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

Optimal. Leaf size=60 \[ -\frac {a^3+i a^3 \tan (c+d x)}{d}+\frac {a^3 \log (\sin (c+d x))}{d}+\frac {3 a^3 \log (\cos (c+d x))}{d}+4 i a^3 x \]

[Out]

4*I*a^3*x+3*a^3*ln(cos(d*x+c))/d+a^3*ln(sin(d*x+c))/d+(-a^3-I*a^3*tan(d*x+c))/d

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Rubi [A]  time = 0.10, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3556, 3589, 3475, 3531} \[ -\frac {a^3+i a^3 \tan (c+d x)}{d}+\frac {a^3 \log (\sin (c+d x))}{d}+\frac {3 a^3 \log (\cos (c+d x))}{d}+4 i a^3 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 3556

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(b^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n - 1)), x] + Dist[a/(d*(m + n - 1
)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[b*c*(m - 2) + a*d*(m + 2*n) + (a*c*(m - 2) +
b*d*(3*m + 2*n - 4))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a
^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 1] && NeQ[m + n - 1, 0] && (IntegerQ[m] || Intege
rsQ[2*m, 2*n])

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]

Rubi steps

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

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Mathematica [A]  time = 1.32, size = 95, normalized size = 1.58 \[ \frac {a^3 \sec (c) \sec (c+d x) \left (\cos (d x) \left (\log \left (\sin ^2(c+d x)\right )+3 \log \left (\cos ^2(c+d x)\right )+8 i d x\right )+\cos (2 c+d x) \left (\log \left (\sin ^2(c+d x)\right )+3 \log \left (\cos ^2(c+d x)\right )+8 i d x\right )-4 i \sin (d x)\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 83, normalized size = 1.38 \[ \frac {2 \, a^{3} + 3 \, {\left (a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + {\left (a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a^3 + 3*(a^3*e^(2*I*d*x + 2*I*c) + a^3)*log(e^(2*I*d*x + 2*I*c) + 1) + (a^3*e^(2*I*d*x + 2*I*c) + a^3)*log(
e^(2*I*d*x + 2*I*c) - 1))/(d*e^(2*I*d*x + 2*I*c) + d)

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giac [B]  time = 1.22, size = 123, normalized size = 2.05 \[ \frac {3 \, a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) - 8 \, a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 3 \, a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right ) + a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - \frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 i \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*a^3*log(tan(1/2*d*x + 1/2*c) + 1) - 8*a^3*log(tan(1/2*d*x + 1/2*c) + I) + 3*a^3*log(tan(1/2*d*x + 1/2*c) -
1) + a^3*log(tan(1/2*d*x + 1/2*c)) - (3*a^3*tan(1/2*d*x + 1/2*c)^2 - 2*I*a^3*tan(1/2*d*x + 1/2*c) - 3*a^3)/(ta
n(1/2*d*x + 1/2*c)^2 - 1))/d

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maple [A]  time = 0.33, size = 63, normalized size = 1.05 \[ 4 i a^{3} x -\frac {i a^{3} \tan \left (d x +c \right )}{d}+\frac {4 i a^{3} c}{d}+\frac {a^{3} \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {3 a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*I*a^3*x-I/d*tan(d*x+c)*a^3+4*I/d*a^3*c+a^3*ln(sin(d*x+c))/d+3*a^3*ln(cos(d*x+c))/d

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maxima [A]  time = 0.64, size = 53, normalized size = 0.88 \[ \frac {4 i \, {\left (d x + c\right )} a^{3} - 2 \, a^{3} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + a^{3} \log \left (\tan \left (d x + c\right )\right ) - i \, a^{3} \tan \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.77, size = 39, normalized size = 0.65 \[ -\frac {a^3\,\left (4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )-\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)*(a + a*tan(c + d*x)*1i)^3,x)

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotInvertible} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotInvertible

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