3.40 \(\int \cot ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx\)

Optimal. Leaf size=103 \[ -\frac{7 a^4 \log (\sin (c+d x))}{d}-\frac{a^4 \log (\cos (c+d x))}{d}-\frac{\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac{3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-8 i a^4 x \]

[Out]

(-8*I)*a^4*x - (a^4*Log[Cos[c + d*x]])/d - (7*a^4*Log[Sin[c + d*x]])/d - (Cot[c + d*x]^2*(a^2 + I*a^2*Tan[c +
d*x])^2)/(2*d) - ((3*I)*Cot[c + d*x]*(a^4 + I*a^4*Tan[c + d*x]))/d

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Rubi [A]  time = 0.217709, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {3553, 3593, 3589, 3475, 3531} \[ -\frac{7 a^4 \log (\sin (c+d x))}{d}-\frac{a^4 \log (\cos (c+d x))}{d}-\frac{\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac{3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-8 i a^4 x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^4,x]

[Out]

(-8*I)*a^4*x - (a^4*Log[Cos[c + d*x]])/d - (7*a^4*Log[Sin[c + d*x]])/d - (Cot[c + d*x]^2*(a^2 + I*a^2*Tan[c +
d*x])^2)/(2*d) - ((3*I)*Cot[c + d*x]*(a^4 + I*a^4*Tan[c + d*x]))/d

Rule 3553

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(a^2*(b*c - a*d)*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(b*c + a*d)*(n + 1)), x] +
Dist[a/(d*(b*c + a*d)*(n + 1)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[b*(b*c*(m
- 2) - a*d*(m - 2*n - 4)) + (a*b*c*(m - 2) + b^2*d*(n + 1) - a^2*d*(m + n - 1))*Tan[e + f*x], x], x], x] /; Fr
eeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && LtQ[
n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3593

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

Rule 3589

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

Rule 3475

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

Rule 3531

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 ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx &=-\frac{\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac{1}{2} \int \cot ^2(c+d x) (a+i a \tan (c+d x))^2 \left (-6 i a^2+2 a^2 \tan (c+d x)\right ) \, dx\\ &=-\frac{\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac{3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-\frac{1}{2} \int \cot (c+d x) (a+i a \tan (c+d x)) \left (14 a^3+2 i a^3 \tan (c+d x)\right ) \, dx\\ &=-\frac{\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac{3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-\frac{1}{2} \int \cot (c+d x) \left (14 a^4+16 i a^4 \tan (c+d x)\right ) \, dx+a^4 \int \tan (c+d x) \, dx\\ &=-8 i a^4 x-\frac{a^4 \log (\cos (c+d x))}{d}-\frac{\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac{3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-\left (7 a^4\right ) \int \cot (c+d x) \, dx\\ &=-8 i a^4 x-\frac{a^4 \log (\cos (c+d x))}{d}-\frac{7 a^4 \log (\sin (c+d x))}{d}-\frac{\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac{3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}\\ \end{align*}

Mathematica [A]  time = 1.94054, size = 133, normalized size = 1.29 \[ \frac{a^4 \csc ^2(c+d x) (\cos (4 d x)+i \sin (4 d x)) \left (-7 \log \left (\sin ^2(c+d x)\right )-\log \left (\cos ^2(c+d x)\right )-8 i \csc (c) \cos (c+2 d x)+\cos (2 (c+d x)) \left (7 \log \left (\sin ^2(c+d x)\right )+\log \left (\cos ^2(c+d x)\right )+16 i d x\right )+8 i \cot (c)-16 i d x-2\right )}{4 d (\cos (d x)+i \sin (d x))^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^4,x]

[Out]

(a^4*Csc[c + d*x]^2*(-2 - (16*I)*d*x + (8*I)*Cot[c] - (8*I)*Cos[c + 2*d*x]*Csc[c] - Log[Cos[c + d*x]^2] - 7*Lo
g[Sin[c + d*x]^2] + Cos[2*(c + d*x)]*((16*I)*d*x + Log[Cos[c + d*x]^2] + 7*Log[Sin[c + d*x]^2]))*(Cos[4*d*x] +
I*Sin[4*d*x]))/(4*d*(Cos[d*x] + I*Sin[d*x])^4)

