3.1.8 \(\int \frac {\sec ^4(x)}{i+\cot (x)} \, dx\) [8]

Optimal. Leaf size=19 \[ \frac {\tan ^2(x)}{2}-\frac {1}{3} i \tan ^3(x) \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 19, 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, 45} \begin {gather*} \frac {\tan ^2(x)}{2}-\frac {1}{3} i \tan ^3(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[x]^4/(I + Cot[x]),x]

[Out]

Tan[x]^2/2 - (I/3)*Tan[x]^3

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

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Mathematica [A]
time = 0.06, size = 24, normalized size = 1.26 \begin {gather*} \frac {1}{6} \left (\sec ^2(x) (3-2 i \tan (x))+2 i \tan (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^4/(I + Cot[x]),x]

[Out]

(Sec[x]^2*(3 - (2*I)*Tan[x]) + (2*I)*Tan[x])/6

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Maple [A]
time = 0.20, size = 18, normalized size = 0.95

method	result	size
default	\(-i \left (\frac {\left (\tan ^{3}\left (x \right )\right )}{3}+\frac {i \left (\tan ^{2}\left (x \right )\right )}{2}\right )\)	\(18\)
risch	\(\frac {2 \,{\mathrm e}^{2 i x}-\frac {2}{3}}{\left ({\mathrm e}^{2 i x}+1\right )^{3}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^4/(I+cot(x)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 13, normalized size = 0.68 \begin {gather*} -\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)^4/(I+cot(x)),x, algorithm="maxima")

[Out]

-1/3*I*tan(x)^3 + 1/2*tan(x)^2

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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. 30 vs. \(2 (13) = 26\).
time = 2.98, size = 30, normalized size = 1.58 \begin {gather*} \frac {2 \, {\left (3 \, e^{\left (2 i \, x\right )} - 1\right )}}{3 \, {\left (e^{\left (6 i \, x\right )} + 3 \, e^{\left (4 i \, x\right )} + 3 \, e^{\left (2 i \, x\right )} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^4/(I+cot(x)),x, algorithm="fricas")

[Out]

2/3*(3*e^(2*I*x) - 1)/(e^(6*I*x) + 3*e^(4*I*x) + 3*e^(2*I*x) + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{4}{\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)**4/(I+cot(x)),x)

[Out]

Integral(sec(x)**4/(cot(x) + I), x)

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Giac [A]
time = 0.42, size = 13, normalized size = 0.68 \begin {gather*} -\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)^4/(I+cot(x)),x, algorithm="giac")

[Out]

-1/3*I*tan(x)^3 + 1/2*tan(x)^2

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Mupad [B]
time = 0.20, size = 13, normalized size = 0.68 \begin {gather*} -\frac {{\mathrm {tan}\left (x\right )}^2\,\left (-3+\mathrm {tan}\left (x\right )\,2{}\mathrm {i}\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^4*(cot(x) + 1i)),x)

[Out]

-(tan(x)^2*(tan(x)*2i - 3))/6

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