3.449 \(\int \cot (c+d x) (a+b \tan (c+d x))^4 \, dx\)

Optimal. Leaf size=92 \[ -\frac{b^2 \left (6 a^2-b^2\right ) \log (\cos (c+d x))}{d}+4 a b x \left (a^2-b^2\right )+\frac{a^4 \log (\sin (c+d x))}{d}+\frac{3 a b^3 \tan (c+d x)}{d}+\frac{b^2 (a+b \tan (c+d x))^2}{2 d} \]

[Out]

4*a*b*(a^2 - b^2)*x - (b^2*(6*a^2 - b^2)*Log[Cos[c + d*x]])/d + (a^4*Log[Sin[c + d*x]])/d + (3*a*b^3*Tan[c + d
*x])/d + (b^2*(a + b*Tan[c + d*x])^2)/(2*d)

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Rubi [A]  time = 0.177631, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3566, 3637, 3624, 3475} \[ -\frac{b^2 \left (6 a^2-b^2\right ) \log (\cos (c+d x))}{d}+4 a b x \left (a^2-b^2\right )+\frac{a^4 \log (\sin (c+d x))}{d}+\frac{3 a b^3 \tan (c+d x)}{d}+\frac{b^2 (a+b \tan (c+d x))^2}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4*a*b*(a^2 - b^2)*x - (b^2*(6*a^2 - b^2)*Log[Cos[c + d*x]])/d + (a^4*Log[Sin[c + d*x]])/d + (3*a*b^3*Tan[c + d
*x])/d + (b^2*(a + b*Tan[c + d*x])^2)/(2*d)

Rule 3566

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(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[1/(d*(m + n -
1)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !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 (c+d x) (a+b \tan (c+d x))^4 \, dx &=\frac{b^2 (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^3+2 b \left (3 a^2-b^2\right ) \tan (c+d x)+6 a b^2 \tan ^2(c+d x)\right ) \, dx\\ &=\frac{3 a b^3 \tan (c+d x)}{d}+\frac{b^2 (a+b \tan (c+d x))^2}{2 d}-\frac{1}{2} \int \cot (c+d x) \left (-2 a^4-8 a b \left (a^2-b^2\right ) \tan (c+d x)-2 b^2 \left (6 a^2-b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=4 a b \left (a^2-b^2\right ) x+\frac{3 a b^3 \tan (c+d x)}{d}+\frac{b^2 (a+b \tan (c+d x))^2}{2 d}+a^4 \int \cot (c+d x) \, dx+\left (b^2 \left (6 a^2-b^2\right )\right ) \int \tan (c+d x) \, dx\\ &=4 a b \left (a^2-b^2\right ) x-\frac{b^2 \left (6 a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac{a^4 \log (\sin (c+d x))}{d}+\frac{3 a b^3 \tan (c+d x)}{d}+\frac{b^2 (a+b \tan (c+d x))^2}{2 d}\\ \end{align*}

Mathematica [C]  time = 0.380198, size = 94, normalized size = 1.02 \[ \frac{2 a^4 \log (\tan (c+d x))+6 a b^3 \tan (c+d x)+b^2 (a+b \tan (c+d x))^2-(a+i b)^4 \log (-\tan (c+d x)+i)-(a-i b)^4 \log (\tan (c+d x)+i)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((a + I*b)^4*Log[I - Tan[c + d*x]]) + 2*a^4*Log[Tan[c + d*x]] - (a - I*b)^4*Log[I + Tan[c + d*x]] + 6*a*b^3*
Tan[c + d*x] + b^2*(a + b*Tan[c + d*x])^2)/(2*d)

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Maple [A]  time = 0.058, size = 113, normalized size = 1.2 \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{2}{b}^{4}}{2\,d}}+{\frac{{b}^{4}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-4\,{b}^{3}ax+4\,{\frac{{b}^{3}a\tan \left ( dx+c \right ) }{d}}-4\,{\frac{a{b}^{3}c}{d}}-6\,{\frac{{a}^{2}{b}^{2}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+4\,x{a}^{3}b+4\,{\frac{{a}^{3}bc}{d}}+{\frac{{a}^{4}\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)*(a+b*tan(d*x+c))^4,x)

