3.11 \(\int \frac{\csc ^7(x)}{i+\cot (x)} \, dx\)

Optimal. Leaf size=40 \[ -\frac{1}{5} \csc ^5(x)+\frac{3}{8} i \tanh ^{-1}(\cos (x))+\frac{1}{4} i \cot (x) \csc ^3(x)+\frac{3}{8} i \cot (x) \csc (x) \]

[Out]

((3*I)/8)*ArcTanh[Cos[x]] + ((3*I)/8)*Cot[x]*Csc[x] + (I/4)*Cot[x]*Csc[x]^3 - Csc[x]^5/5

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Rubi [A]  time = 0.0453658, antiderivative size = 40, normalized size of antiderivative = 1., 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 = {3501, 3768, 3770} \[ -\frac{1}{5} \csc ^5(x)+\frac{3}{8} i \tanh ^{-1}(\cos (x))+\frac{1}{4} i \cot (x) \csc ^3(x)+\frac{3}{8} i \cot (x) \csc (x) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^7/(I + Cot[x]),x]

[Out]

((3*I)/8)*ArcTanh[Cos[x]] + ((3*I)/8)*Cot[x]*Csc[x] + (I/4)*Cot[x]*Csc[x]^3 - Csc[x]^5/5

Rule 3501

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

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\csc ^7(x)}{i+\cot (x)} \, dx &=-\frac{1}{5} \csc ^5(x)-i \int \csc ^5(x) \, dx\\ &=\frac{1}{4} i \cot (x) \csc ^3(x)-\frac{\csc ^5(x)}{5}-\frac{3}{4} i \int \csc ^3(x) \, dx\\ &=\frac{3}{8} i \cot (x) \csc (x)+\frac{1}{4} i \cot (x) \csc ^3(x)-\frac{\csc ^5(x)}{5}-\frac{3}{8} i \int \csc (x) \, dx\\ &=\frac{3}{8} i \tanh ^{-1}(\cos (x))+\frac{3}{8} i \cot (x) \csc (x)+\frac{1}{4} i \cot (x) \csc ^3(x)-\frac{\csc ^5(x)}{5}\\ \end{align*}

Mathematica [B]  time = 0.136459, size = 99, normalized size = 2.48 \[ \frac{1}{640} i \csc ^5(x) \left (140 \sin (2 x)-30 \sin (4 x)+75 \sin (3 x) \log \left (\sin \left (\frac{x}{2}\right )\right )-15 \sin (5 x) \log \left (\sin \left (\frac{x}{2}\right )\right )+150 \sin (x) \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right )-75 \sin (3 x) \log \left (\cos \left (\frac{x}{2}\right )\right )+15 \sin (5 x) \log \left (\cos \left (\frac{x}{2}\right )\right )+128 i\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^7/(I + Cot[x]),x]

[Out]

(I/640)*Csc[x]^5*(128*I + 150*(Log[Cos[x/2]] - Log[Sin[x/2]])*Sin[x] + 140*Sin[2*x] - 75*Log[Cos[x/2]]*Sin[3*x
] + 75*Log[Sin[x/2]]*Sin[3*x] - 30*Sin[4*x] + 15*Log[Cos[x/2]]*Sin[5*x] - 15*Log[Sin[x/2]]*Sin[5*x])

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Maple [B]  time = 0.056, size = 92, normalized size = 2.3 \begin{align*} -{\frac{1}{16}\tan \left ({\frac{x}{2}} \right ) }-{\frac{1}{160} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{5}}-{\frac{i}{64}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{4}-{\frac{1}{32} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}}-{\frac{i}{8}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+{{\frac{i}{64}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-4}}+{{\frac{i}{8}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{1}{16} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{1}{32} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-3}}-{\frac{3\,i}{8}}\ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) -{\frac{1}{160} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^7/(I+cot(x)),x)

[Out]

-1/16*tan(1/2*x)-1/160*tan(1/2*x)^5-1/64*I*tan(1/2*x)^4-1/32*tan(1/2*x)^3-1/8*I*tan(1/2*x)^2+1/64*I/tan(1/2*x)
^4+1/8*I/tan(1/2*x)^2-1/16/tan(1/2*x)-1/32/tan(1/2*x)^3-3/8*I*ln(tan(1/2*x))-1/160/tan(1/2*x)^5

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Maxima [B]  time = 1.25495, size = 177, normalized size = 4.42 \begin{align*} -\frac{{\left (-\frac{15 i \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{30 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{120 i \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{60 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 6\right )}{\left (\cos \left (x\right ) + 1\right )}^{5}}{960 \, \sin \left (x\right )^{5}} - \frac{\sin \left (x\right )}{16 \,{\left (\cos \left (x\right ) + 1\right )}} - \frac{i \, \sin \left (x\right )^{2}}{8 \,{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{\sin \left (x\right )^{3}}{32 \,{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac{i \, \sin \left (x\right )^{4}}{64 \,{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac{\sin \left (x\right )^{5}}{160 \,{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac{3}{8} i \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^7/(I+cot(x)),x, algorithm="maxima")

[Out]

