3.305 \(\int \frac{\cot ^4(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx\)

Optimal. Leaf size=31 \[ -\frac{B \cot ^3(c+d x)}{3 d}+\frac{B \cot (c+d x)}{d}+B x \]

[Out]

B*x + (B*Cot[c + d*x])/d - (B*Cot[c + d*x]^3)/(3*d)

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Rubi [A]  time = 0.0257432, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088, Rules used = {21, 3473, 8} \[ -\frac{B \cot ^3(c+d x)}{3 d}+\frac{B \cot (c+d x)}{d}+B x \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^4*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]

[Out]

B*x + (B*Cot[c + d*x])/d - (B*Cot[c + d*x]^3)/(3*d)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\cot ^4(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx &=B \int \cot ^4(c+d x) \, dx\\ &=-\frac{B \cot ^3(c+d x)}{3 d}-B \int \cot ^2(c+d x) \, dx\\ &=\frac{B \cot (c+d x)}{d}-\frac{B \cot ^3(c+d x)}{3 d}+B \int 1 \, dx\\ &=B x+\frac{B \cot (c+d x)}{d}-\frac{B \cot ^3(c+d x)}{3 d}\\ \end{align*}

Mathematica [C]  time = 0.0152082, size = 34, normalized size = 1.1 \[ -\frac{B \cot ^3(c+d x) \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},-\tan ^2(c+d x)\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]^4*(a*B + b*B*Tan[c + d*x]))/(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)

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Maple [A]  time = 0.044, size = 27, normalized size = 0.9 \begin{align*}{\frac{B}{d} \left ( -{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3}}+\cot \left ( dx+c \right ) +dx+c \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x)

[Out]

1/d*B*(-1/3*cot(d*x+c)^3+cot(d*x+c)+d*x+c)

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Maxima [A]  time = 1.80508, size = 51, normalized size = 1.65 \begin{align*} \frac{3 \,{\left (d x + c\right )} B + \frac{3 \, B \tan \left (d x + c\right )^{2} - B}{\tan \left (d x + c\right )^{3}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/3*(3*(d*x + c)*B + (3*B*tan(d*x + c)^2 - B)/tan(d*x + c)^3)/d

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Fricas [B]  time = 1.67942, size = 212, normalized size = 6.84 \begin{align*} \frac{4 \, B \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, B \cos \left (2 \, d x + 2 \, c\right ) + 3 \,{\left (B d x \cos \left (2 \, d x + 2 \, c\right ) - B d x\right )} \sin \left (2 \, d x + 2 \, c\right ) - 2 \, B}{3 \,{\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )} \sin \left (2 \, d x + 2 \, c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/3*(4*B*cos(2*d*x + 2*c)^2 + 2*B*cos(2*d*x + 2*c) + 3*(B*d*x*cos(2*d*x + 2*c) - B*d*x)*sin(2*d*x + 2*c) - 2*B
)/((d*cos(2*d*x + 2*c) - d)*sin(2*d*x + 2*c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**4*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x)

[Out]

Timed out

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Giac [B]  time = 1.32903, size = 93, normalized size = 3. \begin{align*} \frac{B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 24 \,{\left (d x + c\right )} B - 15 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{15 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - B}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

1/24*(B*tan(1/2*d*x + 1/2*c)^3 + 24*(d*x + c)*B - 15*B*tan(1/2*d*x + 1/2*c) + (15*B*tan(1/2*d*x + 1/2*c)^2 - B
)/tan(1/2*d*x + 1/2*c)^3)/d