3.1.31 \(\int \frac {\cot ^4(c+d x) (B \tan (c+d x)+C \tan ^2(c+d x))}{a+b \tan (c+d x)} \, dx\) [31]
Optimal. Leaf size=137 \[ \frac {(b B-a C) x}{a^2+b^2}+\frac {(b B-a C) \cot (c+d x)}{a^2 d}-\frac {B \cot ^2(c+d x)}{2 a d}-\frac {\left (a^2 B-b^2 B+a b C\right ) \log (\sin (c+d x))}{a^3 d}-\frac {b^3 (b B-a C) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right ) d} \]
[Out]
(B*b-C*a)*x/(a^2+b^2)+(B*b-C*a)*cot(d*x+c)/a^2/d-1/2*B*cot(d*x+c)^2/a/d-(B*a^2-B*b^2+C*a*b)*ln(sin(d*x+c))/a^3
/d-b^3*(B*b-C*a)*ln(a*cos(d*x+c)+b*sin(d*x+c))/a^3/(a^2+b^2)/d
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Rubi [A]
time = 0.43, antiderivative size = 137, 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, 3690,
3730, 3732, 3611, 3556} \begin {gather*} \frac {x (b B-a C)}{a^2+b^2}+\frac {(b B-a C) \cot (c+d x)}{a^2 d}-\frac {\left (a^2 B+a b C-b^2 B\right ) \log (\sin (c+d x))}{a^3 d}-\frac {b^3 (b B-a C) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )}-\frac {B \cot ^2(c+d x)}{2 a d} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(Cot[c + d*x]^4*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x]),x]
[Out]
((b*B - a*C)*x)/(a^2 + b^2) + ((b*B - a*C)*Cot[c + d*x])/(a^2*d) - (B*Cot[c + d*x]^2)/(2*a*d) - ((a^2*B - b^2*
B + a*b*C)*Log[Sin[c + d*x]])/(a^3*d) - (b^3*(b*B - a*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^3*(a^2 + b^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 3611
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[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] && EqQ[a*c + b*d, 0]
Rule 3690
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*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n
+ 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e +
f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(
m + n + 2)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ
[a, 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 3730
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Rule 3732
Int[((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/(((a_) + (b_.)*tan[(e_.) + (f_.)
*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d)
)*(x/((a^2 + b^2)*(c^2 + d^2))), x] + (Dist[(A*b^2 - a*b*B + a^2*C)/((b*c - a*d)*(a^2 + b^2)), Int[(b - a*Tan[
e + f*x])/(a + b*Tan[e + f*x]), x], x] - Dist[(c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)), Int[(d - c*Ta
n[e + f*x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ
[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx &=\int \frac {\cot ^3(c+d x) (B+C \tan (c+d x))}{a+b \tan (c+d x)} \, dx\\ &=-\frac {B \cot ^2(c+d x)}{2 a d}-\frac {\int \frac {\cot ^2(c+d x) \left (2 (b B-a C)+2 a B \tan (c+d x)+2 b B \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a}\\ &=\frac {(b B-a C) \cot (c+d x)}{a^2 d}-\frac {B \cot ^2(c+d x)}{2 a d}+\frac {\int \frac {\cot (c+d x) \left (-2 \left (a^2 B-b^2 B+a b C\right )-2 a^2 C \tan (c+d x)+2 b (b B-a C) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a^2}\\ &=\frac {(b B-a C) x}{a^2+b^2}+\frac {(b B-a C) \cot (c+d x)}{a^2 d}-\frac {B \cot ^2(c+d x)}{2 a d}-\frac {\left (b^3 (b B-a C)\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2 B-b^2 B+a b C\right ) \int \cot (c+d x) \, dx}{a^3}\\ &=\frac {(b B-a C) x}{a^2+b^2}+\frac {(b B-a C) \cot (c+d x)}{a^2 d}-\frac {B \cot ^2(c+d x)}{2 a d}-\frac {\left (a^2 B-b^2 B+a b C\right ) \log (\sin (c+d x))}{a^3 d}-\frac {b^3 (b B-a C) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.95, size = 163, normalized size = 1.19 \begin {gather*} \frac {\frac {2 (b B-a C) \cot (c+d x)}{a^2}-\frac {B \cot ^2(c+d x)}{a}+\frac {(B+i C) \log (i-\tan (c+d x))}{a+i b}-\frac {2 \left (a^2 B-b^2 B+a b C\right ) \log (\tan (c+d x))}{a^3}+\frac {(B-i C) \log (i+\tan (c+d x))}{a-i b}+\frac {2 b^3 (-b B+a C) \log (a+b \tan (c+d x))}{a^3 \left (a^2+b^2\right )}}{2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(Cot[c + d*x]^4*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x]),x]
