3.2.52 \(\int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^4} \, dx\) [152]

Optimal. Leaf size=63 \[ -\frac {4 x}{a^4}-\frac {4 i \log (\cos (c+d x))}{a^4 d}+\frac {\tan (c+d x)}{a^4 d}+\frac {4 i}{d \left (a^4+i a^4 \tan (c+d x)\right )} \]

[Out]

-4*x/a^4-4*I*ln(cos(d*x+c))/a^4/d+tan(d*x+c)/a^4/d+4*I/d/(a^4+I*a^4*tan(d*x+c))

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Rubi [A]
time = 0.04, antiderivative size = 63, 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 {\tan (c+d x)}{a^4 d}+\frac {4 i}{d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac {4 i \log (\cos (c+d x))}{a^4 d}-\frac {4 x}{a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^4,x]

[Out]

(-4*x)/a^4 - ((4*I)*Log[Cos[c + d*x]])/(a^4*d) + Tan[c + d*x]/(a^4*d) + (4*I)/(d*(a^4 + I*a^4*Tan[c + d*x]))

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 \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=-\frac {i \text {Subst}\left (\int \frac {(a-x)^2}{(a+x)^2} \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac {i \text {Subst}\left (\int \left (1+\frac {4 a^2}{(a+x)^2}-\frac {4 a}{a+x}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac {4 x}{a^4}-\frac {4 i \log (\cos (c+d x))}{a^4 d}+\frac {\tan (c+d x)}{a^4 d}+\frac {4 i}{d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(214\) vs. \(2(63)=126\).
time = 0.79, size = 214, normalized size = 3.40 \begin {gather*} \frac {\sec (c) \sec (c+d x) (-\cos (c+d x)+i \sin (c+d x)) (-i \cos (3 c+2 d x)+2 d x \cos (3 c+2 d x)+2 \cos (c+2 d x) (d x+i \log (\cos (c+d x)))+\cos (c) (-3 i+4 d x+4 i \log (\cos (c+d x)))+2 i \cos (3 c+2 d x) \log (\cos (c+d x))+\sin (c)-2 \sin (c+2 d x)+2 i d x \sin (c+2 d x)-2 \log (\cos (c+d x)) \sin (c+2 d x)-\sin (3 c+2 d x)+2 i d x \sin (3 c+2 d x)-2 \log (\cos (c+d x)) \sin (3 c+2 d x))}{2 a^4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^4,x]

[Out]

(Sec[c]*Sec[c + d*x]*(-Cos[c + d*x] + I*Sin[c + d*x])*((-I)*Cos[3*c + 2*d*x] + 2*d*x*Cos[3*c + 2*d*x] + 2*Cos[
c + 2*d*x]*(d*x + I*Log[Cos[c + d*x]]) + Cos[c]*(-3*I + 4*d*x + (4*I)*Log[Cos[c + d*x]]) + (2*I)*Cos[3*c + 2*d
*x]*Log[Cos[c + d*x]] + Sin[c] - 2*Sin[c + 2*d*x] + (2*I)*d*x*Sin[c + 2*d*x] - 2*Log[Cos[c + d*x]]*Sin[c + 2*d
*x] - Sin[3*c + 2*d*x] + (2*I)*d*x*Sin[3*c + 2*d*x] - 2*Log[Cos[c + d*x]]*Sin[3*c + 2*d*x]))/(2*a^4*d)

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Maple [A]
time = 0.29, size = 41, normalized size = 0.65

method	result	size
derivativedivides	\(\frac {\tan \left (d x +c \right )+4 i \ln \left (\tan \left (d x +c \right )-i\right )+\frac {4}{\tan \left (d x +c \right )-i}}{d \,a^{4}}\)	\(41\)
default	\(\frac {\tan \left (d x +c \right )+4 i \ln \left (\tan \left (d x +c \right )-i\right )+\frac {4}{\tan \left (d x +c \right )-i}}{d \,a^{4}}\)	\(41\)
risch	\(\frac {2 i {\mathrm e}^{-2 i \left (d x +c \right )}}{a^{4} d}-\frac {8 x}{a^{4}}-\frac {8 c}{a^{4} d}+\frac {2 i}{d \,a^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {4 i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a^{4} d}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d/a^4*(tan(d*x+c)+4*I*ln(tan(d*x+c)-I)+4/(tan(d*x+c)-I))

