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

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

[Out]

x/a^5-1/9*tan(d*x+c)/d/(a+a*sec(d*x+c))^5-13/63*tan(d*x+c)/a/d/(a+a*sec(d*x+c))^4-34/105*tan(d*x+c)/a^2/d/(a+a
*sec(d*x+c))^3-173/315*tan(d*x+c)/a^3/d/(a+a*sec(d*x+c))^2-488/315*tan(d*x+c)/d/(a^5+a^5*sec(d*x+c))

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Rubi [A]  time = 0.21, antiderivative size = 144, normalized size of antiderivative = 1.00, 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.56, 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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fricas [A]  time = 0.60, size = 188, normalized size = 1.31 \[ \frac {315 \, d x \cos \left (d x + c\right )^{5} + 1575 \, d x \cos \left (d x + c\right )^{4} + 3150 \, d x \cos \left (d x + c\right )^{3} + 3150 \, d x \cos \left (d x + c\right )^{2} + 1575 \, d x \cos \left (d x + c\right ) + 315 \, d x - {\left (863 \, \cos \left (d x + c\right )^{4} + 2740 \, \cos \left (d x + c\right )^{3} + 3549 \, \cos \left (d x + c\right )^{2} + 2125 \, \cos \left (d x + c\right ) + 488\right )} \sin \left (d x + c\right )}{315 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]

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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giac [A]  time = 1.01, size = 100, normalized size = 0.69 \[ \frac {\frac {5040 \, {\left (d x + c\right )}}{a^{5}} - \frac {35 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 270 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1008 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2730 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9765 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{5040 \, d} \]

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

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maple [A]  time = 0.47, size = 113, normalized size = 0.78 \[ -\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 d \,a^{5}}+\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{5}}-\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{5 d \,a^{5}}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{5}}-\frac {31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{5}} \]

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 = 0.68, size = 132, normalized size = 0.92 \[ -\frac {\frac {\frac {9765 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2730 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1008 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {270 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {10080 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{5040 \, d} \]

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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mupad [B]  time = 0.80, size = 125, normalized size = 0.87 \[ \frac {x}{a^5}-\frac {\frac {863\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{315}-\frac {356\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{315}+\frac {169\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{420}-\frac {41\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{504}+\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{144}}{a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + a/cos(c + d*x))^5,x)

[Out]

x/a^5 - (sin(c/2 + (d*x)/2)/144 - (41*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2))/504 + (169*cos(c/2 + (d*x)/2)^4
*sin(c/2 + (d*x)/2))/420 - (356*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2))/315 + (863*cos(c/2 + (d*x)/2)^8*sin(c
/2 + (d*x)/2))/315)/(a^5*d*cos(c/2 + (d*x)/2)^9)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{\sec ^{5}{\left (c + d x \right )} + 5 \sec ^{4}{\left (c + d x \right )} + 10 \sec ^{3}{\left (c + d x \right )} + 10 \sec ^{2}{\left (c + d x \right )} + 5 \sec {\left (c + d x \right )} + 1}\, dx}{a^{5}} \]

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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