3.423 \(\int \cot ^6(c+d x) (a+b \tan (c+d x)) \, dx\)

Optimal. Leaf size=93 \[ -\frac{a \cot ^5(c+d x)}{5 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-a x-\frac{b \cot ^4(c+d x)}{4 d}+\frac{b \cot ^2(c+d x)}{2 d}+\frac{b \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.121125, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3529, 3531, 3475} \[ -\frac{a \cot ^5(c+d x)}{5 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-a x-\frac{b \cot ^4(c+d x)}{4 d}+\frac{b \cot ^2(c+d x)}{2 d}+\frac{b \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*x) - (a*Cot[c + d*x])/d + (b*Cot[c + d*x]^2)/(2*d) + (a*Cot[c + d*x]^3)/(3*d) - (b*Cot[c + d*x]^4)/(4*d) -
(a*Cot[c + d*x]^5)/(5*d) + (b*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 ^6(c+d x) (a+b \tan (c+d x)) \, dx &=-\frac{a \cot ^5(c+d x)}{5 d}+\int \cot ^5(c+d x) (b-a \tan (c+d x)) \, dx\\ &=-\frac{b \cot ^4(c+d x)}{4 d}-\frac{a \cot ^5(c+d x)}{5 d}+\int \cot ^4(c+d x) (-a-b \tan (c+d x)) \, dx\\ &=\frac{a \cot ^3(c+d x)}{3 d}-\frac{b \cot ^4(c+d x)}{4 d}-\frac{a \cot ^5(c+d x)}{5 d}+\int \cot ^3(c+d x) (-b+a \tan (c+d x)) \, dx\\ &=\frac{b \cot ^2(c+d x)}{2 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{b \cot ^4(c+d x)}{4 d}-\frac{a \cot ^5(c+d x)}{5 d}+\int \cot ^2(c+d x) (a+b \tan (c+d x)) \, dx\\ &=-\frac{a \cot (c+d x)}{d}+\frac{b \cot ^2(c+d x)}{2 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{b \cot ^4(c+d x)}{4 d}-\frac{a \cot ^5(c+d x)}{5 d}+\int \cot (c+d x) (b-a \tan (c+d x)) \, dx\\ &=-a x-\frac{a \cot (c+d x)}{d}+\frac{b \cot ^2(c+d x)}{2 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{b \cot ^4(c+d x)}{4 d}-\frac{a \cot ^5(c+d x)}{5 d}+b \int \cot (c+d x) \, dx\\ &=-a x-\frac{a \cot (c+d x)}{d}+\frac{b \cot ^2(c+d x)}{2 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{b \cot ^4(c+d x)}{4 d}-\frac{a \cot ^5(c+d x)}{5 d}+\frac{b \log (\sin (c+d x))}{d}\\ \end{align*}

Mathematica [C]  time = 0.354815, size = 82, normalized size = 0.88 \[ \frac{b \left (-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right )}{4 d}-\frac{a \cot ^5(c+d x) \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(c+d x)\right )}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/(5*d) + (b*(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 = 0.039, size = 93, normalized size = 1. \begin{align*} -{\frac{b \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{b \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{b\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{\cot \left ( dx+c \right ) a}{d}}-ax-{\frac{ac}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^6*(a+b*tan(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.59924, size = 126, normalized size = 1.35 \begin{align*} -\frac{60 \,{\left (d x + c\right )} a + 30 \, b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \, b \log \left (\tan \left (d x + c\right )\right ) + \frac{60 \, a \tan \left (d x + c\right )^{4} - 30 \, b \tan \left (d x + c\right )^{3} - 20 \, a \tan \left (d x + c\right )^{2} + 15 \, b \tan \left (d x + c\right ) + 12 \, a}{\tan \left (d x + c\right )^{5}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(60*(d*x + c)*a + 30*b*log(tan(d*x + c)^2 + 1) - 60*b*log(tan(d*x + c)) + (60*a*tan(d*x + c)^4 - 30*b*ta
n(d*x + c)^3 - 20*a*tan(d*x + c)^2 + 15*b*tan(d*x + c) + 12*a)/tan(d*x + c)^5)/d

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Fricas [A]  time = 1.82352, size = 293, normalized size = 3.15 \begin{align*} \frac{30 \, b \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 )^{5} - 15 \,{\left (4 \, a d x - 3 \, b\right )} \tan \left (d x + c\right )^{5} - 60 \, a \tan \left (d x + c\right )^{4} + 30 \, b \tan \left (d x + c\right )^{3} + 20 \, a \tan \left (d x + c\right )^{2} - 15 \, b \tan \left (d x + c\right ) - 12 \, a}{60 \, d \tan \left (d x + c\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6*(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/60*(30*b*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^5 - 15*(4*a*d*x - 3*b)*tan(d*x + c)^5 - 60*a*
tan(d*x + c)^4 + 30*b*tan(d*x + c)^3 + 20*a*tan(d*x + c)^2 - 15*b*tan(d*x + c) - 12*a)/(d*tan(d*x + c)^5)

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Sympy [A]  time = 9.52609, size = 124, normalized size = 1.33 \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 ^{6}{\left (c \right )} & \text{for}\: d = 0 \\- a x - \frac{a}{d \tan{\left (c + d x \right )}} + \frac{a}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac{a}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac{b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{b \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + \frac{b}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac{b}{4 d \tan ^{4}{\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)**6*(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)**6, Eq(d, 0
)), (-a*x - a/(d*tan(c + d*x)) + a/(3*d*tan(c + d*x)**3) - a/(5*d*tan(c + d*x)**5) - b*log(tan(c + d*x)**2 + 1
)/(2*d) + b*log(tan(c + d*x))/d + b/(2*d*tan(c + d*x)**2) - b/(4*d*tan(c + d*x)**4), True))

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Giac [B]  time = 1.33034, size = 267, normalized size = 2.87 \begin{align*} \frac{6 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 15 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 70 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 180 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 960 \,{\left (d x + c\right )} a - 960 \, b \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) + 960 \, b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 660 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{2192 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 660 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 180 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 70 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 15 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{960 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/960*(6*a*tan(1/2*d*x + 1/2*c)^5 - 15*b*tan(1/2*d*x + 1/2*c)^4 - 70*a*tan(1/2*d*x + 1/2*c)^3 + 180*b*tan(1/2*
d*x + 1/2*c)^2 - 960*(d*x + c)*a - 960*b*log(tan(1/2*d*x + 1/2*c)^2 + 1) + 960*b*log(abs(tan(1/2*d*x + 1/2*c))
) + 660*a*tan(1/2*d*x + 1/2*c) - (2192*b*tan(1/2*d*x + 1/2*c)^5 + 660*a*tan(1/2*d*x + 1/2*c)^4 - 180*b*tan(1/2
*d*x + 1/2*c)^3 - 70*a*tan(1/2*d*x + 1/2*c)^2 + 15*b*tan(1/2*d*x + 1/2*c) + 6*a)/tan(1/2*d*x + 1/2*c)^5)/d