3.1.37 \(\int \sec ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx\) [37]

Optimal. Leaf size=82 \[ -\frac {2 i (a+i a \tan (c+d x))^6}{3 a^3 d}+\frac {4 i (a+i a \tan (c+d x))^7}{7 a^4 d}-\frac {i (a+i a \tan (c+d x))^8}{8 a^5 d} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 82, normalized size of antiderivative = 1.00, 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 = {3568, 45} \begin {gather*} -\frac {i (a+i a \tan (c+d x))^8}{8 a^5 d}+\frac {4 i (a+i a \tan (c+d x))^7}{7 a^4 d}-\frac {2 i (a+i a \tan (c+d x))^6}{3 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^3,x]

[Out]

(((-2*I)/3)*(a + I*a*Tan[c + d*x])^6)/(a^3*d) + (((4*I)/7)*(a + I*a*Tan[c + d*x])^7)/(a^4*d) - ((I/8)*(a + I*a
*Tan[c + d*x])^8)/(a^5*d)

Rule 45

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])

Rule 3568

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]

Rubi steps

\begin {align*} \int \sec ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac {i \text {Subst}\left (\int (a-x)^2 (a+x)^5 \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac {i \text {Subst}\left (\int \left (4 a^2 (a+x)^5-4 a (a+x)^6+(a+x)^7\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac {2 i (a+i a \tan (c+d x))^6}{3 a^3 d}+\frac {4 i (a+i a \tan (c+d x))^7}{7 a^4 d}-\frac {i (a+i a \tan (c+d x))^8}{8 a^5 d}\\ \end {align*}

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Mathematica [A]
time = 0.77, size = 106, normalized size = 1.29 \begin {gather*} \frac {a^3 \sec (c) \sec ^8(c+d x) (35 i \cos (c)+28 i \cos (c+2 d x)+28 i \cos (3 c+2 d x)-35 \sin (c)+28 \sin (c+2 d x)-28 \sin (3 c+2 d x)+28 \sin (3 c+4 d x)+8 \sin (5 c+6 d x)+\sin (7 c+8 d x))}{168 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^3,x]

[Out]

(a^3*Sec[c]*Sec[c + d*x]^8*((35*I)*Cos[c] + (28*I)*Cos[c + 2*d*x] + (28*I)*Cos[3*c + 2*d*x] - 35*Sin[c] + 28*S
in[c + 2*d*x] - 28*Sin[3*c + 2*d*x] + 28*Sin[3*c + 4*d*x] + 8*Sin[5*c + 6*d*x] + Sin[7*c + 8*d*x]))/(168*d)

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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. 173 vs. \(2 (70 ) = 140\).
time = 0.26, size = 174, normalized size = 2.12

method	result	size
risch	\(\frac {32 i a^{3} \left (56 \,{\mathrm e}^{10 i \left (d x +c \right )}+70 \,{\mathrm e}^{8 i \left (d x +c \right )}+56 \,{\mathrm e}^{6 i \left (d x +c \right )}+28 \,{\mathrm e}^{4 i \left (d x +c \right )}+8 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{21 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{8}}\)	\(80\)
derivativedivides	\(\frac {-i a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )-3 a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+\frac {i a^{3}}{2 \cos \left (d x +c \right )^{6}}-a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\)	\(174\)
default	\(\frac {-i a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )-3 a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+\frac {i a^{3}}{2 \cos \left (d x +c \right )^{6}}-a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\)	\(174\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^6*(a+I*a*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-I*a^3*(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)-3*a^
3*(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)+1/2*I*a^3/cos
(d*x+c)^6-a^3*(-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 = 0.29, size = 108, normalized size = 1.32 \begin {gather*} -\frac {21 i \, a^{3} \tan \left (d x + c\right )^{8} + 72 \, a^{3} \tan \left (d x + c\right )^{7} - 28 i \, a^{3} \tan \left (d x + c\right )^{6} + 168 \, a^{3} \tan \left (d x + c\right )^{5} - 210 i \, a^{3} \tan \left (d x + c\right )^{4} + 56 \, a^{3} \tan \left (d x + c\right )^{3} - 252 i \, a^{3} \tan \left (d x + c\right )^{2} - 168 \, a^{3} \tan \left (d x + c\right )}{168 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6*(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/168*(21*I*a^3*tan(d*x + c)^8 + 72*a^3*tan(d*x + c)^7 - 28*I*a^3*tan(d*x + c)^6 + 168*a^3*tan(d*x + c)^5 - 2
10*I*a^3*tan(d*x + c)^4 + 56*a^3*tan(d*x + c)^3 - 252*I*a^3*tan(d*x + c)^2 - 168*a^3*tan(d*x + c))/d

