3.19 \(\int \cot ^2(c+d x) (a+b \tan (c+d x))^3 (B \tan (c+d x)+C \tan ^2(c+d x)) \, dx\)

Optimal. Leaf size=117 \[ -\frac{b \left (3 a^2 C+3 a b B-b^2 C\right ) \log (\cos (c+d x))}{d}+x \left (3 a^2 b B+a^3 C-3 a b^2 C-b^3 B\right )+\frac{a^3 B \log (\sin (c+d x))}{d}+\frac{b^2 (2 a C+b B) \tan (c+d x)}{d}+\frac{b C (a+b \tan (c+d x))^2}{2 d} \]

[Out]

(3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*x - (b*(3*a*b*B + 3*a^2*C - b^2*C)*Log[Cos[c + d*x]])/d + (a^3*B*Log[S
in[c + d*x]])/d + (b^2*(b*B + 2*a*C)*Tan[c + d*x])/d + (b*C*(a + b*Tan[c + d*x])^2)/(2*d)

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Rubi [A]  time = 0.335582, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3632, 3607, 3637, 3624, 3475} \[ -\frac{b \left (3 a^2 C+3 a b B-b^2 C\right ) \log (\cos (c+d x))}{d}+x \left (3 a^2 b B+a^3 C-3 a b^2 C-b^3 B\right )+\frac{a^3 B \log (\sin (c+d x))}{d}+\frac{b^2 (2 a C+b B) \tan (c+d x)}{d}+\frac{b C (a+b \tan (c+d x))^2}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

(3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*x - (b*(3*a*b*B + 3*a^2*C - b^2*C)*Log[Cos[c + d*x]])/d + (a^3*B*Log[S
in[c + d*x]])/d + (b^2*(b*B + 2*a*C)*Tan[c + d*x])/d + (b*C*(a + b*Tan[c + d*x])^2)/(2*d)

Rule 3632

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

Rule 3607

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

Rule 3637

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

Rule 3624

Int[((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/tan[(e_.) + (f_.)*(x_)], x_Symbol
] :> Simp[B*x, x] + (Dist[A, Int[1/Tan[e + f*x], x], x] + Dist[C, Int[Tan[e + f*x], x], x]) /; FreeQ[{e, f, A,
B, C}, x] && NeQ[A, C]

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 ^2(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx &=\int \cot (c+d x) (a+b \tan (c+d x))^3 (B+C \tan (c+d x)) \, dx\\ &=\frac{b C (a+b \tan (c+d x))^2}{2 d}+\frac{1}{2} \int \cot (c+d x) (a+b \tan (c+d x)) \left (2 a^2 B+2 \left (2 a b B+a^2 C-b^2 C\right ) \tan (c+d x)+2 b (b B+2 a C) \tan ^2(c+d x)\right ) \, dx\\ &=\frac{b^2 (b B+2 a C) \tan (c+d x)}{d}+\frac{b C (a+b \tan (c+d x))^2}{2 d}-\frac{1}{2} \int \cot (c+d x) \left (-2 a^3 B-2 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \tan (c+d x)-2 b \left (3 a b B+3 a^2 C-b^2 C\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) x+\frac{b^2 (b B+2 a C) \tan (c+d x)}{d}+\frac{b C (a+b \tan (c+d x))^2}{2 d}+\left (a^3 B\right ) \int \cot (c+d x) \, dx+\left (b \left (3 a b B+3 a^2 C-b^2 C\right )\right ) \int \tan (c+d x) \, dx\\ &=\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) x-\frac{b \left (3 a b B+3 a^2 C-b^2 C\right ) \log (\cos (c+d x))}{d}+\frac{a^3 B \log (\sin (c+d x))}{d}+\frac{b^2 (b B+2 a C) \tan (c+d x)}{d}+\frac{b C (a+b \tan (c+d x))^2}{2 d}\\ \end{align*}

Mathematica [C]  time = 0.455175, size = 113, normalized size = 0.97 \[ \frac{2 a^3 B \log (\tan (c+d x))+2 b^2 (3 a C+b B) \tan (c+d x)-(a+i b)^3 (B+i C) \log (-\tan (c+d x)+i)-(a-i b)^3 (B-i C) \log (\tan (c+d x)+i)+b^3 C \tan ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

(-((a + I*b)^3*(B + I*C)*Log[I - Tan[c + d*x]]) + 2*a^3*B*Log[Tan[c + d*x]] - (a - I*b)^3*(B - I*C)*Log[I + Ta
n[c + d*x]] + 2*b^2*(b*B + 3*a*C)*Tan[c + d*x] + b^3*C*Tan[c + d*x]^2)/(2*d)

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Maple [A]  time = 0.094, size = 183, normalized size = 1.6 \begin{align*} -Bx{b}^{3}+{\frac{B\tan \left ( dx+c \right ){b}^{3}}{d}}-{\frac{B{b}^{3}c}{d}}+{\frac{C{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{C{b}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-3\,{\frac{Ba{b}^{2}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-3\,Ca{b}^{2}x+3\,{\frac{Ca{b}^{2}\tan \left ( dx+c \right ) }{d}}-3\,{\frac{Ca{b}^{2}c}{d}}+3\,B{a}^{2}bx+3\,{\frac{B{a}^{2}bc}{d}}-3\,{\frac{C{a}^{2}b\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{B{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+Cx{a}^{3}+{\frac{C{a}^{3}c}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^2*(a+b*tan(d*x+c))^3*(B*tan(d*x+c)+C*tan(d*x+c)^2),x)