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Maxima [A]
time = 0.30, size = 95, normalized size = 1.51 \begin {gather*} \frac {\frac {4 \, {\left (\tan \left (d x + c\right )^{2} - 2 i \, \tan \left (d x + c\right ) - 1\right )}}{a^{4} \tan \left (d x + c\right )^{3} - 3 i \, a^{4} \tan \left (d x + c\right )^{2} - 3 \, a^{4} \tan \left (d x + c\right ) + i \, a^{4}} + \frac {4 i \, \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{4}} + \frac {\tan \left (d x + c\right )}{a^{4}}}{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))^4,x, algorithm="maxima")

[Out]

(4*(tan(d*x + c)^2 - 2*I*tan(d*x + c) - 1)/(a^4*tan(d*x + c)^3 - 3*I*a^4*tan(d*x + c)^2 - 3*a^4*tan(d*x + c) +
I*a^4) + 4*I*log(I*tan(d*x + c) + 1)/a^4 + tan(d*x + c)/a^4)/d

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Fricas [A]
time = 0.40, size = 102, normalized size = 1.62 \begin {gather*} -\frac {2 \, {\left (4 \, d x e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\left (2 \, d x - i\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \, {\left (i \, e^{\left (4 i \, d x + 4 i \, c\right )} + i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - i\right )}}{a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + a^{4} d e^{\left (2 i \, d x + 2 i \, c\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))^4,x, algorithm="fricas")

[Out]

-2*(4*d*x*e^(4*I*d*x + 4*I*c) + 2*(2*d*x - I)*e^(2*I*d*x + 2*I*c) + 2*(I*e^(4*I*d*x + 4*I*c) + I*e^(2*I*d*x +
2*I*c))*log(e^(2*I*d*x + 2*I*c) + 1) - I)/(a^4*d*e^(4*I*d*x + 4*I*c) + a^4*d*e^(2*I*d*x + 2*I*c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\sec ^{6}{\left (c + d x \right )}}{\tan ^{4}{\left (c + d x \right )} - 4 i \tan ^{3}{\left (c + d x \right )} - 6 \tan ^{2}{\left (c + d x \right )} + 4 i \tan {\left (c + d x \right )} + 1}\, dx}{a^{4}} \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))**4,x)

[Out]

Integral(sec(c + d*x)**6/(tan(c + d*x)**4 - 4*I*tan(c + d*x)**3 - 6*tan(c + d*x)**2 + 4*I*tan(c + d*x) + 1), x
)/a**4

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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. 146 vs. \(2 (57) = 114\).
time = 0.77, size = 146, normalized size = 2.32 \begin {gather*} \frac {2 \, {\left (-\frac {2 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{4}} + \frac {4 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}{a^{4}} - \frac {2 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{4}} + \frac {2 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 i}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{4}} - \frac {2 \, {\left (3 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 i\right )}}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{2}}\right )}}{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))^4,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 3.36, size = 55, normalized size = 0.87 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,4{}\mathrm {i}}{a^4\,d}+\frac {\mathrm {tan}\left (c+d\,x\right )}{a^4\,d}+\frac {4{}\mathrm {i}}{a^4\,d\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^6*(a + a*tan(c + d*x)*1i)^4),x)

[Out]

(log(tan(c + d*x) - 1i)*4i)/(a^4*d) + tan(c + d*x)/(a^4*d) + 4i/(a^4*d*(tan(c + d*x)*1i + 1))

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