3.60 \(\int \sec ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx\)

Optimal. Leaf size=82 \[ -\frac{i (a+i a \tan (c+d x))^{10}}{10 a^5 d}+\frac{4 i (a+i a \tan (c+d x))^9}{9 a^4 d}-\frac{i (a+i a \tan (c+d x))^8}{2 a^3 d} \]

[Out]

((-I/2)*(a + I*a*Tan[c + d*x])^8)/(a^3*d) + (((4*I)/9)*(a + I*a*Tan[c + d*x])^9)/(a^4*d) - ((I/10)*(a + I*a*Ta
n[c + d*x])^10)/(a^5*d)

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Rubi [A]  time = 0.0566016, 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} \[ -\frac{i (a+i a \tan (c+d x))^{10}}{10 a^5 d}+\frac{4 i (a+i a \tan (c+d x))^9}{9 a^4 d}-\frac{i (a+i a \tan (c+d x))^8}{2 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^5,x]

[Out]

((-I/2)*(a + I*a*Tan[c + d*x])^8)/(a^3*d) + (((4*I)/9)*(a + I*a*Tan[c + d*x])^9)/(a^4*d) - ((I/10)*(a + I*a*Ta
n[c + d*x])^10)/(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))^5 \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^2 (a+x)^7 \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (4 a^2 (a+x)^7-4 a (a+x)^8+(a+x)^9\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{i (a+i a \tan (c+d x))^8}{2 a^3 d}+\frac{4 i (a+i a \tan (c+d x))^9}{9 a^4 d}-\frac{i (a+i a \tan (c+d x))^{10}}{10 a^5 d}\\ \end{align*}

Mathematica [A]  time = 2.21879, size = 154, normalized size = 1.88 \[ \frac{a^5 \sec (c) \sec ^{10}(c+d x) (105 \sin (c+2 d x)-105 \sin (3 c+2 d x)+60 \sin (3 c+4 d x)-60 \sin (5 c+4 d x)+45 \sin (5 c+6 d x)+10 \sin (7 c+8 d x)+\sin (9 c+10 d x)+105 i \cos (c+2 d x)+105 i \cos (3 c+2 d x)+60 i \cos (3 c+4 d x)+60 i \cos (5 c+4 d x)-126 \sin (c)+126 i \cos (c))}{360 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^5,x]

[Out]

(a^5*Sec[c]*Sec[c + d*x]^10*((126*I)*Cos[c] + (105*I)*Cos[c + 2*d*x] + (105*I)*Cos[3*c + 2*d*x] + (60*I)*Cos[3
*c + 4*d*x] + (60*I)*Cos[5*c + 4*d*x] - 126*Sin[c] + 105*Sin[c + 2*d*x] - 105*Sin[3*c + 2*d*x] + 60*Sin[3*c +
4*d*x] - 60*Sin[5*c + 4*d*x] + 45*Sin[5*c + 6*d*x] + 10*Sin[7*c + 8*d*x] + Sin[9*c + 10*d*x]))/(360*d)

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Maple [B]  time = 0.079, size = 295, normalized size = 3.6 \begin{align*}{\frac{1}{d} \left ( i{a}^{5} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{10\, \left ( \cos \left ( dx+c \right ) \right ) ^{10}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{20\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{60\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}} \right ) +5\,{a}^{5} \left ( 1/9\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{9}}}+{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{63\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{315\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}} \right ) -10\,i{a}^{5} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{12\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{24\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}} \right ) -10\,{a}^{5} \left ( 1/7\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{105\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) +{\frac{{\frac{5\,i}{6}}{a}^{5}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}-{a}^{5} \left ( -{\frac{8}{15}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \tan \left ( dx+c \right ) \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))^5,x)

[Out]

1/d*(I*a^5*(1/10*sin(d*x+c)^6/cos(d*x+c)^10+1/20*sin(d*x+c)^6/cos(d*x+c)^8+1/60*sin(d*x+c)^6/cos(d*x+c)^6)+5*a
^5*(1/9*sin(d*x+c)^5/cos(d*x+c)^9+4/63*sin(d*x+c)^5/cos(d*x+c)^7+8/315*sin(d*x+c)^5/cos(d*x+c)^5)-10*I*a^5*(1/
8*sin(d*x+c)^4/cos(d*x+c)^8+1/12*sin(d*x+c)^4/cos(d*x+c)^6+1/24*sin(d*x+c)^4/cos(d*x+c)^4)-10*a^5*(1/7*sin(d*x
+c)^3/cos(d*x+c)^7+4/35*sin(d*x+c)^3/cos(d*x+c)^5+8/105*sin(d*x+c)^3/cos(d*x+c)^3)+5/6*I*a^5/cos(d*x+c)^6-a^5*
(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c))

