3.320 \(\int \sec ^6(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx\)

Optimal. Leaf size=88 \[ -\frac{2 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d} \]

[Out]

(((-8*I)/13)*(a + I*a*Tan[c + d*x])^(13/2))/(a^3*d) + (((8*I)/15)*(a + I*a*Tan[c + d*x])^(15/2))/(a^4*d) - (((
2*I)/17)*(a + I*a*Tan[c + d*x])^(17/2))/(a^5*d)

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Rubi [A]  time = 0.0766982, 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} \[ -\frac{2 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^(7/2),x]

[Out]

(((-8*I)/13)*(a + I*a*Tan[c + d*x])^(13/2))/(a^3*d) + (((8*I)/15)*(a + I*a*Tan[c + d*x])^(15/2))/(a^4*d) - (((
2*I)/17)*(a + I*a*Tan[c + d*x])^(17/2))/(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 \sec ^6(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^2 (a+x)^{11/2} \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (4 a^2 (a+x)^{11/2}-4 a (a+x)^{13/2}+(a+x)^{15/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{8 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d}+\frac{8 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d}-\frac{2 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}\\ \end{align*}

Mathematica [A]  time = 0.858398, size = 97, normalized size = 1.1 \[ \frac{2 a^3 \sec ^8(c+d x) \sqrt{a+i a \tan (c+d x)} (-247 i \sin (2 (c+d x))+263 \cos (2 (c+d x))+68) (\sin (6 c+9 d x)-i \cos (6 c+9 d x))}{3315 d (\cos (d x)+i \sin (d x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^(7/2),x]

[Out]

(2*a^3*Sec[c + d*x]^8*(68 + 263*Cos[2*(c + d*x)] - (247*I)*Sin[2*(c + d*x)])*((-I)*Cos[6*c + 9*d*x] + Sin[6*c
+ 9*d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(3315*d*(Cos[d*x] + I*Sin[d*x])^3)

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Maple [B]  time = 0.628, size = 154, normalized size = 1.8 \begin{align*} -{\frac{2\,{a}^{3} \left ( 1024\,i \left ( \cos \left ( dx+c \right ) \right ) ^{8}-1024\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}+128\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}-640\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) +56\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}-504\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) -1072\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}+676\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +195\,i \right ) }{3315\,d \left ( \cos \left ( dx+c \right ) \right ) ^{8}}\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))^(7/2),x)

[Out]

-2/3315/d*a^3*(1024*I*cos(d*x+c)^8-1024*sin(d*x+c)*cos(d*x+c)^7+128*I*cos(d*x+c)^6-640*cos(d*x+c)^5*sin(d*x+c)
+56*I*cos(d*x+c)^4-504*cos(d*x+c)^3*sin(d*x+c)-1072*I*cos(d*x+c)^2+676*cos(d*x+c)*sin(d*x+c)+195*I)*(a*(I*sin(
d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)/cos(d*x+c)^8

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Maxima [A]  time = 1.11964, size = 78, normalized size = 0.89 \begin{align*} -\frac{2 i \,{\left (195 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{17}{2}} - 884 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{15}{2}} a + 1020 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{13}{2}} a^{2}\right )}}{3315 \, a^{5} 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))^(7/2),x, algorithm="maxima")

[Out]

-2/3315*I*(195*(I*a*tan(d*x + c) + a)^(17/2) - 884*(I*a*tan(d*x + c) + a)^(15/2)*a + 1020*(I*a*tan(d*x + c) +
a)^(13/2)*a^2)/(a^5*d)

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Fricas [B]  time = 2.05958, size = 537, normalized size = 6.1 \begin{align*} \frac{\sqrt{2}{\left (-4096 i \, a^{3} e^{\left (16 i \, d x + 16 i \, c\right )} - 34816 i \, a^{3} e^{\left (14 i \, d x + 14 i \, c\right )} - 130560 i \, a^{3} e^{\left (12 i \, d x + 12 i \, c\right )}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}}{3315 \,{\left (d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + 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))^(7/2),x, algorithm="fricas")

[Out]

1/3315*sqrt(2)*(-4096*I*a^3*e^(16*I*d*x + 16*I*c) - 34816*I*a^3*e^(14*I*d*x + 14*I*c) - 130560*I*a^3*e^(12*I*d
*x + 12*I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(I*d*x + I*c)/(d*e^(16*I*d*x + 16*I*c) + 8*d*e^(14*I*d*x + 1
4*I*c) + 28*d*e^(12*I*d*x + 12*I*c) + 56*d*e^(10*I*d*x + 10*I*c) + 70*d*e^(8*I*d*x + 8*I*c) + 56*d*e^(6*I*d*x
+ 6*I*c) + 28*d*e^(4*I*d*x + 4*I*c) + 8*d*e^(2*I*d*x + 2*I*c) + 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))**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{7}{2}} \sec \left (d x + c\right )^{6}\,{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))^(7/2),x, algorithm="giac")

[Out]

integrate((I*a*tan(d*x + c) + a)^(7/2)*sec(d*x + c)^6, x)