3.422 \(\int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx\)

Optimal. Leaf size=75 \[ -\frac{a \cot ^4(c+d x)}{4 d}+\frac{a \cot ^2(c+d x)}{2 d}+\frac{a \log (\sin (c+d x))}{d}-\frac{b \cot ^3(c+d x)}{3 d}+\frac{b \cot (c+d x)}{d}+b x \]

[Out]

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

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Rubi [A]  time = 0.105995, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3529, 3531, 3475} \[ -\frac{a \cot ^4(c+d x)}{4 d}+\frac{a \cot ^2(c+d x)}{2 d}+\frac{a \log (\sin (c+d x))}{d}-\frac{b \cot ^3(c+d x)}{3 d}+\frac{b \cot (c+d x)}{d}+b x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3529

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 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 3475

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

Rubi steps

\begin{align*} \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx &=-\frac{a \cot ^4(c+d x)}{4 d}+\int \cot ^4(c+d x) (b-a \tan (c+d x)) \, dx\\ &=-\frac{b \cot ^3(c+d x)}{3 d}-\frac{a \cot ^4(c+d x)}{4 d}+\int \cot ^3(c+d x) (-a-b \tan (c+d x)) \, dx\\ &=\frac{a \cot ^2(c+d x)}{2 d}-\frac{b \cot ^3(c+d x)}{3 d}-\frac{a \cot ^4(c+d x)}{4 d}+\int \cot ^2(c+d x) (-b+a \tan (c+d x)) \, dx\\ &=\frac{b \cot (c+d x)}{d}+\frac{a \cot ^2(c+d x)}{2 d}-\frac{b \cot ^3(c+d x)}{3 d}-\frac{a \cot ^4(c+d x)}{4 d}+\int \cot (c+d x) (a+b \tan (c+d x)) \, dx\\ &=b x+\frac{b \cot (c+d x)}{d}+\frac{a \cot ^2(c+d x)}{2 d}-\frac{b \cot ^3(c+d x)}{3 d}-\frac{a \cot ^4(c+d x)}{4 d}+a \int \cot (c+d x) \, dx\\ &=b x+\frac{b \cot (c+d x)}{d}+\frac{a \cot ^2(c+d x)}{2 d}-\frac{b \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*}

Mathematica [C]  time = 0.386439, size = 82, normalized size = 1.09 \[ \frac{a \left (-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right )}{4 d}-\frac{b \cot ^3(c+d x) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*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 = 0.037, size = 76, normalized size = 1. \begin{align*} -{\frac{b \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{\cot \left ( dx+c \right ) b}{d}}+bx+{\frac{bc}{d}}-{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{2}a}{2\,d}}+{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.66421, size = 111, normalized size = 1.48 \begin{align*} \frac{12 \,{\left (d x + c\right )} b - 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 \, b \tan \left (d x + c\right )^{3} + 6 \, a \tan \left (d x + c\right )^{2} - 4 \, b \tan \left (d x + c\right ) - 3 \, a}{\tan \left (d x + c\right )^{4}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(12*(d*x + c)*b - 6*a*log(tan(d*x + c)^2 + 1) + 12*a*log(tan(d*x + c)) + (12*b*tan(d*x + c)^3 + 6*a*tan(d
*x + c)^2 - 4*b*tan(d*x + c) - 3*a)/tan(d*x + c)^4)/d

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Fricas [A]  time = 1.79227, size = 257, normalized size = 3.43 \begin{align*} \frac{6 \, a \log \left (\frac{\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{4} + 3 \,{\left (4 \, b d x + 3 \, a\right )} \tan \left (d x + c\right )^{4} + 12 \, b \tan \left (d x + c\right )^{3} + 6 \, a \tan \left (d x + c\right )^{2} - 4 \, b \tan \left (d x + c\right ) - 3 \, a}{12 \, d \tan \left (d x + c\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(6*a*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^4 + 3*(4*b*d*x + 3*a)*tan(d*x + c)^4 + 12*b*ta
n(d*x + c)^3 + 6*a*tan(d*x + c)^2 - 4*b*tan(d*x + c) - 3*a)/(d*tan(d*x + c)^4)

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Sympy [A]  time = 5.00505, size = 110, normalized size = 1.47 \begin{align*} \begin{cases} \tilde{\infty } a x & \text{for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (a + b \tan{\left (c \right )}\right ) \cot ^{5}{\left (c \right )} & \text{for}\: d = 0 \\- \frac{a \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + \frac{a}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac{a}{4 d \tan ^{4}{\left (c + d x \right )}} + b x + \frac{b}{d \tan{\left (c + d x \right )}} - \frac{b}{3 d \tan ^{3}{\left (c + d x \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*a*x, (Eq(c, 0) | Eq(c, -d*x)) & (Eq(d, 0) | Eq(c, -d*x))), (x*(a + b*tan(c))*cot(c)**5, Eq(d, 0
)), (-a*log(tan(c + d*x)**2 + 1)/(2*d) + a*log(tan(c + d*x))/d + a/(2*d*tan(c + d*x)**2) - a/(4*d*tan(c + d*x)
**4) + b*x + b/(d*tan(c + d*x)) - b/(3*d*tan(c + d*x)**3), True))

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Giac [B]  time = 1.37621, size = 228, normalized size = 3.04 \begin{align*} -\frac{3 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 8 \, b \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} - 192 \,{\left (d x + c\right )} b + 192 \, a \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 192 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 120 \, b \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 \, b \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 \, b \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(3*a*tan(1/2*d*x + 1/2*c)^4 - 8*b*tan(1/2*d*x + 1/2*c)^3 - 36*a*tan(1/2*d*x + 1/2*c)^2 - 192*(d*x + c)*
b + 192*a*log(tan(1/2*d*x + 1/2*c)^2 + 1) - 192*a*log(abs(tan(1/2*d*x + 1/2*c))) + 120*b*tan(1/2*d*x + 1/2*c)
+ (400*a*tan(1/2*d*x + 1/2*c)^4 - 120*b*tan(1/2*d*x + 1/2*c)^3 - 36*a*tan(1/2*d*x + 1/2*c)^2 + 8*b*tan(1/2*d*x
+ 1/2*c) + 3*a)/tan(1/2*d*x + 1/2*c)^4)/d