3.334 \(\int \frac{\sec ^6(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx\)

Optimal. Leaf size=88 \[ -\frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{7/2}}{7 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{5/2}}{5 a^3 d} \]

[Out]

(((-8*I)/5)*(a + I*a*Tan[c + d*x])^(5/2))/(a^3*d) + (((8*I)/7)*(a + I*a*Tan[c + d*x])^(7/2))/(a^4*d) - (((2*I)
/9)*(a + I*a*Tan[c + d*x])^(9/2))/(a^5*d)

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Rubi [A]  time = 0.0720852, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3487, 43} \[ -\frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{7/2}}{7 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{5/2}}{5 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^6/Sqrt[a + I*a*Tan[c + d*x]],x]

[Out]

(((-8*I)/5)*(a + I*a*Tan[c + d*x])^(5/2))/(a^3*d) + (((8*I)/7)*(a + I*a*Tan[c + d*x])^(7/2))/(a^4*d) - (((2*I)
/9)*(a + I*a*Tan[c + d*x])^(9/2))/(a^5*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 ^6(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^2 (a+x)^{3/2} \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (4 a^2 (a+x)^{3/2}-4 a (a+x)^{5/2}+(a+x)^{7/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{8 i (a+i a \tan (c+d x))^{5/2}}{5 a^3 d}+\frac{8 i (a+i a \tan (c+d x))^{7/2}}{7 a^4 d}-\frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a^5 d}\\ \end{align*}

Mathematica [A]  time = 0.279938, size = 77, normalized size = 0.88 \[ \frac{2 \sec ^5(c+d x) (-55 i \sin (2 (c+d x))+71 \cos (2 (c+d x))+36) (\sin (3 (c+d x))-i \cos (3 (c+d x)))}{315 d \sqrt{a+i a \tan (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^6/Sqrt[a + I*a*Tan[c + d*x]],x]

[Out]

(2*Sec[c + d*x]^5*(36 + 71*Cos[2*(c + d*x)] - (55*I)*Sin[2*(c + d*x)])*((-I)*Cos[3*(c + d*x)] + Sin[3*(c + d*x
)]))/(315*d*Sqrt[a + I*a*Tan[c + d*x]])

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Maple [A]  time = 0.375, size = 100, normalized size = 1.1 \begin{align*} -{\frac{128\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}-128\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +16\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}-80\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +70\,i}{315\,ad \left ( \cos \left ( dx+c \right ) \right ) ^{4}}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^(1/2),x)

[Out]

-2/315/d/a*(64*I*cos(d*x+c)^4-64*cos(d*x+c)^3*sin(d*x+c)+8*I*cos(d*x+c)^2-40*cos(d*x+c)*sin(d*x+c)+35*I)*(a*(I
*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)/cos(d*x+c)^4

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Maxima [B]  time = 1.13443, size = 228, normalized size = 2.59 \begin{align*} -\frac{2 i \,{\left (315 \, \sqrt{i \, a \tan \left (d x + c\right ) + a} - \frac{42 \,{\left (3 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} - 10 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{i \, a \tan \left (d x + c\right ) + a} a^{2}\right )}}{a^{2}} + \frac{35 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{9}{2}} - 180 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a + 378 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{2} - 420 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{3} + 315 \, \sqrt{i \, a \tan \left (d x + c\right ) + a} a^{4}}{a^{4}}\right )}}{315 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

-2/315*I*(315*sqrt(I*a*tan(d*x + c) + a) - 42*(3*(I*a*tan(d*x + c) + a)^(5/2) - 10*(I*a*tan(d*x + c) + a)^(3/2
)*a + 15*sqrt(I*a*tan(d*x + c) + a)*a^2)/a^2 + (35*(I*a*tan(d*x + c) + a)^(9/2) - 180*(I*a*tan(d*x + c) + a)^(
7/2)*a + 378*(I*a*tan(d*x + c) + a)^(5/2)*a^2 - 420*(I*a*tan(d*x + c) + a)^(3/2)*a^3 + 315*sqrt(I*a*tan(d*x +
c) + a)*a^4)/a^4)/(a*d)

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Fricas [A]  time = 2.21809, size = 362, normalized size = 4.11 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-256 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 1152 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 2016 i \, e^{\left (4 i \, d x + 4 i \, c\right )}\right )} e^{\left (i \, d x + i \, c\right )}}{315 \,{\left (a d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/315*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(-256*I*e^(8*I*d*x + 8*I*c) - 1152*I*e^(6*I*d*x + 6*I*c) - 201
6*I*e^(4*I*d*x + 4*I*c))*e^(I*d*x + I*c)/(a*d*e^(8*I*d*x + 8*I*c) + 4*a*d*e^(6*I*d*x + 6*I*c) + 6*a*d*e^(4*I*d
*x + 4*I*c) + 4*a*d*e^(2*I*d*x + 2*I*c) + a*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**6/(a+I*a*tan(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (d x + c\right )^{6}}{\sqrt{i \, a \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sec(d*x + c)^6/sqrt(I*a*tan(d*x + c) + a), x)