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Maple [A]  time = 0.054, size = 80, normalized size = 0.8 \begin{align*} -{\frac{{a}^{4}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-8\,i{a}^{4}x-{\frac{8\,i{a}^{4}c}{d}}-7\,{\frac{{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{4\,i{a}^{4}\cot \left ( dx+c \right ) }{d}}-{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3*(a+I*a*tan(d*x+c))^4,x)

[Out]

-a^4*ln(cos(d*x+c))/d-8*I*a^4*x-8*I/d*a^4*c-7*a^4*ln(sin(d*x+c))/d-4*I/d*cot(d*x+c)*a^4-1/2*a^4*cot(d*x+c)^2/d

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Maxima [A]  time = 1.63601, size = 92, normalized size = 0.89 \begin{align*} -\frac{16 i \,{\left (d x + c\right )} a^{4} - 8 \, a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 14 \, a^{4} \log \left (\tan \left (d x + c\right )\right ) + \frac{8 i \, a^{4} \tan \left (d x + c\right ) + a^{4}}{\tan \left (d x + c\right )^{2}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(a+I*a*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/2*(16*I*(d*x + c)*a^4 - 8*a^4*log(tan(d*x + c)^2 + 1) + 14*a^4*log(tan(d*x + c)) + (8*I*a^4*tan(d*x + c) +
a^4)/tan(d*x + c)^2)/d

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Fricas [A]  time = 2.25128, size = 373, normalized size = 3.62 \begin{align*} \frac{10 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 8 \, a^{4} -{\left (a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 7 \,{\left (a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

(10*a^4*e^(2*I*d*x + 2*I*c) - 8*a^4 - (a^4*e^(4*I*d*x + 4*I*c) - 2*a^4*e^(2*I*d*x + 2*I*c) + a^4)*log(e^(2*I*d
*x + 2*I*c) + 1) - 7*(a^4*e^(4*I*d*x + 4*I*c) - 2*a^4*e^(2*I*d*x + 2*I*c) + a^4)*log(e^(2*I*d*x + 2*I*c) - 1))
/(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)

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Sympy [A]  time = 3.21115, size = 114, normalized size = 1.11 \begin{align*} \frac{a^{4} \left (- 7 \log{\left (e^{2 i d x} - e^{- 2 i c} \right )} - \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}\right )}{d} + \frac{\frac{10 a^{4} e^{- 2 i c} e^{2 i d x}}{d} - \frac{8 a^{4} e^{- 4 i c}}{d}}{e^{4 i d x} - 2 e^{- 2 i c} e^{2 i d x} + e^{- 4 i c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**3*(a+I*a*tan(d*x+c))**4,x)

[Out]

a**4*(-7*log(exp(2*I*d*x) - exp(-2*I*c)) - log(exp(2*I*d*x) + exp(-2*I*c)))/d + (10*a**4*exp(-2*I*c)*exp(2*I*d
*x)/d - 8*a**4*exp(-4*I*c)/d)/(exp(4*I*d*x) - 2*exp(-2*I*c)*exp(2*I*d*x) + exp(-4*I*c))

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Giac [A]  time = 1.57263, size = 207, normalized size = 2.01 \begin{align*} -\frac{a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 128 \, a^{4} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right ) + 8 \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + 8 \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + 56 \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 16 i \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{84 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 16 i \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a^{4}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(a+I*a*tan(d*x+c))^4,x, algorithm="giac")

[Out]

-1/8*(a^4*tan(1/2*d*x + 1/2*c)^2 - 128*a^4*log(tan(1/2*d*x + 1/2*c) + I) + 8*a^4*log(abs(tan(1/2*d*x + 1/2*c)
+ 1)) + 8*a^4*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 56*a^4*log(abs(tan(1/2*d*x + 1/2*c))) - 16*I*a^4*tan(1/2*d*
x + 1/2*c) - (84*a^4*tan(1/2*d*x + 1/2*c)^2 - 16*I*a^4*tan(1/2*d*x + 1/2*c) - a^4)/tan(1/2*d*x + 1/2*c)^2)/d