3.2.79 \(\int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\) [179]

Optimal. Leaf size=68 \[ \frac {i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8}+\frac {i \sec ^7(c+d x)}{63 a d (a+i a \tan (c+d x))^7} \]

[Out]

1/9*I*sec(d*x+c)^7/d/(a+I*a*tan(d*x+c))^8+1/63*I*sec(d*x+c)^7/a/d/(a+I*a*tan(d*x+c))^7

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Rubi [A]
time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3583, 3569} \begin {gather*} \frac {i \sec ^7(c+d x)}{63 a d (a+i a \tan (c+d x))^7}+\frac {i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^8,x]

[Out]

((I/9)*Sec[c + d*x]^7)/(d*(a + I*a*Tan[c + d*x])^8) + ((I/63)*Sec[c + d*x]^7)/(a*d*(a + I*a*Tan[c + d*x])^7)

Rule 3569

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(d*
Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/(a*f*m)), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] &
& EqQ[Simplify[m + n], 0]

Rule 3583

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(d*
Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/(b*f*(m + 2*n))), x] + Dist[Simplify[m + n]/(a*(m + 2*n)), Int[(d*Sec[
e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0] && NeQ[m + 2*n, 0] && IntegersQ[2*m, 2*n]

Rubi steps

\begin {align*} \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac {i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8}+\frac {\int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^7} \, dx}{9 a}\\ &=\frac {i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8}+\frac {i \sec ^7(c+d x)}{63 a d (a+i a \tan (c+d x))^7}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 40, normalized size = 0.59 \begin {gather*} -\frac {\sec ^7(c+d x) (-8 i+\tan (c+d x))}{63 a^8 d (-i+\tan (c+d x))^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^8,x]

[Out]

-1/63*(Sec[c + d*x]^7*(-8*I + Tan[c + d*x]))/(a^8*d*(-I + Tan[c + d*x])^8)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (60 ) = 120\).
time = 0.34, size = 156, normalized size = 2.29

method	result	size
risch	\(\frac {i {\mathrm e}^{-7 i \left (d x +c \right )}}{14 a^{8} d}+\frac {i {\mathrm e}^{-9 i \left (d x +c \right )}}{18 a^{8} d}\)	\(38\)
derivativedivides	\(\frac {-\frac {172}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {256}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {992 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {152 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {1856}{7 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {272}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}}{a^{8} d}\)	\(156\)
default	\(\frac {-\frac {172}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {256}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {992 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {152 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {1856}{7 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {272}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}}{a^{8} d}\)	\(156\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^7/(a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

2/d/a^8*(-86/3/(-I+tan(1/2*d*x+1/2*c))^3+128/9/(-I+tan(1/2*d*x+1/2*c))^9+496/3*I/(-I+tan(1/2*d*x+1/2*c))^6-76*
I/(-I+tan(1/2*d*x+1/2*c))^4-928/7/(-I+tan(1/2*d*x+1/2*c))^7-64*I/(-I+tan(1/2*d*x+1/2*c))^8+1/(-I+tan(1/2*d*x+1
/2*c))+7*I/(-I+tan(1/2*d*x+1/2*c))^2+136/(-I+tan(1/2*d*x+1/2*c))^5)

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Maxima [A]
time = 0.32, size = 53, normalized size = 0.78 \begin {gather*} \frac {7 i \, \cos \left (9 \, d x + 9 \, c\right ) + 9 i \, \cos \left (7 \, d x + 7 \, c\right ) + 7 \, \sin \left (9 \, d x + 9 \, c\right ) + 9 \, \sin \left (7 \, d x + 7 \, c\right )}{126 \, a^{8} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

1/126*(7*I*cos(9*d*x + 9*c) + 9*I*cos(7*d*x + 7*c) + 7*sin(9*d*x + 9*c) + 9*sin(7*d*x + 7*c))/(a^8*d)

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Fricas [A]
time = 0.38, size = 30, normalized size = 0.44 \begin {gather*} \frac {{\left (9 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i\right )} e^{\left (-9 i \, d x - 9 i \, c\right )}}{126 \, a^{8} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7/(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/126*(9*I*e^(2*I*d*x + 2*I*c) + 7*I)*e^(-9*I*d*x - 9*I*c)/(a^8*d)

