3.1.10 \(\int \frac {\sec ^6(x)}{i+\cot (x)} \, dx\) [10]

Optimal. Leaf size=37 \[ \frac {\tan ^2(x)}{2}-\frac {1}{3} i \tan ^3(x)+\frac {\tan ^4(x)}{4}-\frac {1}{5} i \tan ^5(x) \]

[Out]

1/2*tan(x)^2-1/3*I*tan(x)^3+1/4*tan(x)^4-1/5*I*tan(x)^5

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3597, 862, 76} \begin {gather*} -\frac {1}{5} i \tan ^5(x)+\frac {\tan ^4(x)}{4}-\frac {1}{3} i \tan ^3(x)+\frac {\tan ^2(x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[x]^6/(I + Cot[x]),x]

[Out]

Tan[x]^2/2 - (I/3)*Tan[x]^3 + Tan[x]^4/4 - (I/5)*Tan[x]^5

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rule 862

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rule 3597

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[b/f, Subst[Int
[x^m*((a + x)^n/(b^2 + x^2)^(m/2 + 1)), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/
2]

Rubi steps

\begin {align*} \int \frac {\sec ^6(x)}{i+\cot (x)} \, dx &=-\text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^6 (i+x)} \, dx,x,\cot (x)\right )\\ &=-\text {Subst}\left (\int \frac {(-i+x)^2 (i+x)}{x^6} \, dx,x,\cot (x)\right )\\ &=-\text {Subst}\left (\int \left (-\frac {i}{x^6}+\frac {1}{x^5}-\frac {i}{x^4}+\frac {1}{x^3}\right ) \, dx,x,\cot (x)\right )\\ &=\frac {\tan ^2(x)}{2}-\frac {1}{3} i \tan ^3(x)+\frac {\tan ^4(x)}{4}-\frac {1}{5} i \tan ^5(x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.10, size = 26, normalized size = 0.70 \begin {gather*} \frac {1}{60} \sec ^4(x) \left (15-4 i (4+\cos (2 x)) \sin ^2(x) \tan (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^6/(I + Cot[x]),x]

[Out]

(Sec[x]^4*(15 - (4*I)*(4 + Cos[2*x])*Sin[x]^2*Tan[x]))/60

________________________________________________________________________________________

Maple [A]
time = 0.25, size = 31, normalized size = 0.84

method	result	size
risch	\(\frac {\frac {16 \,{\mathrm e}^{4 i x}}{3}-\frac {4 \,{\mathrm e}^{2 i x}}{3}-\frac {4}{15}}{\left ({\mathrm e}^{2 i x}+1\right )^{5}}\)	\(28\)
default	\(i \left (-\frac {\left (\tan ^{5}\left (x \right )\right )}{5}-\frac {i \left (\tan ^{4}\left (x \right )\right )}{4}-\frac {\left (\tan ^{3}\left (x \right )\right )}{3}-\frac {i \left (\tan ^{2}\left (x \right )\right )}{2}\right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^6/(I+cot(x)),x,method=_RETURNVERBOSE)

[Out]

I*(-1/5*tan(x)^5-1/4*I*tan(x)^4-1/3*tan(x)^3-1/2*I*tan(x)^2)

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 25, normalized size = 0.68 \begin {gather*} -\frac {1}{5} i \, \tan \left (x\right )^{5} + \frac {1}{4} \, \tan \left (x\right )^{4} - \frac {1}{3} i \, \tan \left (x\right )^{3} + \frac {1}{2} \, \tan \left (x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^6/(I+cot(x)),x, algorithm="maxima")

[Out]

-1/5*I*tan(x)^5 + 1/4*tan(x)^4 - 1/3*I*tan(x)^3 + 1/2*tan(x)^2

________________________________________________________________________________________

Fricas [A]
time = 2.55, size = 48, normalized size = 1.30 \begin {gather*} \frac {4 \, {\left (20 \, e^{\left (4 i \, x\right )} - 5 \, e^{\left (2 i \, x\right )} - 1\right )}}{15 \, {\left (e^{\left (10 i \, x\right )} + 5 \, e^{\left (8 i \, x\right )} + 10 \, e^{\left (6 i \, x\right )} + 10 \, e^{\left (4 i \, x\right )} + 5 \, e^{\left (2 i \, x\right )} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^6/(I+cot(x)),x, algorithm="fricas")

[Out]

4/15*(20*e^(4*I*x) - 5*e^(2*I*x) - 1)/(e^(10*I*x) + 5*e^(8*I*x) + 10*e^(6*I*x) + 10*e^(4*I*x) + 5*e^(2*I*x) +
1)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{6}{\left (x \right )}}{\cot {\left (x \right )} + i}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**6/(I+cot(x)),x)

[Out]

Integral(sec(x)**6/(cot(x) + I), x)

________________________________________________________________________________________

Giac [A]
time = 0.42, size = 25, normalized size = 0.68 \begin {gather*} -\frac {1}{5} i \, \tan \left (x\right )^{5} + \frac {1}{4} \, \tan \left (x\right )^{4} - \frac {1}{3} i \, \tan \left (x\right )^{3} + \frac {1}{2} \, \tan \left (x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^6/(I+cot(x)),x, algorithm="giac")

[Out]

-1/5*I*tan(x)^5 + 1/4*tan(x)^4 - 1/3*I*tan(x)^3 + 1/2*tan(x)^2

________________________________________________________________________________________

Mupad [B]
time = 0.21, size = 27, normalized size = 0.73 \begin {gather*} -\frac {{\mathrm {tan}\left (x\right )}^5\,1{}\mathrm {i}}{5}+\frac {{\mathrm {tan}\left (x\right )}^4}{4}-\frac {{\mathrm {tan}\left (x\right )}^3\,1{}\mathrm {i}}{3}+\frac {{\mathrm {tan}\left (x\right )}^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^6*(cot(x) + 1i)),x)

[Out]

tan(x)^2/2 - (tan(x)^3*1i)/3 + tan(x)^4/4 - (tan(x)^5*1i)/5

________________________________________________________________________________________