3.5.42 \(\int \cot ^4(c+d x) (a+b \tan (c+d x))^3 \, dx\) [442]

Optimal. Leaf size=104 \[ a \left (a^2-3 b^2\right ) x+\frac {a \left (a^2-3 b^2\right ) \cot (c+d x)}{d}-\frac {7 a^2 b \cot ^2(c+d x)}{6 d}-\frac {b \left (3 a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^3(c+d x) (a+b \tan (c+d x))}{3 d} \]

[Out]

a*(a^2-3*b^2)*x+a*(a^2-3*b^2)*cot(d*x+c)/d-7/6*a^2*b*cot(d*x+c)^2/d-b*(3*a^2-b^2)*ln(sin(d*x+c))/d-1/3*a^2*cot
(d*x+c)^3*(a+b*tan(d*x+c))/d

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3646, 3709, 3610, 3612, 3556} \begin {gather*} \frac {a \left (a^2-3 b^2\right ) \cot (c+d x)}{d}-\frac {b \left (3 a^2-b^2\right ) \log (\sin (c+d x))}{d}+a x \left (a^2-3 b^2\right )-\frac {7 a^2 b \cot ^2(c+d x)}{6 d}-\frac {a^2 \cot ^3(c+d x) (a+b \tan (c+d x))}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^3,x]

[Out]

a*(a^2 - 3*b^2)*x + (a*(a^2 - 3*b^2)*Cot[c + d*x])/d - (7*a^2*b*Cot[c + d*x]^2)/(6*d) - (b*(3*a^2 - b^2)*Log[S
in[c + d*x]])/d - (a^2*Cot[c + d*x]^3*(a + b*Tan[c + d*x]))/(3*d)

Rule 3556

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

Rule 3610

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 3612

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 3646

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3709

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

Rubi steps

\begin {align*} \int \cot ^4(c+d x) (a+b \tan (c+d x))^3 \, dx &=-\frac {a^2 \cot ^3(c+d x) (a+b \tan (c+d x))}{3 d}+\frac {1}{3} \int \cot ^3(c+d x) \left (7 a^2 b-3 a \left (a^2-3 b^2\right ) \tan (c+d x)-b \left (2 a^2-3 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac {7 a^2 b \cot ^2(c+d x)}{6 d}-\frac {a^2 \cot ^3(c+d x) (a+b \tan (c+d x))}{3 d}+\frac {1}{3} \int \cot ^2(c+d x) \left (-3 a \left (a^2-3 b^2\right )-3 b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac {a \left (a^2-3 b^2\right ) \cot (c+d x)}{d}-\frac {7 a^2 b \cot ^2(c+d x)}{6 d}-\frac {a^2 \cot ^3(c+d x) (a+b \tan (c+d x))}{3 d}+\frac {1}{3} \int \cot (c+d x) \left (-3 b \left (3 a^2-b^2\right )+3 a \left (a^2-3 b^2\right ) \tan (c+d x)\right ) \, dx\\ &=a \left (a^2-3 b^2\right ) x+\frac {a \left (a^2-3 b^2\right ) \cot (c+d x)}{d}-\frac {7 a^2 b \cot ^2(c+d x)}{6 d}-\frac {a^2 \cot ^3(c+d x) (a+b \tan (c+d x))}{3 d}-\left (b \left (3 a^2-b^2\right )\right ) \int \cot (c+d x) \, dx\\ &=a \left (a^2-3 b^2\right ) x+\frac {a \left (a^2-3 b^2\right ) \cot (c+d x)}{d}-\frac {7 a^2 b \cot ^2(c+d x)}{6 d}-\frac {b \left (3 a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^3(c+d x) (a+b \tan (c+d x))}{3 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 1.15, size = 120, normalized size = 1.15 \begin {gather*} \frac {6 a \left (a^2-3 b^2\right ) \cot (c+d x)-9 a^2 b \cot ^2(c+d x)-2 a^3 \cot ^3(c+d x)+3 (i a-b)^3 \log (i-\tan (c+d x))+6 b \left (-3 a^2+b^2\right ) \log (\tan (c+d x))-3 (i a+b)^3 \log (i+\tan (c+d x))}{6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^3,x]

[Out]

(6*a*(a^2 - 3*b^2)*Cot[c + d*x] - 9*a^2*b*Cot[c + d*x]^2 - 2*a^3*Cot[c + d*x]^3 + 3*(I*a - b)^3*Log[I - Tan[c
+ d*x]] + 6*b*(-3*a^2 + b^2)*Log[Tan[c + d*x]] - 3*(I*a + b)^3*Log[I + Tan[c + d*x]])/(6*d)