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (54) = 108\).
time = 11.31, size = 311, normalized size = 4.57 \begin {gather*} \begin {cases} - \frac {\tan {\left (c + d x \right )} \sec ^{7}{\left (c + d x \right )}}{63 a^{8} d \tan ^{8}{\left (c + d x \right )} - 504 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 1764 a^{8} d \tan ^{6}{\left (c + d x \right )} + 3528 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 4410 a^{8} d \tan ^{4}{\left (c + d x \right )} - 3528 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 1764 a^{8} d \tan ^{2}{\left (c + d x \right )} + 504 i a^{8} d \tan {\left (c + d x \right )} + 63 a^{8} d} + \frac {8 i \sec ^{7}{\left (c + d x \right )}}{63 a^{8} d \tan ^{8}{\left (c + d x \right )} - 504 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 1764 a^{8} d \tan ^{6}{\left (c + d x \right )} + 3528 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 4410 a^{8} d \tan ^{4}{\left (c + d x \right )} - 3528 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 1764 a^{8} d \tan ^{2}{\left (c + d x \right )} + 504 i a^{8} d \tan {\left (c + d x \right )} + 63 a^{8} d} & \text {for}\: d \neq 0 \\\frac {x \sec ^{7}{\left (c \right )}}{\left (i a \tan {\left (c \right )} + a\right )^{8}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**7/(a+I*a*tan(d*x+c))**8,x)

[Out]

Piecewise((-tan(c + d*x)*sec(c + d*x)**7/(63*a**8*d*tan(c + d*x)**8 - 504*I*a**8*d*tan(c + d*x)**7 - 1764*a**8
*d*tan(c + d*x)**6 + 3528*I*a**8*d*tan(c + d*x)**5 + 4410*a**8*d*tan(c + d*x)**4 - 3528*I*a**8*d*tan(c + d*x)*
*3 - 1764*a**8*d*tan(c + d*x)**2 + 504*I*a**8*d*tan(c + d*x) + 63*a**8*d) + 8*I*sec(c + d*x)**7/(63*a**8*d*tan
(c + d*x)**8 - 504*I*a**8*d*tan(c + d*x)**7 - 1764*a**8*d*tan(c + d*x)**6 + 3528*I*a**8*d*tan(c + d*x)**5 + 44
10*a**8*d*tan(c + d*x)**4 - 3528*I*a**8*d*tan(c + d*x)**3 - 1764*a**8*d*tan(c + d*x)**2 + 504*I*a**8*d*tan(c +
d*x) + 63*a**8*d), Ne(d, 0)), (x*sec(c)**7/(I*a*tan(c) + a)**8, True))

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (56) = 112\).
time = 1.41, size = 125, normalized size = 1.84 \begin {gather*} \frac {2 \, {\left (63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 63 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 483 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 315 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 693 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 189 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 225 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8\right )}}{63 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7/(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

2/63*(63*tan(1/2*d*x + 1/2*c)^8 - 63*I*tan(1/2*d*x + 1/2*c)^7 - 483*tan(1/2*d*x + 1/2*c)^6 + 315*I*tan(1/2*d*x
+ 1/2*c)^5 + 693*tan(1/2*d*x + 1/2*c)^4 - 189*I*tan(1/2*d*x + 1/2*c)^3 - 225*tan(1/2*d*x + 1/2*c)^2 + 9*I*tan
(1/2*d*x + 1/2*c) + 8)/(a^8*d*(tan(1/2*d*x + 1/2*c) - I)^9)

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Mupad [B]
time = 3.74, size = 37, normalized size = 0.54 \begin {gather*} \frac {2\,\left (\frac {{\mathrm {e}}^{-c\,7{}\mathrm {i}-d\,x\,7{}\mathrm {i}}\,9{}\mathrm {i}}{4}+\frac {{\mathrm {e}}^{-c\,9{}\mathrm {i}-d\,x\,9{}\mathrm {i}}\,7{}\mathrm {i}}{4}\right )}{63\,a^8\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^7*(a + a*tan(c + d*x)*1i)^8),x)

[Out]

(2*((exp(- c*7i - d*x*7i)*9i)/4 + (exp(- c*9i - d*x*9i)*7i)/4))/(63*a^8*d)

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