3.186 \(\int \frac{\sec ^6(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=104 \[ \frac{i \sec ^5(c+d x)}{5 a^3 d}-\frac{4 i \sec ^3(c+d x)}{3 a^3 d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}-\frac{3 \tan (c+d x) \sec ^3(c+d x)}{4 a^3 d}+\frac{7 \tan (c+d x) \sec (c+d x)}{8 a^3 d} \]

[Out]

(7*ArcTanh[Sin[c + d*x]])/(8*a^3*d) - (((4*I)/3)*Sec[c + d*x]^3)/(a^3*d) + ((I/5)*Sec[c + d*x]^5)/(a^3*d) + (7
*Sec[c + d*x]*Tan[c + d*x])/(8*a^3*d) - (3*Sec[c + d*x]^3*Tan[c + d*x])/(4*a^3*d)

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Rubi [A]  time = 0.229947, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {3092, 3090, 3768, 3770, 2606, 30, 2611, 14} \[ \frac{i \sec ^5(c+d x)}{5 a^3 d}-\frac{4 i \sec ^3(c+d x)}{3 a^3 d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}-\frac{3 \tan (c+d x) \sec ^3(c+d x)}{4 a^3 d}+\frac{7 \tan (c+d x) \sec (c+d x)}{8 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^6/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]

[Out]

(7*ArcTanh[Sin[c + d*x]])/(8*a^3*d) - (((4*I)/3)*Sec[c + d*x]^3)/(a^3*d) + ((I/5)*Sec[c + d*x]^5)/(a^3*d) + (7
*Sec[c + d*x]*Tan[c + d*x])/(8*a^3*d) - (3*Sec[c + d*x]^3*Tan[c + d*x])/(4*a^3*d)

Rule 3092

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symb
ol] :> Dist[a^n*b^n, Int[Cos[c + d*x]^m/(b*Cos[c + d*x] + a*Sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, m},
x] && EqQ[a^2 + b^2, 0] && ILtQ[n, 0]

Rule 3090

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_.), x_Sym
bol] :> Int[ExpandTrig[cos[c + d*x]^m*(a*cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] &&
IntegerQ[m] && IGtQ[n, 0]

Rule 3768

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

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{\sec ^6(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx &=\frac{i \int \sec ^6(c+d x) (i a \cos (c+d x)+a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac{i \int \left (-i a^3 \sec ^3(c+d x)-3 a^3 \sec ^3(c+d x) \tan (c+d x)+3 i a^3 \sec ^3(c+d x) \tan ^2(c+d x)+a^3 \sec ^3(c+d x) \tan ^3(c+d x)\right ) \, dx}{a^6}\\ &=\frac{i \int \sec ^3(c+d x) \tan ^3(c+d x) \, dx}{a^3}-\frac{(3 i) \int \sec ^3(c+d x) \tan (c+d x) \, dx}{a^3}+\frac{\int \sec ^3(c+d x) \, dx}{a^3}-\frac{3 \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx}{a^3}\\ &=\frac{\sec (c+d x) \tan (c+d x)}{2 a^3 d}-\frac{3 \sec ^3(c+d x) \tan (c+d x)}{4 a^3 d}+\frac{\int \sec (c+d x) \, dx}{2 a^3}+\frac{3 \int \sec ^3(c+d x) \, dx}{4 a^3}+\frac{i \operatorname{Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}-\frac{(3 i) \operatorname{Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=\frac{\tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{i \sec ^3(c+d x)}{a^3 d}+\frac{7 \sec (c+d x) \tan (c+d x)}{8 a^3 d}-\frac{3 \sec ^3(c+d x) \tan (c+d x)}{4 a^3 d}+\frac{3 \int \sec (c+d x) \, dx}{8 a^3}+\frac{i \operatorname{Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=\frac{7 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}-\frac{4 i \sec ^3(c+d x)}{3 a^3 d}+\frac{i \sec ^5(c+d x)}{5 a^3 d}+\frac{7 \sec (c+d x) \tan (c+d x)}{8 a^3 d}-\frac{3 \sec ^3(c+d x) \tan (c+d x)}{4 a^3 d}\\ \end{align*}

Mathematica [A]  time = 0.443505, size = 115, normalized size = 1.11 \[ \frac{i \sec ^8(c+d x) (\sin (3 (c+d x))-i \cos (3 (c+d x))) \left (-150 i \sin (2 (c+d x))+105 i \sin (4 (c+d x))+640 \cos (2 (c+d x))+1680 i \cos ^5(c+d x) \tanh ^{-1}\left (\cos (c) \tan \left (\frac{d x}{2}\right )+\sin (c)\right )+448\right )}{960 a^3 d (\tan (c+d x)-i)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^6/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]

[Out]

((I/960)*Sec[c + d*x]^8*((-I)*Cos[3*(c + d*x)] + Sin[3*(c + d*x)])*(448 + (1680*I)*ArcTanh[Sin[c] + Cos[c]*Tan
[(d*x)/2]]*Cos[c + d*x]^5 + 640*Cos[2*(c + d*x)] - (150*I)*Sin[2*(c + d*x)] + (105*I)*Sin[4*(c + d*x)]))/(a^3*
d*(-I + Tan[c + d*x])^3)

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Maple [B]  time = 0.224, size = 430, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^6/(a*cos(d*x+c)+I*a*sin(d*x+c))^3,x)

[Out]

