3.5.44 \(\int \cot ^6(c+d x) (a+b \tan (c+d x))^3 \, dx\) [444]

Optimal. Leaf size=157 \[ -a \left (a^2-3 b^2\right ) x-\frac {a \left (a^2-3 b^2\right ) \cot (c+d x)}{d}+\frac {b \left (3 a^2-b^2\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (a^2-3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {11 a^2 b \cot ^4(c+d x)}{20 d}+\frac {b \left (3 a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d} \]

[Out]

-a*(a^2-3*b^2)*x-a*(a^2-3*b^2)*cot(d*x+c)/d+1/2*b*(3*a^2-b^2)*cot(d*x+c)^2/d+1/3*a*(a^2-3*b^2)*cot(d*x+c)^3/d-
11/20*a^2*b*cot(d*x+c)^4/d+b*(3*a^2-b^2)*ln(sin(d*x+c))/d-1/5*a^2*cot(d*x+c)^5*(a+b*tan(d*x+c))/d

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Rubi [A]
time = 0.20, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 7, 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 ^3(c+d x)}{3 d}+\frac {b \left (3 a^2-b^2\right ) \cot ^2(c+d x)}{2 d}-\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 {11 a^2 b \cot ^4(c+d x)}{20 d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*(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 + (b*(3*a^2 - b^2)*Cot[c + d*x]^2)/(2*d) + (a*(a^2 - 3
*b^2)*Cot[c + d*x]^3)/(3*d) - (11*a^2*b*Cot[c + d*x]^4)/(20*d) + (b*(3*a^2 - b^2)*Log[Sin[c + d*x]])/d - (a^2*
Cot[c + d*x]^5*(a + b*Tan[c + d*x]))/(5*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 ^6(c+d x) (a+b \tan (c+d x))^3 \, dx &=-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d}+\frac {1}{5} \int \cot ^5(c+d x) \left (11 a^2 b-5 a \left (a^2-3 b^2\right ) \tan (c+d x)-b \left (4 a^2-5 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac {11 a^2 b \cot ^4(c+d x)}{20 d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d}+\frac {1}{5} \int \cot ^4(c+d x) \left (-5 a \left (a^2-3 b^2\right )-5 b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac {a \left (a^2-3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {11 a^2 b \cot ^4(c+d x)}{20 d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d}+\frac {1}{5} \int \cot ^3(c+d x) \left (-5 b \left (3 a^2-b^2\right )+5 a \left (a^2-3 b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac {b \left (3 a^2-b^2\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (a^2-3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {11 a^2 b \cot ^4(c+d x)}{20 d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d}+\frac {1}{5} \int \cot ^2(c+d x) \left (5 a \left (a^2-3 b^2\right )+5 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 {b \left (3 a^2-b^2\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (a^2-3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {11 a^2 b \cot ^4(c+d x)}{20 d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d}+\frac {1}{5} \int \cot (c+d x) \left (5 b \left (3 a^2-b^2\right )-5 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 {b \left (3 a^2-b^2\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (a^2-3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {11 a^2 b \cot ^4(c+d x)}{20 d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 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 {b \left (3 a^2-b^2\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (a^2-3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {11 a^2 b \cot ^4(c+d x)}{20 d}+\frac {b \left (3 a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.00, size = 152, normalized size = 0.97 \begin {gather*} -\frac {a \left (a^2-3 b^2\right ) \cot (c+d x)-\frac {1}{2} b \left (3 a^2-b^2\right ) \cot ^2(c+d x)-\frac {1}{3} a \left (a^2-3 b^2\right ) \cot ^3(c+d x)+\frac {3}{4} a^2 b \cot ^4(c+d x)+\frac {1}{5} a^3 \cot ^5(c+d x)-\frac {1}{2} (i a+b)^3 \log (i-\cot (c+d x))+\frac {1}{2} (i a-b)^3 \log (i+\cot (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*(a + b*Tan[c + d*x])^3,x]

[Out]