[Out]

-B*x*b^3+1/d*B*tan(d*x+c)*b^3-1/d*B*b^3*c+1/2/d*C*b^3*tan(d*x+c)^2+b^3*C*ln(cos(d*x+c))/d-3/d*B*a*b^2*ln(cos(d
*x+c))-3*C*a*b^2*x+3/d*C*a*b^2*tan(d*x+c)-3/d*C*a*b^2*c+3*B*a^2*b*x+3/d*B*a^2*b*c-3/d*C*a^2*b*ln(cos(d*x+c))+1
/d*B*a^3*ln(sin(d*x+c))+C*x*a^3+1/d*C*a^3*c

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Maxima [A]  time = 1.79214, size = 167, normalized size = 1.43 \begin{align*} \frac{C b^{3} \tan \left (d x + c\right )^{2} + 2 \, B a^{3} \log \left (\tan \left (d x + c\right )\right ) + 2 \,{\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )}{\left (d x + c\right )} -{\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \,{\left (3 \, C a b^{2} + B b^{3}\right )} \tan \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(a+b*tan(d*x+c))^3*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="maxima")

[Out]

1/2*(C*b^3*tan(d*x + c)^2 + 2*B*a^3*log(tan(d*x + c)) + 2*(C*a^3 + 3*B*a^2*b - 3*C*a*b^2 - B*b^3)*(d*x + c) -
(B*a^3 - 3*C*a^2*b - 3*B*a*b^2 + C*b^3)*log(tan(d*x + c)^2 + 1) + 2*(3*C*a*b^2 + B*b^3)*tan(d*x + c))/d

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Fricas [A]  time = 1.83794, size = 305, normalized size = 2.61 \begin{align*} \frac{C b^{3} \tan \left (d x + c\right )^{2} + B a^{3} \log \left (\frac{\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \,{\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} d x -{\left (3 \, C a^{2} b + 3 \, B a b^{2} - C b^{3}\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \,{\left (3 \, C a b^{2} + B b^{3}\right )} \tan \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(a+b*tan(d*x+c))^3*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="fricas")

[Out]

1/2*(C*b^3*tan(d*x + c)^2 + B*a^3*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) + 2*(C*a^3 + 3*B*a^2*b - 3*C*a*b^2
- B*b^3)*d*x - (3*C*a^2*b + 3*B*a*b^2 - C*b^3)*log(1/(tan(d*x + c)^2 + 1)) + 2*(3*C*a*b^2 + B*b^3)*tan(d*x + c
))/d

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Sympy [A]  time = 26.6943, size = 211, normalized size = 1.8 \begin{align*} \begin{cases} - \frac{B a^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{B a^{3} \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + 3 B a^{2} b x + \frac{3 B a b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - B b^{3} x + \frac{B b^{3} \tan{\left (c + d x \right )}}{d} + C a^{3} x + \frac{3 C a^{2} b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - 3 C a b^{2} x + \frac{3 C a b^{2} \tan{\left (c + d x \right )}}{d} - \frac{C b^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{C b^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a + b \tan{\left (c \right )}\right )^{3} \left (B \tan{\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**2*(a+b*tan(d*x+c))**3*(B*tan(d*x+c)+C*tan(d*x+c)**2),x)

[Out]

Piecewise((-B*a**3*log(tan(c + d*x)**2 + 1)/(2*d) + B*a**3*log(tan(c + d*x))/d + 3*B*a**2*b*x + 3*B*a*b**2*log
(tan(c + d*x)**2 + 1)/(2*d) - B*b**3*x + B*b**3*tan(c + d*x)/d + C*a**3*x + 3*C*a**2*b*log(tan(c + d*x)**2 + 1
)/(2*d) - 3*C*a*b**2*x + 3*C*a*b**2*tan(c + d*x)/d - C*b**3*log(tan(c + d*x)**2 + 1)/(2*d) + C*b**3*tan(c + d*
x)**2/(2*d), Ne(d, 0)), (x*(a + b*tan(c))**3*(B*tan(c) + C*tan(c)**2)*cot(c)**2, True))

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Giac [A]  time = 2.51779, size = 174, normalized size = 1.49 \begin{align*} \frac{C b^{3} \tan \left (d x + c\right )^{2} + 2 \, B a^{3} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + 6 \, C a b^{2} \tan \left (d x + c\right ) + 2 \, B b^{3} \tan \left (d x + c\right ) + 2 \,{\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )}{\left (d x + c\right )} -{\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(a+b*tan(d*x+c))^3*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="giac")

[Out]

1/2*(C*b^3*tan(d*x + c)^2 + 2*B*a^3*log(abs(tan(d*x + c))) + 6*C*a*b^2*tan(d*x + c) + 2*B*b^3*tan(d*x + c) + 2
*(C*a^3 + 3*B*a^2*b - 3*C*a*b^2 - B*b^3)*(d*x + c) - (B*a^3 - 3*C*a^2*b - 3*B*a*b^2 + C*b^3)*log(tan(d*x + c)^
2 + 1))/d