[Out]

1/2/d*tan(d*x+c)^2*b^4+b^4*ln(cos(d*x+c))/d-4*b^3*a*x+4*a*b^3*tan(d*x+c)/d-4/d*a*b^3*c-6/d*a^2*b^2*ln(cos(d*x+
c))+4*x*a^3*b+4/d*a^3*b*c+a^4*ln(sin(d*x+c))/d

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Maxima [A]  time = 1.63779, size = 120, normalized size = 1.3 \begin{align*} \frac{b^{4} \tan \left (d x + c\right )^{2} + 2 \, a^{4} \log \left (\tan \left (d x + c\right )\right ) + 8 \, a b^{3} \tan \left (d x + c\right ) + 8 \,{\left (a^{3} b - a b^{3}\right )}{\left (d x + c\right )} -{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\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)*(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

1/2*(b^4*tan(d*x + c)^2 + 2*a^4*log(tan(d*x + c)) + 8*a*b^3*tan(d*x + c) + 8*(a^3*b - a*b^3)*(d*x + c) - (a^4
- 6*a^2*b^2 + b^4)*log(tan(d*x + c)^2 + 1))/d

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Fricas [A]  time = 2.22296, size = 230, normalized size = 2.5 \begin{align*} \frac{b^{4} \tan \left (d x + c\right )^{2} + a^{4} \log \left (\frac{\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 8 \, a b^{3} \tan \left (d x + c\right ) + 8 \,{\left (a^{3} b - a b^{3}\right )} d x -{\left (6 \, a^{2} b^{2} - b^{4}\right )} \log \left (\frac{1}{\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)*(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/2*(b^4*tan(d*x + c)^2 + a^4*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) + 8*a*b^3*tan(d*x + c) + 8*(a^3*b - a*b
^3)*d*x - (6*a^2*b^2 - b^4)*log(1/(tan(d*x + c)^2 + 1)))/d

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Sympy [A]  time = 2.68075, size = 133, normalized size = 1.45 \begin{align*} \begin{cases} - \frac{a^{4} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a^{4} \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + 4 a^{3} b x + \frac{3 a^{2} b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - 4 a b^{3} x + \frac{4 a b^{3} \tan{\left (c + d x \right )}}{d} - \frac{b^{4} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{b^{4} \tan ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a + b \tan{\left (c \right )}\right )^{4} \cot{\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)*(a+b*tan(d*x+c))**4,x)

[Out]

Piecewise((-a**4*log(tan(c + d*x)**2 + 1)/(2*d) + a**4*log(tan(c + d*x))/d + 4*a**3*b*x + 3*a**2*b**2*log(tan(
c + d*x)**2 + 1)/d - 4*a*b**3*x + 4*a*b**3*tan(c + d*x)/d - b**4*log(tan(c + d*x)**2 + 1)/(2*d) + b**4*tan(c +
d*x)**2/(2*d), Ne(d, 0)), (x*(a + b*tan(c))**4*cot(c), True))

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Giac [A]  time = 2.68803, size = 122, normalized size = 1.33 \begin{align*} \frac{b^{4} \tan \left (d x + c\right )^{2} + 2 \, a^{4} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + 8 \, a b^{3} \tan \left (d x + c\right ) + 8 \,{\left (a^{3} b - a b^{3}\right )}{\left (d x + c\right )} -{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\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)*(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

1/2*(b^4*tan(d*x + c)^2 + 2*a^4*log(abs(tan(d*x + c))) + 8*a*b^3*tan(d*x + c) + 8*(a^3*b - a*b^3)*(d*x + c) -
(a^4 - 6*a^2*b^2 + b^4)*log(tan(d*x + c)^2 + 1))/d