[Out]
((2*(b*B - a*C)*Cot[c + d*x])/a^2 - (B*Cot[c + d*x]^2)/a + ((B + I*C)*Log[I - Tan[c + d*x]])/(a + I*b) - (2*(a
^2*B - b^2*B + a*b*C)*Log[Tan[c + d*x]])/a^3 + ((B - I*C)*Log[I + Tan[c + d*x]])/(a - I*b) + (2*b^3*(-(b*B) +
a*C)*Log[a + b*Tan[c + d*x]])/(a^3*(a^2 + b^2)))/(2*d)
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Maple [A]
time = 0.48, size = 152, normalized size = 1.11
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\frac {\frac {\frac {\left (a B +C b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (B b -C a \right ) \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {\left (B b -C a \right ) b^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{3} \left (a^{2}+b^{2}\right )}-\frac {B}{2 a \tan \left (d x +c \right )^{2}}-\frac {-B b +C a}{a^{2} \tan \left (d x +c \right )}+\frac {\left (-a^{2} B +b^{2} B -C a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}}{d}\) |
\(152\) |
| default |
\(\frac {\frac {\frac {\left (a B +C b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (B b -C a \right ) \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {\left (B b -C a \right ) b^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{3} \left (a^{2}+b^{2}\right )}-\frac {B}{2 a \tan \left (d x +c \right )^{2}}-\frac {-B b +C a}{a^{2} \tan \left (d x +c \right )}+\frac {\left (-a^{2} B +b^{2} B -C a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}}{d}\) |
\(152\) |
| norman |
\(\frac {\frac {\left (B b -C a \right ) \left (\tan ^{2}\left (d x +c \right )\right )}{a^{2} d}+\frac {\left (B b -C a \right ) x \left (\tan ^{3}\left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {B \tan \left (d x +c \right )}{2 a d}}{\tan \left (d x +c \right )^{3}}+\frac {\left (a B +C b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{2}+b^{2}\right )}-\frac {\left (a^{2} B -b^{2} B +C a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}-\frac {b^{3} \left (B b -C a \right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a^{3} d}\) |
\(179\) |
| risch |
\(\frac {2 i C b x}{a^{2}}+\frac {x C}{i b -a}+\frac {2 i b^{4} B x}{\left (a^{2}+b^{2}\right ) a^{3}}-\frac {2 i \left (i B a \,{\mathrm e}^{2 i \left (d x +c \right )}-B b \,{\mathrm e}^{2 i \left (d x +c \right )}+C a \,{\mathrm e}^{2 i \left (d x +c \right )}+B b -C a \right )}{a^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {2 i b^{3} C c}{\left (a^{2}+b^{2}\right ) a^{2} d}+\frac {2 i C b c}{a^{2} d}-\frac {2 i b^{3} C x}{\left (a^{2}+b^{2}\right ) a^{2}}+\frac {2 i x B}{a}-\frac {2 i B \,b^{2} x}{a^{3}}+\frac {i x B}{i b -a}-\frac {2 i B \,b^{2} c}{d \,a^{3}}+\frac {2 i b^{4} B c}{\left (a^{2}+b^{2}\right ) d \,a^{3}}+\frac {2 i B c}{a d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{a d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B \,b^{2}}{d \,a^{3}}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C b}{a^{2} d}-\frac {b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{\left (a^{2}+b^{2}\right ) d \,a^{3}}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C}{\left (a^{2}+b^{2}\right ) a^{2} d}\) |