________________________________________________________________________________________

Maple [A]
time = 0.17, size = 90, normalized size = 0.87

method	result	size
derivativedivides	\(\frac {a^{3} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+3 a^{2} b \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+3 b^{2} a \left (-\cot \left (d x +c \right )-d x -c \right )+b^{3} \ln \left (\sin \left (d x +c \right )\right )}{d}\)	\(90\)
default	\(\frac {a^{3} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+3 a^{2} b \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+3 b^{2} a \left (-\cot \left (d x +c \right )-d x -c \right )+b^{3} \ln \left (\sin \left (d x +c \right )\right )}{d}\)	\(90\)
norman	\(\frac {\frac {a \left (a^{2}-3 b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d}+a \left (a^{2}-3 b^{2}\right ) x \left (\tan ^{3}\left (d x +c \right )\right )-\frac {a^{3}}{3 d}-\frac {3 a^{2} b \tan \left (d x +c \right )}{2 d}}{\tan \left (d x +c \right )^{3}}-\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(129\)
risch	\(3 i x b \,a^{2}-i b^{3} x +a^{3} x -3 a \,b^{2} x +\frac {6 i b \,a^{2} c}{d}-\frac {2 i b^{3} c}{d}+\frac {2 i a \left (6 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-9 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-9 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}-6 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+18 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+9 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+4 a^{2}-9 b^{2}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {3 a^{2} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(205\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4*(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.52, size = 117, normalized size = 1.12 \begin {gather*} \frac {6 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )} + 3 \, {\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, {\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac {9 \, a^{2} b \tan \left (d x + c\right ) + 2 \, a^{3} - 6 \, {\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{3}}}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

1/6*(6*(a^3 - 3*a*b^2)*(d*x + c) + 3*(3*a^2*b - b^3)*log(tan(d*x + c)^2 + 1) - 6*(3*a^2*b - b^3)*log(tan(d*x +
c)) - (9*a^2*b*tan(d*x + c) + 2*a^3 - 6*(a^3 - 3*a*b^2)*tan(d*x + c)^2)/tan(d*x + c)^3)/d

________________________________________________________________________________________

Fricas [A]
time = 1.06, size = 126, normalized size = 1.21 \begin {gather*} -\frac {3 \, {\left (3 \, a^{2} b - b^{3}\right )} \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 )^{3} + 9 \, a^{2} b \tan \left (d x + c\right ) + 3 \, {\left (3 \, a^{2} b - 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 2 \, a^{3} - 6 \, {\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{6 \, d \tan \left (d x + c\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/6*(3*(3*a^2*b - b^3)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^3 + 9*a^2*b*tan(d*x + c) + 3*(3*
a^2*b - 2*(a^3 - 3*a*b^2)*d*x)*tan(d*x + c)^3 + 2*a^3 - 6*(a^3 - 3*a*b^2)*tan(d*x + c)^2)/(d*tan(d*x + c)^3)

________________________________________________________________________________________

Sympy [A]
time = 1.38, size = 177, normalized size = 1.70 \begin {gather*} \begin {cases} \tilde {\infty } a^{3} 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 )^{3} \cot ^{4}{\left (c \right )} & \text {for}\: d = 0 \\a^{3} x + \frac {a^{3}}{d \tan {\left (c + d x \right )}} - \frac {a^{3}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {3 a^{2} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {3 a^{2} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {3 a^{2} b}{2 d \tan ^{2}{\left (c + d x \right )}} - 3 a b^{2} x - \frac {3 a b^{2}}{d \tan {\left (c + d x \right )}} - \frac {b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {b^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**4*(a+b*tan(d*x+c))**3,x)

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (100) = 200\).
time = 1.50, size = 236, normalized size = 2.27 \begin {gather*} \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )} + 24 \, {\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 24 \, {\left (3 \, a^{2} b - b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {132 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 44 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

1/24*(a^3*tan(1/2*d*x + 1/2*c)^3 - 9*a^2*b*tan(1/2*d*x + 1/2*c)^2 - 15*a^3*tan(1/2*d*x + 1/2*c) + 36*a*b^2*tan
(1/2*d*x + 1/2*c) + 24*(a^3 - 3*a*b^2)*(d*x + c) + 24*(3*a^2*b - b^3)*log(tan(1/2*d*x + 1/2*c)^2 + 1) - 24*(3*
a^2*b - b^3)*log(abs(tan(1/2*d*x + 1/2*c))) + (132*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 44*b^3*tan(1/2*d*x + 1/2*c)^
3 + 15*a^3*tan(1/2*d*x + 1/2*c)^2 - 36*a*b^2*tan(1/2*d*x + 1/2*c)^2 - 9*a^2*b*tan(1/2*d*x + 1/2*c) - a^3)/tan(
1/2*d*x + 1/2*c)^3)/d

________________________________________________________________________________________

Mupad [B]
time = 3.96, size = 124, normalized size = 1.19 \begin {gather*} -\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,a^2\,b-b^3\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^3}{2\,d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2\,\left (3\,a\,b^2-a^3\right )+\frac {a^3}{3}+\frac {3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )}{2}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^4*(a + b*tan(c + d*x))^3,x)

[Out]

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

________________________________________________________________________________________