3.1.22 \(\int \cot ^5(c+d x) (a+b \tan (c+d x))^3 (B \tan (c+d x)+C \tan ^2(c+d x)) \, dx\) [22]

Optimal. Leaf size=154 \[ \left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) x+\frac {a \left (3 a^2 B-8 b^2 B-9 a b C\right ) \cot (c+d x)}{3 d}-\frac {a^2 (5 b B+3 a C) \cot ^2(c+d x)}{6 d}-\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \log (\sin (c+d x))}{d}-\frac {a B \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d} \]

[Out]

(B*a^3-3*B*a*b^2-3*C*a^2*b+C*b^3)*x+1/3*a*(3*B*a^2-8*B*b^2-9*C*a*b)*cot(d*x+c)/d-1/6*a^2*(5*B*b+3*C*a)*cot(d*x
+c)^2/d-(3*B*a^2*b-B*b^3+C*a^3-3*C*a*b^2)*ln(sin(d*x+c))/d-1/3*a*B*cot(d*x+c)^3*(a+b*tan(d*x+c))^2/d

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Rubi [A]
time = 0.30, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3713, 3686, 3716, 3709, 3612, 3556} \begin {gather*} \frac {a \left (3 a^2 B-9 a b C-8 b^2 B\right ) \cot (c+d x)}{3 d}-\frac {a^2 (3 a C+5 b B) \cot ^2(c+d x)}{6 d}-\frac {\left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right ) \log (\sin (c+d x))}{d}+x \left (a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right )-\frac {a B \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

(a^3*B - 3*a*b^2*B - 3*a^2*b*C + b^3*C)*x + (a*(3*a^2*B - 8*b^2*B - 9*a*b*C)*Cot[c + d*x])/(3*d) - (a^2*(5*b*B
+ 3*a*C)*Cot[c + d*x]^2)/(6*d) - ((3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*Log[Sin[c + d*x]])/d - (a*B*Cot[c +
d*x]^3*(a + b*Tan[c + d*x])^2)/(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 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 3686

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*c - a*d)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e
+ f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m -
2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a*A*d*(b*d*(m - 1) - a*c*(n + 1)) + (b*B*c - (A*b + a*B)*d)*(b*c*(m - 1)
+ a*d*(n + 1)) - d*((a*A - b*B)*(b*c - a*d) + (A*b + a*B)*(a*c + b*d))*(n + 1)*Tan[e + f*x] - b*(d*(A*b*c + a
*B*c - a*A*d)*(m + n) - b*B*(c^2*(m - 1) - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f
, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (Inte
gerQ[m] || IntegersQ[2*m, 2*n])

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]

Rule 3713

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 3716

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 - a*d))*(c^2*C - B*c*d + A*d^2)
*((c + d*Tan[e + f*x])^(n + 1)/(d^2*f*(n + 1)*(c^2 + d^2))), x] + Dist[1/(d*(c^2 + d^2)), Int[(c + d*Tan[e + f
*x])^(n + 1)*Simp[a*d*(A*c - c*C + B*d) + b*(c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d
+ a*C*d)*Tan[e + f*x] + b*C*(c^2 + d^2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] &
& NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]

