3.1.11 \(\int \cot ^5(c+d x) (a+i a \tan (c+d x)) \, dx\) [11]

Optimal. Leaf size=83 \[ i a x+\frac {i a \cot (c+d x)}{d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {i a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3610, 3612, 3556} \begin {gather*} -\frac {a \cot ^4(c+d x)}{4 d}-\frac {i a \cot ^3(c+d x)}{3 d}+\frac {a \cot ^2(c+d x)}{2 d}+\frac {i a \cot (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d}+i a x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x]),x]

[Out]

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

Rubi steps

\begin {align*} \int \cot ^5(c+d x) (a+i a \tan (c+d x)) \, dx &=-\frac {a \cot ^4(c+d x)}{4 d}+\int \cot ^4(c+d x) (i a-a \tan (c+d x)) \, dx\\ &=-\frac {i a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^4(c+d x)}{4 d}+\int \cot ^3(c+d x) (-a-i a \tan (c+d x)) \, dx\\ &=\frac {a \cot ^2(c+d x)}{2 d}-\frac {i a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^4(c+d x)}{4 d}+\int \cot ^2(c+d x) (-i a+a \tan (c+d x)) \, dx\\ &=\frac {i a \cot (c+d x)}{d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {i a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^4(c+d x)}{4 d}+\int \cot (c+d x) (a+i a \tan (c+d x)) \, dx\\ &=i a x+\frac {i a \cot (c+d x)}{d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {i a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^4(c+d x)}{4 d}+a \int \cot (c+d x) \, dx\\ &=i a x+\frac {i a \cot (c+d x)}{d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {i a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \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 = 0.39, size = 84, normalized size = 1.01 \begin {gather*} -\frac {i a \cot ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(c+d x)\right )}{3 d}+\frac {a \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 )}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x]),x]

[Out]

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

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Maple [A]
time = 0.20, size = 61, normalized size = 0.73

method	result	size
derivativedivides	\(\frac {i a \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+a \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(61\)
default	\(\frac {i a \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+a \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(61\)
risch	\(-\frac {2 i a c}{d}-\frac {4 a \left (6 \,{\mathrm e}^{6 i \left (d x +c \right )}-9 \,{\mathrm e}^{4 i \left (d x +c \right )}+8 \,{\mathrm e}^{2 i \left (d x +c \right )}-2\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(81\)
norman	\(\frac {\frac {i a \left (\tan ^{3}\left (d x +c \right )\right )}{d}+i a x \left (\tan ^{4}\left (d x +c \right )\right )-\frac {a}{4 d}+\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {i a \tan \left (d x +c \right )}{3 d}}{\tan \left (d x +c \right )^{4}}+\frac {a \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^5*(a+I*a*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 83, normalized size = 1.00 \begin {gather*} -\frac {-12 i \, {\left (d x + c\right )} a + 6 \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 12 \, a \log \left (\tan \left (d x + c\right )\right ) - \frac {12 i \, a \tan \left (d x + c\right )^{3} + 6 \, a \tan \left (d x + c\right )^{2} - 4 i \, a \tan \left (d x + c\right ) - 3 \, a}{\tan \left (d x + c\right )^{4}}}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/12*(-12*I*(d*x + c)*a + 6*a*log(tan(d*x + c)^2 + 1) - 12*a*log(tan(d*x + c)) - (12*I*a*tan(d*x + c)^3 + 6*a
*tan(d*x + c)^2 - 4*I*a*tan(d*x + c) - 3*a)/tan(d*x + c)^4)/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. 156 vs. \(2 (71) = 142\).
time = 0.55, size = 156, normalized size = 1.88 \begin {gather*} -\frac {24 \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 36 \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 32 \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 3 \, {\left (a e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 8 \, a}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, 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)^5*(a+I*a*tan(d*x+c)),x, algorithm="fricas")

[Out]

-1/3*(24*a*e^(6*I*d*x + 6*I*c) - 36*a*e^(4*I*d*x + 4*I*c) + 32*a*e^(2*I*d*x + 2*I*c) - 3*(a*e^(8*I*d*x + 8*I*c
) - 4*a*e^(6*I*d*x + 6*I*c) + 6*a*e^(4*I*d*x + 4*I*c) - 4*a*e^(2*I*d*x + 2*I*c) + a)*log(e^(2*I*d*x + 2*I*c) -
1) - 8*a)/(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*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. 158 vs. \(2 (70) = 140\).
time = 0.27, size = 158, normalized size = 1.90 \begin {gather*} \frac {a \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 24 a e^{6 i c} e^{6 i d x} + 36 a e^{4 i c} e^{4 i d x} - 32 a e^{2 i c} e^{2 i d x} + 8 a}{3 d e^{8 i c} e^{8 i d x} - 12 d e^{6 i c} e^{6 i d x} + 18 d e^{4 i c} e^{4 i d x} - 12 d e^{2 i c} e^{2 i d x} + 3 d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**5*(a+I*a*tan(d*x+c)),x)

[Out]

a*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-24*a*exp(6*I*c)*exp(6*I*d*x) + 36*a*exp(4*I*c)*exp(4*I*d*x) - 32*a*exp
(2*I*c)*exp(2*I*d*x) + 8*a)/(3*d*exp(8*I*c)*exp(8*I*d*x) - 12*d*exp(6*I*c)*exp(6*I*d*x) + 18*d*exp(4*I*c)*exp(
4*I*d*x) - 12*d*exp(2*I*c)*exp(2*I*d*x) + 3*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. 158 vs. \(2 (71) = 142\).
time = 0.68, size = 158, normalized size = 1.90 \begin {gather*} -\frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 384 \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 192 \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 120 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {400 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

-1/192*(3*a*tan(1/2*d*x + 1/2*c)^4 - 8*I*a*tan(1/2*d*x + 1/2*c)^3 - 36*a*tan(1/2*d*x + 1/2*c)^2 + 384*a*log(ta
n(1/2*d*x + 1/2*c) + I) - 192*a*log(tan(1/2*d*x + 1/2*c)) + 120*I*a*tan(1/2*d*x + 1/2*c) + (400*a*tan(1/2*d*x
+ 1/2*c)^4 - 120*I*a*tan(1/2*d*x + 1/2*c)^3 - 36*a*tan(1/2*d*x + 1/2*c)^2 + 8*I*a*tan(1/2*d*x + 1/2*c) + 3*a)/
tan(1/2*d*x + 1/2*c)^4)/d

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Mupad [B]
time = 3.98, size = 70, normalized size = 0.84 \begin {gather*} \frac {a\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d}-\frac {-1{}\mathrm {i}\,a\,{\mathrm {tan}\left (c+d\,x\right )}^3-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {1{}\mathrm {i}\,a\,\mathrm {tan}\left (c+d\,x\right )}{3}+\frac {a}{4}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^5*(a + a*tan(c + d*x)*1i),x)

[Out]

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

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