3.185 \(\int \frac{\sec ^5(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=34 \[ \frac{i \tan ^4(c+d x) (-\cot (c+d x)+i)^4}{4 a^3 d} \]

[Out]

((I/4)*(I - Cot[c + d*x])^4*Tan[c + d*x]^4)/(a^3*d)

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Rubi [A]  time = 0.062903, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3088, 848, 37} \[ \frac{i \tan ^4(c+d x) (-\cot (c+d x)+i)^4}{4 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]

[Out]

((I/4)*(I - Cot[c + d*x])^4*Tan[c + d*x]^4)/(a^3*d)

Rule 3088

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symb
ol] :> -Dist[d^(-1), Subst[Int[(x^m*(b + a*x)^n)/(1 + x^2)^((m + n + 2)/2), x], x, Cot[c + d*x]], x] /; FreeQ[
{a, b, c, d}, x] && IntegerQ[n] && IntegerQ[(m + n)/2] && NeQ[n, -1] &&  !(GtQ[n, 0] && GtQ[m, 1])

Rule 848

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rule 37

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

Rubi steps

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

Mathematica [B]  time = 0.465381, size = 90, normalized size = 2.65 \[ -\frac{i \sec (c) \sec ^4(c+d x) (2 i \sin (c+2 d x)-2 i \sin (3 c+2 d x)+i \sin (3 c+4 d x)+2 \cos (c+2 d x)+2 \cos (3 c+2 d x)-3 i \sin (c)+3 \cos (c))}{4 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]

[Out]

((-I/4)*Sec[c]*Sec[c + d*x]^4*(3*Cos[c] + 2*Cos[c + 2*d*x] + 2*Cos[3*c + 2*d*x] - (3*I)*Sin[c] + (2*I)*Sin[c +
2*d*x] - (2*I)*Sin[3*c + 2*d*x] + I*Sin[3*c + 4*d*x]))/(a^3*d)

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Maple [A]  time = 0.197, size = 47, normalized size = 1.4 \begin{align*}{\frac{\tan \left ( dx+c \right ) +{\frac{i}{4}} \left ( \tan \left ( dx+c \right ) \right ) ^{4}- \left ( \tan \left ( dx+c \right ) \right ) ^{3}-{\frac{3\,i}{2}} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^5/(a*cos(d*x+c)+I*a*sin(d*x+c))^3,x)

[Out]

1/d/a^3*(tan(d*x+c)+1/4*I*tan(d*x+c)^4-tan(d*x+c)^3-3/2*I*tan(d*x+c)^2)

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Maxima [B]  time = 1.08891, size = 324, normalized size = 9.53 \begin{align*} \frac{2 \,{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{3 i \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{8 i \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{7 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{3 i \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{\sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{{\left (a^{3} - \frac{4 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{4 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a*cos(d*x+c)+I*a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

2*(sin(d*x + c)/(cos(d*x + c) + 1) - 3*I*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 7*sin(d*x + c)^3/(cos(d*x + c)
+ 1)^3 + 8*I*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 7*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 3*I*sin(d*x + c)^6/
(cos(d*x + c) + 1)^6 - sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/((a^3 - 4*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2
+ 6*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 4*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a^3*sin(d*x + c)^8/(
cos(d*x + c) + 1)^8)*d)

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Fricas [B]  time = 0.448625, size = 177, normalized size = 5.21 \begin{align*} \frac{4 i}{a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a*cos(d*x+c)+I*a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

4*I/(a^3*d*e^(8*I*d*x + 8*I*c) + 4*a^3*d*e^(6*I*d*x + 6*I*c) + 6*a^3*d*e^(4*I*d*x + 4*I*c) + 4*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)**5/(a*cos(d*x+c)+I*a*sin(d*x+c))**3,x)

[Out]

Exception raised: AttributeError

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Giac [A]  time = 1.21989, size = 63, normalized size = 1.85 \begin{align*} -\frac{-i \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{3} + 6 i \, \tan \left (d x + c\right )^{2} - 4 \, \tan \left (d x + c\right )}{4 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a*cos(d*x+c)+I*a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/4*(-I*tan(d*x + c)^4 + 4*tan(d*x + c)^3 + 6*I*tan(d*x + c)^2 - 4*tan(d*x + c))/(a^3*d)