3.1.23 \(\int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx\) [23]

Optimal. Leaf size=112 \[ -2 a^2 x-\frac {2 a^2 \cot (c+d x)}{d}+\frac {i a^2 \cot ^2(c+d x)}{d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {i a^2 \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {2 i a^2 \log (\sin (c+d x))}{d} \]

[Out]

-2*a^2*x-2*a^2*cot(d*x+c)/d+I*a^2*cot(d*x+c)^2/d+2/3*a^2*cot(d*x+c)^3/d-1/2*I*a^2*cot(d*x+c)^4/d-1/5*a^2*cot(d
*x+c)^5/d+2*I*a^2*ln(sin(d*x+c))/d

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Rubi [A]
time = 0.12, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3623, 3610, 3612, 3556} \begin {gather*} -\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {i a^2 \cot ^4(c+d x)}{2 d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}+\frac {i a^2 \cot ^2(c+d x)}{d}-\frac {2 a^2 \cot (c+d x)}{d}+\frac {2 i a^2 \log (\sin (c+d x))}{d}-2 a^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^2,x]

[Out]

-2*a^2*x - (2*a^2*Cot[c + d*x])/d + (I*a^2*Cot[c + d*x]^2)/d + (2*a^2*Cot[c + d*x]^3)/(3*d) - ((I/2)*a^2*Cot[c
+ d*x]^4)/d - (a^2*Cot[c + d*x]^5)/(5*d) + ((2*I)*a^2*Log[Sin[c + d*x]])/d

Rule 3556

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

Rule 3610

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b
*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3612

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c +
b*d)*(x/(a^2 + b^2)), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rule 3623

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

Rubi steps

\begin {align*} \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx &=-\frac {a^2 \cot ^5(c+d x)}{5 d}+\int \cot ^5(c+d x) \left (2 i a^2-2 a^2 \tan (c+d x)\right ) \, dx\\ &=-\frac {i a^2 \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\int \cot ^4(c+d x) \left (-2 a^2-2 i a^2 \tan (c+d x)\right ) \, dx\\ &=\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {i a^2 \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\int \cot ^3(c+d x) \left (-2 i a^2+2 a^2 \tan (c+d x)\right ) \, dx\\ &=\frac {i a^2 \cot ^2(c+d x)}{d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {i a^2 \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\int \cot ^2(c+d x) \left (2 a^2+2 i a^2 \tan (c+d x)\right ) \, dx\\ &=-\frac {2 a^2 \cot (c+d x)}{d}+\frac {i a^2 \cot ^2(c+d x)}{d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {i a^2 \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\int \cot (c+d x) \left (2 i a^2-2 a^2 \tan (c+d x)\right ) \, dx\\ &=-2 a^2 x-\frac {2 a^2 \cot (c+d x)}{d}+\frac {i a^2 \cot ^2(c+d x)}{d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {i a^2 \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\left (2 i a^2\right ) \int \cot (c+d x) \, dx\\ &=-2 a^2 x-\frac {2 a^2 \cot (c+d x)}{d}+\frac {i a^2 \cot ^2(c+d x)}{d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {i a^2 \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {2 i a^2 \log (\sin (c+d x))}{d}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.83, size = 124, normalized size = 1.11 \begin {gather*} -\frac {a^2 \cot ^5(c+d x) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\tan ^2(c+d x)\right )}{5 d}+\frac {a^2 \cot ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(c+d x)\right )}{3 d}+\frac {i a^2 \left (2 \cot ^2(c+d x)-\cot ^4(c+d x)+4 \log (\cos (c+d x))+4 \log (\tan (c+d x))\right )}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^2,x]

[Out]

-1/5*(a^2*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/d + (a^2*Cot[c + d*x]^3*Hypergeome
tric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) + ((I/2)*a^2*(2*Cot[c + d*x]^2 - Cot[c + d*x]^4 + 4*Log[Cos[c +
d*x]] + 4*Log[Tan[c + d*x]]))/d

