3.148 \(\int \frac{\sec ^{14}(c+d x)}{(a+i a \tan (c+d x))^4} \, dx\)

Optimal. Leaf size=82 \[ \frac{i (a-i a \tan (c+d x))^9}{9 a^{13} d}-\frac{i (a-i a \tan (c+d x))^8}{2 a^{12} d}+\frac{4 i (a-i a \tan (c+d x))^7}{7 a^{11} d} \]

[Out]

(((4*I)/7)*(a - I*a*Tan[c + d*x])^7)/(a^11*d) - ((I/2)*(a - I*a*Tan[c + d*x])^8)/(a^12*d) + ((I/9)*(a - I*a*Ta
n[c + d*x])^9)/(a^13*d)

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Rubi [A]  time = 0.0636081, 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} \[ \frac{i (a-i a \tan (c+d x))^9}{9 a^{13} d}-\frac{i (a-i a \tan (c+d x))^8}{2 a^{12} d}+\frac{4 i (a-i a \tan (c+d x))^7}{7 a^{11} d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^14/(a + I*a*Tan[c + d*x])^4,x]

[Out]

(((4*I)/7)*(a - I*a*Tan[c + d*x])^7)/(a^11*d) - ((I/2)*(a - I*a*Tan[c + d*x])^8)/(a^12*d) + ((I/9)*(a - I*a*Ta
n[c + d*x])^9)/(a^13*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 ^{14}(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^6 (a+x)^2 \, dx,x,i a \tan (c+d x)\right )}{a^{13} d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (4 a^2 (a-x)^6-4 a (a-x)^7+(a-x)^8\right ) \, dx,x,i a \tan (c+d x)\right )}{a^{13} d}\\ &=\frac{4 i (a-i a \tan (c+d x))^7}{7 a^{11} d}-\frac{i (a-i a \tan (c+d x))^8}{2 a^{12} d}+\frac{i (a-i a \tan (c+d x))^9}{9 a^{13} d}\\ \end{align*}

Mathematica [A]  time = 0.559942, size = 136, normalized size = 1.66 \[ \frac{\sec (c) \sec ^9(c+d x) (-63 \sin (2 c+d x)+42 \sin (2 c+3 d x)-42 \sin (4 c+3 d x)+36 \sin (4 c+5 d x)+9 \sin (6 c+7 d x)+\sin (8 c+9 d x)-63 i \cos (2 c+d x)-42 i \cos (2 c+3 d x)-42 i \cos (4 c+3 d x)+63 \sin (d x)-63 i \cos (d x))}{252 a^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^14/(a + I*a*Tan[c + d*x])^4,x]

[Out]

(Sec[c]*Sec[c + d*x]^9*((-63*I)*Cos[d*x] - (63*I)*Cos[2*c + d*x] - (42*I)*Cos[2*c + 3*d*x] - (42*I)*Cos[4*c +
3*d*x] + 63*Sin[d*x] - 63*Sin[2*c + d*x] + 42*Sin[2*c + 3*d*x] - 42*Sin[4*c + 3*d*x] + 36*Sin[4*c + 5*d*x] + 9
*Sin[6*c + 7*d*x] + Sin[8*c + 9*d*x]))/(252*a^4*d)

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Maple [A]  time = 0.087, size = 99, normalized size = 1.2 \begin{align*}{\frac{1}{{a}^{4}d} \left ( \tan \left ( dx+c \right ) +{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{9}}{9}}+{\frac{i}{2}} \left ( \tan \left ( dx+c \right ) \right ) ^{8}-{\frac{4\, \left ( \tan \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{2\,i}{3}} \left ( \tan \left ( dx+c \right ) \right ) ^{6}-2\, \left ( \tan \left ( dx+c \right ) \right ) ^{5}-i \left ( \tan \left ( dx+c \right ) \right ) ^{4}-{\frac{4\, \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3}}-2\,i \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)^14/(a+I*a*tan(d*x+c))^4,x)

[Out]