-1/960*(-15*I*sin(x)/(cos(x) + 1) + 30*sin(x)^2/(cos(x) + 1)^2 - 120*I*sin(x)^3/(cos(x) + 1)^3 + 60*sin(x)^4/(
cos(x) + 1)^4 + 6)*(cos(x) + 1)^5/sin(x)^5 - 1/16*sin(x)/(cos(x) + 1) - 1/8*I*sin(x)^2/(cos(x) + 1)^2 - 1/32*s
in(x)^3/(cos(x) + 1)^3 - 1/64*I*sin(x)^4/(cos(x) + 1)^4 - 1/160*sin(x)^5/(cos(x) + 1)^5 - 3/8*I*log(sin(x)/(co
s(x) + 1))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (240 \,{\left (e^{\left (12 i \, x\right )} - 6 \, e^{\left (10 i \, x\right )} + 15 \, e^{\left (8 i \, x\right )} - 20 \, e^{\left (6 i \, x\right )} + 15 \, e^{\left (4 i \, x\right )} - 6 \, e^{\left (2 i \, x\right )} + 1\right )} e^{\left (2 i \, x\right )}{\rm integral}\left (\frac{{\left (7 \, e^{\left (13 i \, x\right )} - 42 \, e^{\left (11 i \, x\right )} - 919 \, e^{\left (9 i \, x\right )} - 140 \, e^{\left (7 i \, x\right )} + 105 \, e^{\left (5 i \, x\right )} - 42 \, e^{\left (3 i \, x\right )} + 7 \, e^{\left (i \, x\right )}\right )} e^{\left (-2 i \, x\right )}}{16 \,{\left (e^{\left (14 i \, x\right )} - 7 \, e^{\left (12 i \, x\right )} + 21 \, e^{\left (10 i \, x\right )} - 35 \, e^{\left (8 i \, x\right )} + 35 \, e^{\left (6 i \, x\right )} - 21 \, e^{\left (4 i \, x\right )} + 7 \, e^{\left (2 i \, x\right )} - 1\right )}}, x\right ) - 35 i \, e^{\left (11 i \, x\right )} + 189 i \, e^{\left (9 i \, x\right )} - 414 i \, e^{\left (7 i \, x\right )} + 2170 i \, e^{\left (5 i \, x\right )} - 735 i \, e^{\left (3 i \, x\right )} + 105 i \, e^{\left (i \, x\right )}\right )} e^{\left (-2 i \, x\right )}}{240 \,{\left (e^{\left (12 i \, x\right )} - 6 \, e^{\left (10 i \, x\right )} + 15 \, e^{\left (8 i \, x\right )} - 20 \, e^{\left (6 i \, x\right )} + 15 \, e^{\left (4 i \, x\right )} - 6 \, e^{\left (2 i \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^7/(I+cot(x)),x, algorithm="fricas")

[Out]

1/240*(240*(e^(12*I*x) - 6*e^(10*I*x) + 15*e^(8*I*x) - 20*e^(6*I*x) + 15*e^(4*I*x) - 6*e^(2*I*x) + 1)*e^(2*I*x
)*integral(1/16*(7*e^(13*I*x) - 42*e^(11*I*x) - 919*e^(9*I*x) - 140*e^(7*I*x) + 105*e^(5*I*x) - 42*e^(3*I*x) +
7*e^(I*x))*e^(-2*I*x)/(e^(14*I*x) - 7*e^(12*I*x) + 21*e^(10*I*x) - 35*e^(8*I*x) + 35*e^(6*I*x) - 21*e^(4*I*x)
+ 7*e^(2*I*x) - 1), x) - 35*I*e^(11*I*x) + 189*I*e^(9*I*x) - 414*I*e^(7*I*x) + 2170*I*e^(5*I*x) - 735*I*e^(3*
I*x) + 105*I*e^(I*x))*e^(-2*I*x)/(e^(12*I*x) - 6*e^(10*I*x) + 15*e^(8*I*x) - 20*e^(6*I*x) + 15*e^(4*I*x) - 6*e
^(2*I*x) + 1)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)**7/(I+cot(x)),x)

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.32942, size = 128, normalized size = 3.2 \begin{align*} -\frac{1}{160} \, \tan \left (\frac{1}{2} \, x\right )^{5} - \frac{1}{64} i \, \tan \left (\frac{1}{2} \, x\right )^{4} - \frac{1}{32} \, \tan \left (\frac{1}{2} \, x\right )^{3} - \frac{1}{8} i \, \tan \left (\frac{1}{2} \, x\right )^{2} - \frac{-274 i \, \tan \left (\frac{1}{2} \, x\right )^{5} + 20 \, \tan \left (\frac{1}{2} \, x\right )^{4} - 40 i \, \tan \left (\frac{1}{2} \, x\right )^{3} + 10 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 5 i \, \tan \left (\frac{1}{2} \, x\right ) + 2}{320 \, \tan \left (\frac{1}{2} \, x\right )^{5}} - \frac{3}{8} i \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right ) - \frac{1}{16} \, \tan \left (\frac{1}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^7/(I+cot(x)),x, algorithm="giac")

[Out]

-1/160*tan(1/2*x)^5 - 1/64*I*tan(1/2*x)^4 - 1/32*tan(1/2*x)^3 - 1/8*I*tan(1/2*x)^2 - 1/320*(-274*I*tan(1/2*x)^
5 + 20*tan(1/2*x)^4 - 40*I*tan(1/2*x)^3 + 10*tan(1/2*x)^2 - 5*I*tan(1/2*x) + 2)/tan(1/2*x)^5 - 3/8*I*log(abs(t
an(1/2*x))) - 1/16*tan(1/2*x)