-((a*(a^2 - 3*b^2)*Cot[c + d*x] - (b*(3*a^2 - b^2)*Cot[c + d*x]^2)/2 - (a*(a^2 - 3*b^2)*Cot[c + d*x]^3)/3 + (3
*a^2*b*Cot[c + d*x]^4)/4 + (a^3*Cot[c + d*x]^5)/5 - ((I*a + b)^3*Log[I - Cot[c + d*x]])/2 + ((I*a - b)^3*Log[I
+ Cot[c + d*x]])/2)/d)

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Maple [A]
time = 0.20, size = 131, normalized size = 0.83

method	result	size
derivativedivides	\(\frac {a^{3} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+3 a^{2} b \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+3 b^{2} a \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+b^{3} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(131\)
default	\(\frac {a^{3} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+3 a^{2} b \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+3 b^{2} a \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+b^{3} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(131\)
norman	\(\frac {-\frac {a^{3}}{5 d}+\frac {a \left (a^{2}-3 b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}-\frac {a \left (a^{2}-3 b^{2}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{d}-a \left (a^{2}-3 b^{2}\right ) x \left (\tan ^{5}\left (d x +c \right )\right )-\frac {3 a^{2} b \tan \left (d x +c \right )}{4 d}+\frac {b \left (3 a^{2}-b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{2 d}}{\tan \left (d x +c \right )^{5}}+\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(178\)
risch	\(-3 i x b \,a^{2}+i b^{3} x -a^{3} x +3 a \,b^{2} x -\frac {6 i b \,a^{2} c}{d}+\frac {2 i b^{3} c}{d}+\frac {44 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-\frac {46 i a^{3}}{15}-12 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+2 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-6 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+\frac {28 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{3}+24 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-6 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+8 i a \,b^{2}-24 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+6 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-36 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+12 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+12 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-28 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-\frac {56 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}}{3}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}-\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(364\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^6*(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^3*(-1/5*cot(d*x+c)^5+1/3*cot(d*x+c)^3-cot(d*x+c)-d*x-c)+3*a^2*b*(-1/4*cot(d*x+c)^4+1/2*cot(d*x+c)^2+ln(
sin(d*x+c)))+3*b^2*a*(-1/3*cot(d*x+c)^3+cot(d*x+c)+d*x+c)+b^3*(-1/2*cot(d*x+c)^2-ln(sin(d*x+c))))

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Maxima [A]
time = 0.52, size = 158, normalized size = 1.01 \begin {gather*} -\frac {60 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )} + 30 \, {\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \, {\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {60 \, {\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{4} + 45 \, a^{2} b \tan \left (d x + c\right ) - 30 \, {\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{3} + 12 \, a^{3} - 20 \, {\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{5}}}{60 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6*(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/60*(60*(a^3 - 3*a*b^2)*(d*x + c) + 30*(3*a^2*b - b^3)*log(tan(d*x + c)^2 + 1) - 60*(3*a^2*b - b^3)*log(tan(
d*x + c)) + (60*(a^3 - 3*a*b^2)*tan(d*x + c)^4 + 45*a^2*b*tan(d*x + c) - 30*(3*a^2*b - b^3)*tan(d*x + c)^3 + 1
2*a^3 - 20*(a^3 - 3*a*b^2)*tan(d*x + c)^2)/tan(d*x + c)^5)/d

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Fricas [A]
time = 1.09, size = 173, normalized size = 1.10 \begin {gather*} \frac {30 \, {\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 )^{5} + 15 \, {\left (9 \, a^{2} b - 2 \, b^{3} - 4 \, {\left (a^{3} - 3 \, a b^{2}\right )} d x\right )} \tan \left (d x + c\right )^{5} - 60 \, {\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{4} - 45 \, a^{2} b \tan \left (d x + c\right ) + 30 \, {\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{3} - 12 \, a^{3} + 20 \, {\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{60 \, d \tan \left (d x + c\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6*(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