1/d/a^4*(tan(d*x+c)+1/9*tan(d*x+c)^9+1/2*I*tan(d*x+c)^8-4/7*tan(d*x+c)^7+2/3*I*tan(d*x+c)^6-2*tan(d*x+c)^5-I*t
an(d*x+c)^4-4/3*tan(d*x+c)^3-2*I*tan(d*x+c)^2)

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Maxima [A]  time = 0.984934, size = 131, normalized size = 1.6 \begin{align*} \frac{14 \, \tan \left (d x + c\right )^{9} + 63 i \, \tan \left (d x + c\right )^{8} - 72 \, \tan \left (d x + c\right )^{7} + 84 i \, \tan \left (d x + c\right )^{6} - 252 \, \tan \left (d x + c\right )^{5} - 126 i \, \tan \left (d x + c\right )^{4} - 168 \, \tan \left (d x + c\right )^{3} - 252 i \, \tan \left (d x + c\right )^{2} + 126 \, \tan \left (d x + c\right )}{126 \, a^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^14/(a+I*a*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

1/126*(14*tan(d*x + c)^9 + 63*I*tan(d*x + c)^8 - 72*tan(d*x + c)^7 + 84*I*tan(d*x + c)^6 - 252*tan(d*x + c)^5
- 126*I*tan(d*x + c)^4 - 168*tan(d*x + c)^3 - 252*I*tan(d*x + c)^2 + 126*tan(d*x + c))/(a^4*d)

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Fricas [B]  time = 2.8042, size = 494, normalized size = 6.02 \begin{align*} \frac{4608 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 1152 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 128 i}{63 \,{\left (a^{4} d e^{\left (18 i \, d x + 18 i \, c\right )} + 9 \, a^{4} d e^{\left (16 i \, d x + 16 i \, c\right )} + 36 \, a^{4} d e^{\left (14 i \, d x + 14 i \, c\right )} + 84 \, a^{4} d e^{\left (12 i \, d x + 12 i \, c\right )} + 126 \, a^{4} d e^{\left (10 i \, d x + 10 i \, c\right )} + 126 \, a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )} + 84 \, a^{4} d e^{\left (6 i \, d x + 6 i \, c\right )} + 36 \, a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + 9 \, a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^14/(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/63*(4608*I*e^(4*I*d*x + 4*I*c) + 1152*I*e^(2*I*d*x + 2*I*c) + 128*I)/(a^4*d*e^(18*I*d*x + 18*I*c) + 9*a^4*d*
e^(16*I*d*x + 16*I*c) + 36*a^4*d*e^(14*I*d*x + 14*I*c) + 84*a^4*d*e^(12*I*d*x + 12*I*c) + 126*a^4*d*e^(10*I*d*
x + 10*I*c) + 126*a^4*d*e^(8*I*d*x + 8*I*c) + 84*a^4*d*e^(6*I*d*x + 6*I*c) + 36*a^4*d*e^(4*I*d*x + 4*I*c) + 9*
a^4*d*e^(2*I*d*x + 2*I*c) + a^4*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)**14/(a+I*a*tan(d*x+c))**4,x)

[Out]

Exception raised: AttributeError

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Giac [A]  time = 1.19941, size = 131, normalized size = 1.6 \begin{align*} \frac{14 \, \tan \left (d x + c\right )^{9} + 63 i \, \tan \left (d x + c\right )^{8} - 72 \, \tan \left (d x + c\right )^{7} + 84 i \, \tan \left (d x + c\right )^{6} - 252 \, \tan \left (d x + c\right )^{5} - 126 i \, \tan \left (d x + c\right )^{4} - 168 \, \tan \left (d x + c\right )^{3} - 252 i \, \tan \left (d x + c\right )^{2} + 126 \, \tan \left (d x + c\right )}{126 \, a^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^14/(a+I*a*tan(d*x+c))^4,x, algorithm="giac")

[Out]

1/126*(14*tan(d*x + c)^9 + 63*I*tan(d*x + c)^8 - 72*tan(d*x + c)^7 + 84*I*tan(d*x + c)^6 - 252*tan(d*x + c)^5
- 126*I*tan(d*x + c)^4 - 168*tan(d*x + c)^3 - 252*I*tan(d*x + c)^2 + 126*tan(d*x + c))/(a^4*d)