Rubi steps

\begin {align*} \int \cot ^5(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 ^4(c+d x) (a+b \tan (c+d x))^3 (B+C \tan (c+d x)) \, dx\\ &=-\frac {a B \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d}+\frac {1}{3} \int \cot ^3(c+d x) (a+b \tan (c+d x)) \left (a (5 b B+3 a C)-3 \left (a^2 B-b^2 B-2 a b C\right ) \tan (c+d x)-b (a B-3 b C) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac {a^2 (5 b B+3 a C) \cot ^2(c+d x)}{6 d}-\frac {a B \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d}+\frac {1}{3} \int \cot ^2(c+d x) \left (-a \left (3 a^2 B-8 b^2 B-9 a b C\right )-3 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \tan (c+d x)-b^2 (a B-3 b C) \tan ^2(c+d x)\right ) \, dx\\ &=\frac {a \left (3 a^2 B-8 b^2 B-9 a b C\right ) \cot (c+d x)}{3 d}-\frac {a^2 (5 b B+3 a C) \cot ^2(c+d x)}{6 d}-\frac {a B \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d}+\frac {1}{3} \int \cot (c+d x) \left (-3 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right )+3 \left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \tan (c+d x)\right ) \, dx\\ &=\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) x+\frac {a \left (3 a^2 B-8 b^2 B-9 a b C\right ) \cot (c+d x)}{3 d}-\frac {a^2 (5 b B+3 a C) \cot ^2(c+d x)}{6 d}-\frac {a B \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d}+\left (-3 a^2 b B+b^3 B-a^3 C+3 a b^2 C\right ) \int \cot (c+d x) \, dx\\ &=\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) x+\frac {a \left (3 a^2 B-8 b^2 B-9 a b C\right ) \cot (c+d x)}{3 d}-\frac {a^2 (5 b B+3 a C) \cot ^2(c+d x)}{6 d}-\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \log (\sin (c+d x))}{d}-\frac {a B \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.84, size = 164, normalized size = 1.06 \begin {gather*} \frac {6 a \left (a^2 B-3 b^2 B-3 a b C\right ) \cot (c+d x)-3 a^2 (3 b B+a C) \cot ^2(c+d x)-2 a^3 B \cot ^3(c+d x)+3 (a+i b)^3 (-i B+C) \log (i-\tan (c+d x))-6 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \log (\tan (c+d x))+3 (a-i b)^3 (i B+C) \log (i+\tan (c+d x))}{6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

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

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Maple [A]
time = 0.32, size = 166, normalized size = 1.08

method	result	size
derivativedivides	\(\frac {B \,b^{3} \ln \left (\sin \left (d x +c \right )\right )+C \,b^{3} \left (d x +c \right )+3 B a \,b^{2} \left (-\cot \left (d x +c \right )-d x -c \right )+3 C a \,b^{2} \ln \left (\sin \left (d x +c \right )\right )+3 B \,a^{2} b \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+3 C \,a^{2} b \left (-\cot \left (d x +c \right )-d x -c \right )+B \,a^{3} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+C \,a^{3} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(166\)
default	\(\frac {B \,b^{3} \ln \left (\sin \left (d x +c \right )\right )+C \,b^{3} \left (d x +c \right )+3 B a \,b^{2} \left (-\cot \left (d x +c \right )-d x -c \right )+3 C a \,b^{2} \ln \left (\sin \left (d x +c \right )\right )+3 B \,a^{2} b \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+3 C \,a^{2} b \left (-\cot \left (d x +c \right )-d x -c \right )+B \,a^{3} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+C \,a^{3} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(166\)
norman	\(\frac {\left (B \,a^{3}-3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) x \left (\tan ^{4}\left (d x +c \right )\right )+\frac {a \left (a^{2} B -3 b^{2} B -3 C a b \right ) \left (\tan ^{3}\left (d x +c \right )\right )}{d}-\frac {B \,a^{3} \tan \left (d x +c \right )}{3 d}-\frac {a^{2} \left (3 B b +C a \right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}}{\tan \left (d x +c \right )^{4}}-\frac {\left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {\left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(196\)
risch	\(B \,a^{3} x -3 B a \,b^{2} x -3 C \,a^{2} b x +C \,b^{3} x -3 i C a \,b^{2} x +\frac {6 i B \,a^{2} b c}{d}-\frac {2 i a \left (9 i B a b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 i C \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 a^{2} B \,{\mathrm e}^{4 i \left (d x +c \right )}+9 b^{2} B \,{\mathrm e}^{4 i \left (d x +c \right )}+9 C a b \,{\mathrm e}^{4 i \left (d x +c \right )}-9 i B a b \,{\mathrm e}^{2 i \left (d x +c \right )}-3 i C \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+6 a^{2} B \,{\mathrm e}^{2 i \left (d x +c \right )}-18 B \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-18 C a b \,{\mathrm e}^{2 i \left (d x +c \right )}-4 a^{2} B +9 b^{2} B +9 C a b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+i C \,a^{3} x -\frac {6 i C a \,b^{2} c}{d}+\frac {2 i C \,a^{3} c}{d}-\frac {2 i B \,b^{3} c}{d}+3 i B \,a^{2} b x -i B \,b^{3} x -\frac {3 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B b}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B \,b^{3}}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C}{d}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C a \,b^{2}}{d}\)	\(383\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^5*(a+b*tan(d*x+c))^3*(B*tan(d*x+c)+C*tan(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 180, normalized size = 1.17 \begin {gather*} \frac {6 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} {\left (d x + c\right )} + 3 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac {2 \, B a^{3} - 6 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \tan \left (d x + c\right )}{\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)^5*(a+b*tan(d*x+c))^3*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 2.89, size = 181, normalized size = 1.18 \begin {gather*} -\frac {3 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{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} + 2 \, B a^{3} + 3 \, {\left (C a^{3} + 3 \, B a^{2} b - 2 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{3} - 6 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \tan \left (d x + c\right )}{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)^5*(a+b*tan(d*x+c))^3*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (150) = 300\).
time = 4.91, size = 330, normalized size = 2.14 \begin {gather*} \begin {cases} \text {NaN} & \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} \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot ^{5}{\left (c \right )} & \text {for}\: d = 0 \\B a^{3} x + \frac {B a^{3}}{d \tan {\left (c + d x \right )}} - \frac {B a^{3}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {3 B a^{2} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {3 B a^{2} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {3 B a^{2} b}{2 d \tan ^{2}{\left (c + d x \right )}} - 3 B a b^{2} x - \frac {3 B a b^{2}}{d \tan {\left (c + d x \right )}} - \frac {B b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B b^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {C a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {C a^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {C a^{3}}{2 d \tan ^{2}{\left (c + d x \right )}} - 3 C a^{2} b x - \frac {3 C a^{2} b}{d \tan {\left (c + d x \right )}} - \frac {3 C a b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {3 C a b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + C 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)**5*(a+b*tan(d*x+c))**3*(B*tan(d*x+c)+C*tan(d*x+c)**2),x)