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Maple [A]
time = 0.20, size = 106, normalized size = 0.95

method	result	size
risch	\(\frac {4 a^{2} c}{d}-\frac {2 i a^{2} \left (135 \,{\mathrm e}^{8 i \left (d x +c \right )}-300 \,{\mathrm e}^{6 i \left (d x +c \right )}+370 \,{\mathrm e}^{4 i \left (d x +c \right )}-200 \,{\mathrm e}^{2 i \left (d x +c \right )}+43\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {2 i a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(100\)
derivativedivides	\(\frac {-a^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+2 i a^{2} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\)	\(106\)
default	\(\frac {-a^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+2 i a^{2} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\)	\(106\)
norman	\(\frac {\frac {i a^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{d}-\frac {a^{2}}{5 d}-2 a^{2} x \left (\tan ^{5}\left (d x +c \right )\right )+\frac {2 a^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}-\frac {2 a^{2} \left (\tan ^{4}\left (d x +c \right )\right )}{d}-\frac {i a^{2} \tan \left (d x +c \right )}{2 d}}{\tan \left (d x +c \right )^{5}}+\frac {2 i a^{2} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {i a^{2} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\)	\(134\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-a^2*(-1/3*cot(d*x+c)^3+cot(d*x+c)+d*x+c)+2*I*a^2*(-1/4*cot(d*x+c)^4+1/2*cot(d*x+c)^2+ln(sin(d*x+c)))+a^2
*(-1/5*cot(d*x+c)^5+1/3*cot(d*x+c)^3-cot(d*x+c)-d*x-c))

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Maxima [A]
time = 0.67, size = 109, normalized size = 0.97 \begin {gather*} -\frac {60 \, {\left (d x + c\right )} a^{2} + 30 i \, a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 i \, a^{2} \log \left (\tan \left (d x + c\right )\right ) + \frac {60 \, a^{2} \tan \left (d x + c\right )^{4} - 30 i \, a^{2} \tan \left (d x + c\right )^{3} - 20 \, a^{2} \tan \left (d x + c\right )^{2} + 15 i \, a^{2} \tan \left (d x + c\right ) + 6 \, a^{2}}{\tan \left (d x + c\right )^{5}}}{30 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/30*(60*(d*x + c)*a^2 + 30*I*a^2*log(tan(d*x + c)^2 + 1) - 60*I*a^2*log(tan(d*x + c)) + (60*a^2*tan(d*x + c)
^4 - 30*I*a^2*tan(d*x + c)^3 - 20*a^2*tan(d*x + c)^2 + 15*I*a^2*tan(d*x + c) + 6*a^2)/tan(d*x + c)^5)/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. 219 vs. \(2 (100) = 200\).
time = 0.43, size = 219, normalized size = 1.96 \begin {gather*} -\frac {2 \, {\left (135 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 300 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 370 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 200 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 43 i \, a^{2} + 15 \, {\left (-i \, a^{2} e^{\left (10 i \, d x + 10 i \, c\right )} + 5 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 10 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 10 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 5 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{15 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} - 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} - 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, 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(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

-2/15*(135*I*a^2*e^(8*I*d*x + 8*I*c) - 300*I*a^2*e^(6*I*d*x + 6*I*c) + 370*I*a^2*e^(4*I*d*x + 4*I*c) - 200*I*a
^2*e^(2*I*d*x + 2*I*c) + 43*I*a^2 + 15*(-I*a^2*e^(10*I*d*x + 10*I*c) + 5*I*a^2*e^(8*I*d*x + 8*I*c) - 10*I*a^2*
e^(6*I*d*x + 6*I*c) + 10*I*a^2*e^(4*I*d*x + 4*I*c) - 5*I*a^2*e^(2*I*d*x + 2*I*c) + I*a^2)*log(e^(2*I*d*x + 2*I
*c) - 1))/(d*e^(10*I*d*x + 10*I*c) - 5*d*e^(8*I*d*x + 8*I*c) + 10*d*e^(6*I*d*x + 6*I*c) - 10*d*e^(4*I*d*x + 4*
I*c) + 5*d*e^(2*I*d*x + 2*I*c) - d)

