3.131 \(\int \frac{\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=82 \[ \frac{i (a-i a \tan (c+d x))^8}{8 a^{11} d}-\frac{4 i (a-i a \tan (c+d x))^7}{7 a^{10} d}+\frac{2 i (a-i a \tan (c+d x))^6}{3 a^9 d} \]

[Out]

(((2*I)/3)*(a - I*a*Tan[c + d*x])^6)/(a^9*d) - (((4*I)/7)*(a - I*a*Tan[c + d*x])^7)/(a^10*d) + ((I/8)*(a - I*a
*Tan[c + d*x])^8)/(a^11*d)

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Rubi [A]  time = 0.0613849, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3487, 43} \[ \frac{i (a-i a \tan (c+d x))^8}{8 a^{11} d}-\frac{4 i (a-i a \tan (c+d x))^7}{7 a^{10} d}+\frac{2 i (a-i a \tan (c+d x))^6}{3 a^9 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^12/(a + I*a*Tan[c + d*x])^3,x]

[Out]

(((2*I)/3)*(a - I*a*Tan[c + d*x])^6)/(a^9*d) - (((4*I)/7)*(a - I*a*Tan[c + d*x])^7)/(a^10*d) + ((I/8)*(a - I*a
*Tan[c + d*x])^8)/(a^11*d)

Rule 3487

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^5 (a+x)^2 \, dx,x,i a \tan (c+d x)\right )}{a^{11} d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (4 a^2 (a-x)^5-4 a (a-x)^6+(a-x)^7\right ) \, dx,x,i a \tan (c+d x)\right )}{a^{11} d}\\ &=\frac{2 i (a-i a \tan (c+d x))^6}{3 a^9 d}-\frac{4 i (a-i a \tan (c+d x))^7}{7 a^{10} d}+\frac{i (a-i a \tan (c+d x))^8}{8 a^{11} d}\\ \end{align*}

Mathematica [A]  time = 0.468781, size = 106, normalized size = 1.29 \[ \frac{\sec (c) \sec ^8(c+d x) (28 \sin (c+2 d x)-28 \sin (3 c+2 d x)+28 \sin (3 c+4 d x)+8 \sin (5 c+6 d x)+\sin (7 c+8 d x)-28 i \cos (c+2 d x)-28 i \cos (3 c+2 d x)-35 \sin (c)-35 i \cos (c))}{168 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^12/(a + I*a*Tan[c + d*x])^3,x]

[Out]

(Sec[c]*Sec[c + d*x]^8*((-35*I)*Cos[c] - (28*I)*Cos[c + 2*d*x] - (28*I)*Cos[3*c + 2*d*x] - 35*Sin[c] + 28*Sin[
c + 2*d*x] - 28*Sin[3*c + 2*d*x] + 28*Sin[3*c + 4*d*x] + 8*Sin[5*c + 6*d*x] + Sin[7*c + 8*d*x]))/(168*a^3*d)

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Maple [A]  time = 0.079, size = 89, normalized size = 1.1 \begin{align*}{\frac{1}{d{a}^{3}} \left ( \tan \left ( dx+c \right ) +{\frac{i}{8}} \left ( \tan \left ( dx+c \right ) \right ) ^{8}-{\frac{3\, \left ( \tan \left ( dx+c \right ) \right ) ^{7}}{7}}-{\frac{i}{6}} \left ( \tan \left ( dx+c \right ) \right ) ^{6}- \left ( \tan \left ( dx+c \right ) \right ) ^{5}-{\frac{5\,i}{4}} \left ( \tan \left ( dx+c \right ) \right ) ^{4}-{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3}}-{\frac{3\,i}{2}} \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^12/(a+I*a*tan(d*x+c))^3,x)

[Out]

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

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Maxima [A]  time = 0.986612, size = 117, normalized size = 1.43 \begin{align*} -\frac{-21 i \, \tan \left (d x + c\right )^{8} + 72 \, \tan \left (d x + c\right )^{7} + 28 i \, \tan \left (d x + c\right )^{6} + 168 \, \tan \left (d x + c\right )^{5} + 210 i \, \tan \left (d x + c\right )^{4} + 56 \, \tan \left (d x + c\right )^{3} + 252 i \, \tan \left (d x + c\right )^{2} - 168 \, \tan \left (d x + c\right )}{168 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^12/(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/168*(-21*I*tan(d*x + c)^8 + 72*tan(d*x + c)^7 + 28*I*tan(d*x + c)^6 + 168*tan(d*x + c)^5 + 210*I*tan(d*x +
c)^4 + 56*tan(d*x + c)^3 + 252*I*tan(d*x + c)^2 - 168*tan(d*x + c))/(a^3*d)

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Fricas [B]  time = 2.34707, size = 443, normalized size = 5.4 \begin{align*} \frac{896 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 256 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 32 i}{21 \,{\left (a^{3} d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, a^{3} d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, a^{3} d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, a^{3} d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, 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)^12/(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

1/21*(896*I*e^(4*I*d*x + 4*I*c) + 256*I*e^(2*I*d*x + 2*I*c) + 32*I)/(a^3*d*e^(16*I*d*x + 16*I*c) + 8*a^3*d*e^(
14*I*d*x + 14*I*c) + 28*a^3*d*e^(12*I*d*x + 12*I*c) + 56*a^3*d*e^(10*I*d*x + 10*I*c) + 70*a^3*d*e^(8*I*d*x + 8
*I*c) + 56*a^3*d*e^(6*I*d*x + 6*I*c) + 28*a^3*d*e^(4*I*d*x + 4*I*c) + 8*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)**12/(a+I*a*tan(d*x+c))**3,x)

[Out]

Exception raised: AttributeError

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Giac [A]  time = 1.20249, size = 117, normalized size = 1.43 \begin{align*} -\frac{-21 i \, \tan \left (d x + c\right )^{8} + 72 \, \tan \left (d x + c\right )^{7} + 28 i \, \tan \left (d x + c\right )^{6} + 168 \, \tan \left (d x + c\right )^{5} + 210 i \, \tan \left (d x + c\right )^{4} + 56 \, \tan \left (d x + c\right )^{3} + 252 i \, \tan \left (d x + c\right )^{2} - 168 \, \tan \left (d x + c\right )}{168 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^12/(a+I*a*tan(d*x+c))^3,x, algorithm="giac")

[Out]

-1/168*(-21*I*tan(d*x + c)^8 + 72*tan(d*x + c)^7 + 28*I*tan(d*x + c)^6 + 168*tan(d*x + c)^5 + 210*I*tan(d*x +
c)^4 + 56*tan(d*x + c)^3 + 252*I*tan(d*x + c)^2 - 168*tan(d*x + c))/(a^3*d)