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (64) = 128\).
time = 0.36, size = 177, normalized size = 2.16 \begin {gather*} -\frac {32 \, {\left (-56 i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 70 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 56 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 28 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 8 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3}\right )}}{21 \, {\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6*(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

-32/21*(-56*I*a^3*e^(10*I*d*x + 10*I*c) - 70*I*a^3*e^(8*I*d*x + 8*I*c) - 56*I*a^3*e^(6*I*d*x + 6*I*c) - 28*I*a
^3*e^(4*I*d*x + 4*I*c) - 8*I*a^3*e^(2*I*d*x + 2*I*c) - I*a^3)/(d*e^(16*I*d*x + 16*I*c) + 8*d*e^(14*I*d*x + 14*
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - i a^{3} \left (\int i \sec ^{6}{\left (c + d x \right )}\, dx + \int \left (- 3 \tan {\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\right )\, dx + \int \tan ^{3}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int \left (- 3 i \tan ^{2}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\right )\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**6*(a+I*a*tan(d*x+c))**3,x)

[Out]

-I*a**3*(Integral(I*sec(c + d*x)**6, x) + Integral(-3*tan(c + d*x)*sec(c + d*x)**6, x) + Integral(tan(c + d*x)
**3*sec(c + d*x)**6, x) + Integral(-3*I*tan(c + d*x)**2*sec(c + d*x)**6, x))

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Giac [A]
time = 0.65, size = 108, normalized size = 1.32 \begin {gather*} -\frac {21 i \, a^{3} \tan \left (d x + c\right )^{8} + 72 \, a^{3} \tan \left (d x + c\right )^{7} - 28 i \, a^{3} \tan \left (d x + c\right )^{6} + 168 \, a^{3} \tan \left (d x + c\right )^{5} - 210 i \, a^{3} \tan \left (d x + c\right )^{4} + 56 \, a^{3} \tan \left (d x + c\right )^{3} - 252 i \, a^{3} \tan \left (d x + c\right )^{2} - 168 \, a^{3} \tan \left (d x + c\right )}{168 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6*(a+I*a*tan(d*x+c))^3,x, algorithm="giac")

[Out]

-1/168*(21*I*a^3*tan(d*x + c)^8 + 72*a^3*tan(d*x + c)^7 - 28*I*a^3*tan(d*x + c)^6 + 168*a^3*tan(d*x + c)^5 - 2
10*I*a^3*tan(d*x + c)^4 + 56*a^3*tan(d*x + c)^3 - 252*I*a^3*tan(d*x + c)^2 - 168*a^3*tan(d*x + c))/d

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Mupad [B]
time = 3.29, size = 151, normalized size = 1.84 \begin {gather*} -\frac {a^3\,\sin \left (c+d\,x\right )\,\left (-168\,{\cos \left (c+d\,x\right )}^7-{\cos \left (c+d\,x\right )}^6\,\sin \left (c+d\,x\right )\,252{}\mathrm {i}+56\,{\cos \left (c+d\,x\right )}^5\,{\sin \left (c+d\,x\right )}^2-{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^3\,210{}\mathrm {i}+168\,{\cos \left (c+d\,x\right )}^3\,{\sin \left (c+d\,x\right )}^4-{\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^5\,28{}\mathrm {i}+72\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^6+{\sin \left (c+d\,x\right )}^7\,21{}\mathrm {i}\right )}{168\,d\,{\cos \left (c+d\,x\right )}^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*tan(c + d*x)*1i)^3/cos(c + d*x)^6,x)

[Out]

-(a^3*sin(c + d*x)*(72*cos(c + d*x)*sin(c + d*x)^6 - cos(c + d*x)^6*sin(c + d*x)*252i - 168*cos(c + d*x)^7 + s
in(c + d*x)^7*21i - cos(c + d*x)^2*sin(c + d*x)^5*28i + 168*cos(c + d*x)^3*sin(c + d*x)^4 - cos(c + d*x)^4*sin
(c + d*x)^3*210i + 56*cos(c + d*x)^5*sin(c + d*x)^2))/(168*d*cos(c + d*x)^8)

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