[Out]

Piecewise((nan, (Eq(c, 0) | Eq(c, -d*x)) & (Eq(d, 0) | Eq(c, -d*x))), (x*(a + b*tan(c))**3*(B*tan(c) + C*tan(c
)**2)*cot(c)**5, Eq(d, 0)), (B*a**3*x + B*a**3/(d*tan(c + d*x)) - B*a**3/(3*d*tan(c + d*x)**3) + 3*B*a**2*b*lo
g(tan(c + d*x)**2 + 1)/(2*d) - 3*B*a**2*b*log(tan(c + d*x))/d - 3*B*a**2*b/(2*d*tan(c + d*x)**2) - 3*B*a*b**2*
x - 3*B*a*b**2/(d*tan(c + d*x)) - B*b**3*log(tan(c + d*x)**2 + 1)/(2*d) + B*b**3*log(tan(c + d*x))/d + C*a**3*
log(tan(c + d*x)**2 + 1)/(2*d) - C*a**3*log(tan(c + d*x))/d - C*a**3/(2*d*tan(c + d*x)**2) - 3*C*a**2*b*x - 3*
C*a**2*b/(d*tan(c + d*x)) - 3*C*a*b**2*log(tan(c + d*x)**2 + 1)/(2*d) + 3*C*a*b**2*log(tan(c + d*x))/d + C*b**
3*x, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 390 vs. \(2 (148) = 296\).
time = 1.54, size = 390, normalized size = 2.53 \begin {gather*} \frac {B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} {\left (d x + c\right )} + 24 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 24 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {44 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 132 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 132 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 44 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B 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)^5*(a+b*tan(d*x+c))^3*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 9.00, size = 169, normalized size = 1.10 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-C\,a^3-3\,B\,a^2\,b+3\,C\,a\,b^2+B\,b^3\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {C\,a^3}{2}+\frac {3\,B\,b\,a^2}{2}\right )+\frac {B\,a^3}{3}+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-B\,a^3+3\,C\,a^2\,b+3\,B\,a\,b^2\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\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)^5*(B*tan(c + d*x) + C*tan(c + d*x)^2)*(a + b*tan(c + d*x))^3,x)

[Out]

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

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