3.5.41 \(\int \cot ^3(c+d x) (a+b \tan (c+d x))^3 \, dx\) [441]

Optimal. Leaf size=83 \[ -b \left (3 a^2-b^2\right ) x-\frac {5 a^2 b \cot (c+d x)}{2 d}-\frac {a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^2(c+d x) (a+b \tan (c+d x))}{2 d} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3646, 3709, 3612, 3556} \begin {gather*} -\frac {a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}-b x \left (3 a^2-b^2\right )-\frac {5 a^2 b \cot (c+d x)}{2 d}-\frac {a^2 \cot ^2(c+d x) (a+b \tan (c+d x))}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(3*a^2 - b^2)*x) - (5*a^2*b*Cot[c + d*x])/(2*d) - (a*(a^2 - 3*b^2)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x
]^2*(a + b*Tan[c + d*x]))/(2*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 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 ^3(c+d x) (a+b \tan (c+d x))^3 \, dx &=-\frac {a^2 \cot ^2(c+d x) (a+b \tan (c+d x))}{2 d}+\frac {1}{2} \int \cot ^2(c+d x) \left (5 a^2 b-2 a \left (a^2-3 b^2\right ) \tan (c+d x)-b \left (a^2-2 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac {5 a^2 b \cot (c+d x)}{2 d}-\frac {a^2 \cot ^2(c+d x) (a+b \tan (c+d x))}{2 d}+\frac {1}{2} \int \cot (c+d x) \left (-2 a \left (a^2-3 b^2\right )-2 b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-b \left (3 a^2-b^2\right ) x-\frac {5 a^2 b \cot (c+d x)}{2 d}-\frac {a^2 \cot ^2(c+d x) (a+b \tan (c+d x))}{2 d}-\left (a \left (a^2-3 b^2\right )\right ) \int \cot (c+d x) \, dx\\ &=-b \left (3 a^2-b^2\right ) x-\frac {5 a^2 b \cot (c+d x)}{2 d}-\frac {a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^2(c+d x) (a+b \tan (c+d x))}{2 d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.33, size = 96, normalized size = 1.16 \begin {gather*} \frac {-6 a^2 b \cot (c+d x)-a^3 \cot ^2(c+d x)+(a+i b)^3 \log (i-\tan (c+d x))-2 a \left (a^2-3 b^2\right ) \log (\tan (c+d x))+(a-i b)^3 \log (i+\tan (c+d x))}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.21, size = 74, normalized size = 0.89

method	result	size
derivativedivides	\(\frac {a^{3} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+3 a^{2} b \left (-\cot \left (d x +c \right )-d x -c \right )+3 b^{2} a \ln \left (\sin \left (d x +c \right )\right )+b^{3} \left (d x +c \right )}{d}\)	\(74\)
default	\(\frac {a^{3} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+3 a^{2} b \left (-\cot \left (d x +c \right )-d x -c \right )+3 b^{2} a \ln \left (\sin \left (d x +c \right )\right )+b^{3} \left (d x +c \right )}{d}\)	\(74\)
norman	\(\frac {\left (-3 a^{2} b +b^{3}\right ) x \left (\tan ^{2}\left (d x +c \right )\right )-\frac {a^{3}}{2 d}-\frac {3 a^{2} b \tan \left (d x +c \right )}{d}}{\tan \left (d x +c \right )^{2}}-\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(103\)
risch	\(-3 a^{2} b x +b^{3} x +i a^{3} x -3 i a \,b^{2} x +\frac {2 i a^{3} c}{d}-\frac {6 i a \,b^{2} c}{d}+\frac {2 a^{2} \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}-3 i b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 i b \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{d}\)	\(141\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.53, size = 92, normalized size = 1.11 \begin {gather*} -\frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )} - {\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {6 \, a^{2} b \tan \left (d x + c\right ) + a^{3}}{\tan \left (d x + c\right )^{2}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.35, size = 99, normalized size = 1.19 \begin {gather*} -\frac {6 \, a^{2} b \tan \left (d x + c\right ) + {\left (a^{3} - 3 \, a b^{2}\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 )^{2} + a^{3} + {\left (a^{3} + 2 \, {\left (3 \, a^{2} b - b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{2}}{2 \, d \tan \left (d x + c\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.95, size = 146, normalized size = 1.76 \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 ^{3}{\left (c \right )} & \text {for}\: d = 0 \\\frac {a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {a^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {a^{3}}{2 d \tan ^{2}{\left (c + d x \right )}} - 3 a^{2} b x - \frac {3 a^{2} b}{d \tan {\left (c + d x \right )}} - \frac {3 a b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {3 a b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + b^{3} x & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**3*(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)**3, E
q(d, 0)), (a**3*log(tan(c + d*x)**2 + 1)/(2*d) - a**3*log(tan(c + d*x))/d - a**3/(2*d*tan(c + d*x)**2) - 3*a**
2*b*x - 3*a**2*b/(d*tan(c + d*x)) - 3*a*b**2*log(tan(c + d*x)**2 + 1)/(2*d) + 3*a*b**2*log(tan(c + d*x))/d + b
**3*x, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (79) = 158\).
time = 1.25, size = 171, normalized size = 2.06 \begin {gather*} -\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, {\left (3 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )} - 8 \, {\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 8 \, {\left (a^{3} - 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {12 \, 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} - 12 \, 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 )^{2}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(a^3*tan(1/2*d*x + 1/2*c)^2 - 12*a^2*b*tan(1/2*d*x + 1/2*c) + 8*(3*a^2*b - b^3)*(d*x + c) - 8*(a^3 - 3*a*
b^2)*log(tan(1/2*d*x + 1/2*c)^2 + 1) + 8*(a^3 - 3*a*b^2)*log(abs(tan(1/2*d*x + 1/2*c))) - (12*a^3*tan(1/2*d*x
+ 1/2*c)^2 - 36*a*b^2*tan(1/2*d*x + 1/2*c)^2 - 12*a^2*b*tan(1/2*d*x + 1/2*c) - a^3)/tan(1/2*d*x + 1/2*c)^2)/d

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Mupad [B]
time = 4.07, size = 102, normalized size = 1.23 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,a\,b^2-a^3\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^3}{2\,d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^2\,\left (\frac {a^3}{2}+3\,b\,\mathrm {tan}\left (c+d\,x\right )\,a^2\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (b+a\,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)^3*(a + b*tan(c + d*x))^3,x)

[Out]

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

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