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Maxima [A]  time = 1.07371, size = 146, normalized size = 1.78 \begin{align*} \frac{126 i \, a^{5} \tan \left (d x + c\right )^{10} + 700 \, a^{5} \tan \left (d x + c\right )^{9} - 1260 i \, a^{5} \tan \left (d x + c\right )^{8} - 2940 i \, a^{5} \tan \left (d x + c\right )^{6} - 3528 \, a^{5} \tan \left (d x + c\right )^{5} - 3360 \, a^{5} \tan \left (d x + c\right )^{3} + 3150 i \, a^{5} \tan \left (d x + c\right )^{2} + 1260 \, a^{5} \tan \left (d x + c\right )}{1260 \, 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))^5,x, algorithm="maxima")

[Out]

1/1260*(126*I*a^5*tan(d*x + c)^10 + 700*a^5*tan(d*x + c)^9 - 1260*I*a^5*tan(d*x + c)^8 - 2940*I*a^5*tan(d*x +
c)^6 - 3528*a^5*tan(d*x + c)^5 - 3360*a^5*tan(d*x + c)^3 + 3150*I*a^5*tan(d*x + c)^2 + 1260*a^5*tan(d*x + c))/
d

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Fricas [B]  time = 1.14535, size = 740, normalized size = 9.02 \begin{align*} \frac{15360 i \, a^{5} e^{\left (14 i \, d x + 14 i \, c\right )} + 26880 i \, a^{5} e^{\left (12 i \, d x + 12 i \, c\right )} + 32256 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} + 26880 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} + 15360 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} + 5760 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 1280 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + 128 i \, a^{5}}{45 \,{\left (d e^{\left (20 i \, d x + 20 i \, c\right )} + 10 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 45 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 120 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 210 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 252 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 210 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 120 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 45 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 10 \, 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))^5,x, algorithm="fricas")

[Out]

1/45*(15360*I*a^5*e^(14*I*d*x + 14*I*c) + 26880*I*a^5*e^(12*I*d*x + 12*I*c) + 32256*I*a^5*e^(10*I*d*x + 10*I*c
) + 26880*I*a^5*e^(8*I*d*x + 8*I*c) + 15360*I*a^5*e^(6*I*d*x + 6*I*c) + 5760*I*a^5*e^(4*I*d*x + 4*I*c) + 1280*
I*a^5*e^(2*I*d*x + 2*I*c) + 128*I*a^5)/(d*e^(20*I*d*x + 20*I*c) + 10*d*e^(18*I*d*x + 18*I*c) + 45*d*e^(16*I*d*
x + 16*I*c) + 120*d*e^(14*I*d*x + 14*I*c) + 210*d*e^(12*I*d*x + 12*I*c) + 252*d*e^(10*I*d*x + 10*I*c) + 210*d*
e^(8*I*d*x + 8*I*c) + 120*d*e^(6*I*d*x + 6*I*c) + 45*d*e^(4*I*d*x + 4*I*c) + 10*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))**5,x)

[Out]

Timed out

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Giac [A]  time = 1.48996, size = 146, normalized size = 1.78 \begin{align*} -\frac{-9 i \, a^{5} \tan \left (d x + c\right )^{10} - 50 \, a^{5} \tan \left (d x + c\right )^{9} + 90 i \, a^{5} \tan \left (d x + c\right )^{8} + 210 i \, a^{5} \tan \left (d x + c\right )^{6} + 252 \, a^{5} \tan \left (d x + c\right )^{5} + 240 \, a^{5} \tan \left (d x + c\right )^{3} - 225 i \, a^{5} \tan \left (d x + c\right )^{2} - 90 \, a^{5} \tan \left (d x + c\right )}{90 \, 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))^5,x, algorithm="giac")

[Out]

-1/90*(-9*I*a^5*tan(d*x + c)^10 - 50*a^5*tan(d*x + c)^9 + 90*I*a^5*tan(d*x + c)^8 + 210*I*a^5*tan(d*x + c)^6 +
252*a^5*tan(d*x + c)^5 + 240*a^5*tan(d*x + c)^3 - 225*I*a^5*tan(d*x + c)^2 - 90*a^5*tan(d*x + c))/d