1/60*(30*(3*a^2*b - b^3)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^5 + 15*(9*a^2*b - 2*b^3 - 4*(a^
3 - 3*a*b^2)*d*x)*tan(d*x + c)^5 - 60*(a^3 - 3*a*b^2)*tan(d*x + c)^4 - 45*a^2*b*tan(d*x + c) + 30*(3*a^2*b - b
^3)*tan(d*x + c)^3 - 12*a^3 + 20*(a^3 - 3*a*b^2)*tan(d*x + c)^2)/(d*tan(d*x + c)^5)

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Sympy [A]
time = 3.05, size = 241, normalized size = 1.54 \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 ^{6}{\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 {a^{3}}{5 d \tan ^{5}{\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 )}} - \frac {3 a^{2} b}{4 d \tan ^{4}{\left (c + d x \right )}} + 3 a b^{2} x + \frac {3 a b^{2}}{d \tan {\left (c + d x \right )}} - \frac {a b^{2}}{d \tan ^{3}{\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} - \frac {b^{3}}{2 d \tan ^{2}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**6*(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)**6, E
q(d, 0)), (-a**3*x - a**3/(d*tan(c + d*x)) + a**3/(3*d*tan(c + d*x)**3) - a**3/(5*d*tan(c + d*x)**5) - 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**2*b/(4
*d*tan(c + d*x)**4) + 3*a*b**2*x + 3*a*b**2/(d*tan(c + d*x)) - a*b**2/(d*tan(c + d*x)**3) + b**3*log(tan(c + d
*x)**2 + 1)/(2*d) - b**3*log(tan(c + d*x))/d - b**3/(2*d*tan(c + d*x)**2), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (149) = 298\).
time = 2.81, size = 370, normalized size = 2.36 \begin {gather*} \frac {6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 540 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 660 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1800 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 960 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )} - 960 \, {\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 ) + 960 \, {\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 {6576 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2192 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 660 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1800 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 540 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 70 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 45 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6*(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

1/960*(6*a^3*tan(1/2*d*x + 1/2*c)^5 - 45*a^2*b*tan(1/2*d*x + 1/2*c)^4 - 70*a^3*tan(1/2*d*x + 1/2*c)^3 + 120*a*
b^2*tan(1/2*d*x + 1/2*c)^3 + 540*a^2*b*tan(1/2*d*x + 1/2*c)^2 - 120*b^3*tan(1/2*d*x + 1/2*c)^2 + 660*a^3*tan(1
/2*d*x + 1/2*c) - 1800*a*b^2*tan(1/2*d*x + 1/2*c) - 960*(a^3 - 3*a*b^2)*(d*x + c) - 960*(3*a^2*b - b^3)*log(ta
n(1/2*d*x + 1/2*c)^2 + 1) + 960*(3*a^2*b - b^3)*log(abs(tan(1/2*d*x + 1/2*c))) - (6576*a^2*b*tan(1/2*d*x + 1/2
*c)^5 - 2192*b^3*tan(1/2*d*x + 1/2*c)^5 + 660*a^3*tan(1/2*d*x + 1/2*c)^4 - 1800*a*b^2*tan(1/2*d*x + 1/2*c)^4 -
540*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 120*b^3*tan(1/2*d*x + 1/2*c)^3 - 70*a^3*tan(1/2*d*x + 1/2*c)^2 + 120*a*b^2
*tan(1/2*d*x + 1/2*c)^2 + 45*a^2*b*tan(1/2*d*x + 1/2*c) + 6*a^3)/tan(1/2*d*x + 1/2*c)^5)/d

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Mupad [B]
time = 4.10, size = 166, normalized size = 1.06 \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 )}^5\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a\,b^2-\frac {a^3}{3}\right )-{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (3\,a\,b^2-a^3\right )-{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {3\,a^2\,b}{2}-\frac {b^3}{2}\right )+\frac {a^3}{5}+\frac {3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )}{4}\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)^6*(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) + 1
i)*(a*1i + b)^3)/(2*d) - (cot(c + d*x)^5*(tan(c + d*x)^2*(a*b^2 - a^3/3) - tan(c + d*x)^4*(3*a*b^2 - a^3) - ta
n(c + d*x)^3*((3*a^2*b)/2 - b^3/2) + a^3/5 + (3*a^2*b*tan(c + d*x))/4))/d

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