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (100) = 200\).
time = 0.33, size = 218, normalized size = 1.95 \begin {gather*} \frac {2 i a^{2} \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 270 i a^{2} e^{8 i c} e^{8 i d x} + 600 i a^{2} e^{6 i c} e^{6 i d x} - 740 i a^{2} e^{4 i c} e^{4 i d x} + 400 i a^{2} e^{2 i c} e^{2 i d x} - 86 i a^{2}}{15 d e^{10 i c} e^{10 i d x} - 75 d e^{8 i c} e^{8 i d x} + 150 d e^{6 i c} e^{6 i d x} - 150 d e^{4 i c} e^{4 i d x} + 75 d e^{2 i c} e^{2 i d x} - 15 d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**6*(a+I*a*tan(d*x+c))**2,x)

[Out]

2*I*a**2*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-270*I*a**2*exp(8*I*c)*exp(8*I*d*x) + 600*I*a**2*exp(6*I*c)*exp(
6*I*d*x) - 740*I*a**2*exp(4*I*c)*exp(4*I*d*x) + 400*I*a**2*exp(2*I*c)*exp(2*I*d*x) - 86*I*a**2)/(15*d*exp(10*I
*c)*exp(10*I*d*x) - 75*d*exp(8*I*c)*exp(8*I*d*x) + 150*d*exp(6*I*c)*exp(6*I*d*x) - 150*d*exp(4*I*c)*exp(4*I*d*
x) + 75*d*exp(2*I*c)*exp(2*I*d*x) - 15*d)

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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. 212 vs. \(2 (100) = 200\).
time = 1.13, size = 212, normalized size = 1.89 \begin {gather*} \frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 55 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 180 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1920 i \, a^{2} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 960 i \, a^{2} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {-2192 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 180 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 55 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^2,x, algorithm="giac")

[Out]

1/480*(3*a^2*tan(1/2*d*x + 1/2*c)^5 - 15*I*a^2*tan(1/2*d*x + 1/2*c)^4 - 55*a^2*tan(1/2*d*x + 1/2*c)^3 + 180*I*
a^2*tan(1/2*d*x + 1/2*c)^2 - 1920*I*a^2*log(tan(1/2*d*x + 1/2*c) + I) + 960*I*a^2*log(tan(1/2*d*x + 1/2*c)) +
630*a^2*tan(1/2*d*x + 1/2*c) + (-2192*I*a^2*tan(1/2*d*x + 1/2*c)^5 - 630*a^2*tan(1/2*d*x + 1/2*c)^4 + 180*I*a^
2*tan(1/2*d*x + 1/2*c)^3 + 55*a^2*tan(1/2*d*x + 1/2*c)^2 - 15*I*a^2*tan(1/2*d*x + 1/2*c) - 3*a^2)/tan(1/2*d*x
+ 1/2*c)^5)/d

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Mupad [B]
time = 4.16, size = 92, normalized size = 0.82 \begin {gather*} -\frac {4\,a^2\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{d}-\frac {2\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^4-a^2\,{\mathrm {tan}\left (c+d\,x\right )}^3\,1{}\mathrm {i}-\frac {2\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^2}{3}+\frac {a^2\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{2}+\frac {a^2}{5}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^6*(a + a*tan(c + d*x)*1i)^2,x)

[Out]

- (4*a^2*atan(2*tan(c + d*x) + 1i))/d - ((a^2*tan(c + d*x)*1i)/2 + a^2/5 - (2*a^2*tan(c + d*x)^2)/3 - a^2*tan(
c + d*x)^3*1i + 2*a^2*tan(c + d*x)^4)/(d*tan(c + d*x)^5)

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