{945 a^8}\\ &=\frac{x}{a^5}-\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{13 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac{34 \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}-\frac{173 \tan (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac{488 \int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{315 a^4}\\ &=\frac{x}{a^5}-\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{13 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac{34 \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}-\frac{173 \tan (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac{488 \tan (c+d x)}{315 d \left (a^5+a^5 \sec (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.522156, size = 280, normalized size = 1.94 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^9\left (\frac{1}{2} (c+d x)\right ) \left (100800 \sin \left (c+\frac{d x}{2}\right )-88284 \sin \left (c+\frac{3 d x}{2}\right )+56700 \sin \left (2 c+\frac{3 d x}{2}\right )-43236 \sin \left (2 c+\frac{5 d x}{2}\right )+18900 \sin \left (3 c+\frac{5 d x}{2}\right )-12384 \sin \left (3 c+\frac{7 d x}{2}\right )+3150 \sin \left (4 c+\frac{7 d x}{2}\right )-1726 \sin \left (4 c+\frac{9 d x}{2}\right )+39690 d x \cos \left (c+\frac{d x}{2}\right )+26460 d x \cos \left (c+\frac{3 d x}{2}\right )+26460 d x \cos \left (2 c+\frac{3 d x}{2}\right )+11340 d x \cos \left (2 c+\frac{5 d x}{2}\right )+11340 d x \cos \left (3 c+\frac{5 d x}{2}\right )+2835 d x \cos \left (3 c+\frac{7 d x}{2}\right )+2835 d x \cos \left (4 c+\frac{7 d x}{2}\right )+315 d x \cos \left (4 c+\frac{9 d x}{2}\right )+315 d x \cos \left (5 c+\frac{9 d x}{2}\right )-116676 \sin \left (\frac{d x}{2}\right )+39690 d x \cos \left (\frac{d x}{2}\right )\right )}{161280 a^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]^9*(39690*d*x*Cos[(d*x)/2] + 39690*d*x*Cos[c + (d*x)/2] + 26460*d*x*Cos[c + (3*d*x)/
2] + 26460*d*x*Cos[2*c + (3*d*x)/2] + 11340*d*x*Cos[2*c + (5*d*x)/2] + 11340*d*x*Cos[3*c + (5*d*x)/2] + 2835*d
*x*Cos[3*c + (7*d*x)/2] + 2835*d*x*Cos[4*c + (7*d*x)/2] + 315*d*x*Cos[4*c + (9*d*x)/2] + 315*d*x*Cos[5*c + (9*
d*x)/2] - 116676*Sin[(d*x)/2] + 100800*Sin[c + (d*x)/2] - 88284*Sin[c + (3*d*x)/2] + 56700*Sin[2*c + (3*d*x)/2
] - 43236*Sin[2*c + (5*d*x)/2] + 18900*Sin[3*c + (5*d*x)/2] - 12384*Sin[3*c + (7*d*x)/2] + 3150*Sin[4*c + (7*d
*x)/2] - 1726*Sin[4*c + (9*d*x)/2]))/(161280*a^5*d)

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Maple [A]  time = 0.044, size = 113, 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{3}{56\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{1}{5\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{13}{24\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{31}{16\,d{a}^{5}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\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(1/(a+a*sec(d*x+c))^5,x)

[Out]

-1/144/d/a^5*tan(1/2*d*x+1/2*c)^9+3/56/d/a^5*tan(1/2*d*x+1/2*c)^7-1/5/d/a^5*tan(1/2*d*x+1/2*c)^5+13/24/d/a^5*t
an(1/2*d*x+1/2*c)^3-31/16/d/a^5*tan(1/2*d*x+1/2*c)+2/d/a^5*arctan(tan(1/2*d*x+1/2*c))

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Maxima [A]  time = 1.66243, size = 178, normalized size = 1.24 \begin{align*} -\frac{\frac{\frac{9765 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{2730 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{1008 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{270 \, \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{10080 \, \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(1/(a+a*sec(d*x+c))^5,x, algorithm="maxima")

[Out]

-1/5040*((9765*sin(d*x + c)/(cos(d*x + c) + 1) - 2730*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 1008*sin(d*x + c)^
5/(cos(d*x + c) + 1)^5 - 270*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
- 10080*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^5)/d

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Fricas [A]  time = 1.71136, size = 514, normalized size = 3.57 \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 (863 \, \cos \left (d x + c\right )^{4} + 2740 \, \cos \left (d x + c\right )^{3} + 3549 \, \cos \left (d x + c\right )^{2} + 2125 \, \cos \left (d x + c\right ) + 488\right )} \sin \left (d x + c\right )}{315 \,{\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(1/(a+a*sec(d*x+c))^5,x, algorithm="fricas")

[Out]

1/315*(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 - (863*cos(d*x + c)^4 + 2740*cos(d*x + c)^3 + 3549*cos(d*x + c)^2 + 2125*cos(d
*x + c) + 488)*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]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{\sec ^{5}{\left (c + d x \right )} + 5 \sec ^{4}{\left (c + d x \right )} + 10 \sec ^{3}{\left (c + d x \right )} + 10 \sec ^{2}{\left (c + d x \right )} + 5 \sec{\left (c + d x \right )} + 1}\, dx}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sec(d*x+c))**5,x)

[Out]

Integral(1/(sec(c + d*x)**5 + 5*sec(c + d*x)**4 + 10*sec(c + d*x)**3 + 10*sec(c + d*x)**2 + 5*sec(c + d*x) + 1
), x)/a**5

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Giac [A]  time = 1.3747, size = 135, normalized size = 0.94 \begin{align*} \frac{\frac{5040 \,{\left (d x + c\right )}}{a^{5}} - \frac{35 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 270 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 1008 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 2730 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9765 \, 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(1/(a+a*sec(d*x+c))^5,x, algorithm="giac")

[Out]

1/5040*(5040*(d*x + c)/a^5 - (35*a^40*tan(1/2*d*x + 1/2*c)^9 - 270*a^40*tan(1/2*d*x + 1/2*c)^7 + 1008*a^40*tan
(1/2*d*x + 1/2*c)^5 - 2730*a^40*tan(1/2*d*x + 1/2*c)^3 + 9765*a^40*tan(1/2*d*x + 1/2*c))/a^45)/d