-1/5*I/d/a^3/(tan(1/2*d*x+1/2*c)-1)^5+5/8/d/a^3/(tan(1/2*d*x+1/2*c)+1)^2-13/8*I/d/a^3/(tan(1/2*d*x+1/2*c)+1)+3
/4/d/a^3/(tan(1/2*d*x+1/2*c)+1)^4+11/8*I/d/a^3/(tan(1/2*d*x+1/2*c)+1)^2+1/8/d/a^3/(tan(1/2*d*x+1/2*c)+1)-1/2*I
/d/a^3/(tan(1/2*d*x+1/2*c)-1)^4-3/2/d/a^3/(tan(1/2*d*x+1/2*c)+1)^3+7/12*I/d/a^3/(tan(1/2*d*x+1/2*c)-1)^3+7/8/d
/a^3*ln(tan(1/2*d*x+1/2*c)+1)+11/8*I/d/a^3/(tan(1/2*d*x+1/2*c)-1)^2+1/8/d/a^3/(tan(1/2*d*x+1/2*c)-1)+13/8*I/d/
a^3/(tan(1/2*d*x+1/2*c)-1)-3/4/d/a^3/(tan(1/2*d*x+1/2*c)-1)^4+1/5*I/d/a^3/(tan(1/2*d*x+1/2*c)+1)^5-5/8/d/a^3/(
tan(1/2*d*x+1/2*c)-1)^2-7/12*I/d/a^3/(tan(1/2*d*x+1/2*c)+1)^3-3/2/d/a^3/(tan(1/2*d*x+1/2*c)-1)^3-1/2*I/d/a^3/(
tan(1/2*d*x+1/2*c)+1)^4-7/8/d/a^3*ln(tan(1/2*d*x+1/2*c)-1)

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Maxima [B]  time = 1.10627, size = 460, normalized size = 4.42 \begin{align*} \frac{\frac{16 \,{\left (-\frac{15 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{320 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{390 i \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{400 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{960 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{390 i \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{360 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{15 i \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 136\right )}}{-120 i \, a^{3} + \frac{600 i \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{1200 i \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{1200 i \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{600 i \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{120 i \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} + \frac{7 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} - \frac{7 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6/(a*cos(d*x+c)+I*a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/8*(16*(-15*I*sin(d*x + c)/(cos(d*x + c) + 1) + 320*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 390*I*sin(d*x + c)^
3/(cos(d*x + c) + 1)^3 - 400*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 960*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 3
90*I*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 360*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 15*I*sin(d*x + c)^9/(cos(
d*x + c) + 1)^9 - 136)/(-120*I*a^3 + 600*I*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1200*I*a^3*sin(d*x + c)^4
/(cos(d*x + c) + 1)^4 + 1200*I*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 600*I*a^3*sin(d*x + c)^8/(cos(d*x + c
) + 1)^8 + 120*I*a^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^10) + 7*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^3 -
7*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^3)/d

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Fricas [B]  time = 0.501209, size = 836, normalized size = 8.04 \begin{align*} \frac{105 \,{\left (e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 105 \,{\left (e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 210 i \, e^{\left (9 i \, d x + 9 i \, c\right )} - 980 i \, e^{\left (7 i \, d x + 7 i \, c\right )} - 1792 i \, e^{\left (5 i \, d x + 5 i \, c\right )} - 1580 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 210 i \, e^{\left (i \, d x + i \, c\right )}}{120 \,{\left (a^{3} d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6/(a*cos(d*x+c)+I*a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/120*(105*(e^(10*I*d*x + 10*I*c) + 5*e^(8*I*d*x + 8*I*c) + 10*e^(6*I*d*x + 6*I*c) + 10*e^(4*I*d*x + 4*I*c) +
5*e^(2*I*d*x + 2*I*c) + 1)*log(e^(I*d*x + I*c) + I) - 105*(e^(10*I*d*x + 10*I*c) + 5*e^(8*I*d*x + 8*I*c) + 10*
e^(6*I*d*x + 6*I*c) + 10*e^(4*I*d*x + 4*I*c) + 5*e^(2*I*d*x + 2*I*c) + 1)*log(e^(I*d*x + I*c) - I) - 210*I*e^(
9*I*d*x + 9*I*c) - 980*I*e^(7*I*d*x + 7*I*c) - 1792*I*e^(5*I*d*x + 5*I*c) - 1580*I*e^(3*I*d*x + 3*I*c) + 210*I
*e^(I*d*x + I*c))/(a^3*d*e^(10*I*d*x + 10*I*c) + 5*a^3*d*e^(8*I*d*x + 8*I*c) + 10*a^3*d*e^(6*I*d*x + 6*I*c) +
10*a^3*d*e^(4*I*d*x + 4*I*c) + 5*a^3*d*e^(2*I*d*x + 2*I*c) + a^3*d)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**6/(a*cos(d*x+c)+I*a*sin(d*x+c))**3,x)

[Out]

Exception raised: AttributeError

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Giac [A]  time = 1.21218, size = 224, normalized size = 2.15 \begin{align*} \frac{\frac{105 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac{105 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} + \frac{2 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 360 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 390 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 960 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 400 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 390 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 320 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 136 i\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{5} a^{3}}}{120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6/(a*cos(d*x+c)+I*a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/120*(105*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^3 - 105*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^3 + 2*(15*tan(1/2
*d*x + 1/2*c)^9 + 360*I*tan(1/2*d*x + 1/2*c)^8 - 390*tan(1/2*d*x + 1/2*c)^7 - 960*I*tan(1/2*d*x + 1/2*c)^6 + 4
00*I*tan(1/2*d*x + 1/2*c)^4 + 390*tan(1/2*d*x + 1/2*c)^3 - 320*I*tan(1/2*d*x + 1/2*c)^2 - 15*tan(1/2*d*x + 1/2
*c) + 136*I)/((tan(1/2*d*x + 1/2*c)^2 - 1)^5*a^3))/d