\(415\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(d*x+c)^4*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)
[Out]
1/d*(1/(a^2+b^2)*(1/2*(B*a+C*b)*ln(1+tan(d*x+c)^2)+(B*b-C*a)*arctan(tan(d*x+c)))-(B*b-C*a)*b^3/a^3/(a^2+b^2)*l
n(a+b*tan(d*x+c))-1/2*B/a/tan(d*x+c)^2-(-B*b+C*a)/a^2/tan(d*x+c)+1/a^3*(-B*a^2+B*b^2-C*a*b)*ln(tan(d*x+c)))
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Maxima [A]
time = 0.51, size = 158, normalized size = 1.15 \begin {gather*} -\frac {\frac {2 \, {\left (C a - B b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} - \frac {2 \, {\left (C a b^{3} - B b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{5} + a^{3} b^{2}} - \frac {{\left (B a + C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {2 \, {\left (B a^{2} + C a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{3}} + \frac {B a + 2 \, {\left (C a - B b\right )} \tan \left (d x + c\right )}{a^{2} \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)^4*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c)),x, algorithm="maxima")
[Out]
-1/2*(2*(C*a - B*b)*(d*x + c)/(a^2 + b^2) - 2*(C*a*b^3 - B*b^4)*log(b*tan(d*x + c) + a)/(a^5 + a^3*b^2) - (B*a
+ C*b)*log(tan(d*x + c)^2 + 1)/(a^2 + b^2) + 2*(B*a^2 + C*a*b - B*b^2)*log(tan(d*x + c))/a^3 + (B*a + 2*(C*a
- B*b)*tan(d*x + c))/(a^2*tan(d*x + c)^2))/d
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Fricas [A]
time = 4.00, size = 234, normalized size = 1.71 \begin {gather*} -\frac {B a^{4} + B a^{2} b^{2} + {\left (B a^{4} + C a^{3} b + C a b^{3} - B b^{4}\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} - {\left (C a b^{3} - B b^{4}\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{2} + {\left (B a^{4} + B a^{2} b^{2} + 2 \, {\left (C a^{4} - B a^{3} b\right )} d x\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (C a^{4} - B a^{3} b + C a^{2} b^{2} - B a b^{3}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{5} + a^{3} b^{2}\right )} 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)^4*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c)),x, algorithm="fricas")
[Out]
-1/2*(B*a^4 + B*a^2*b^2 + (B*a^4 + C*a^3*b + C*a*b^3 - B*b^4)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x
+ c)^2 - (C*a*b^3 - B*b^4)*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1))*tan(d*x
+ c)^2 + (B*a^4 + B*a^2*b^2 + 2*(C*a^4 - B*a^3*b)*d*x)*tan(d*x + c)^2 + 2*(C*a^4 - B*a^3*b + C*a^2*b^2 - B*a*b
^3)*tan(d*x + c))/((a^5 + a^3*b^2)*d*tan(d*x + c)^2)
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Sympy [C] Result contains complex when optimal does not.
time = 8.45, size = 2592, normalized size = 18.92 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)**4*(B*tan(d*x+c)+C*tan(d*x+c)**2)/(a+b*tan(d*x+c)),x)
[Out]
Piecewise((nan, Eq(a, 0) & Eq(b, 0) & Eq(c, 0) & Eq(d, 0)), ((B*log(tan(c + d*x)**2 + 1)/(2*d) - B*log(tan(c +
d*x))/d - B/(2*d*tan(c + d*x)**2) - C*x - C/(d*tan(c + d*x)))/a, Eq(b, 0)), ((B*x + B/(d*tan(c + d*x)) - B/(3
*d*tan(c + d*x)**3) + C*log(tan(c + d*x)**2 + 1)/(2*d) - C*log(tan(c + d*x))/d - C/(2*d*tan(c + d*x)**2))/b, E
q(a, 0)), (-3*B*d*x*tan(c + d*x)**3/(2*b*d*tan(c + d*x)**3 - 2*I*b*d*tan(c + d*x)**2) + 3*I*B*d*x*tan(c + d*x)
**2/(2*b*d*tan(c + d*x)**3 - 2*I*b*d*tan(c + d*x)**2) + 2*I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(2*b*d*
tan(c + d*x)**3 - 2*I*b*d*tan(c + d*x)**2) + 2*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(2*b*d*tan(c + d*x)*
*3 - 2*I*b*d*tan(c + d*x)**2) - 4*I*B*log(tan(c + d*x))*tan(c + d*x)**3/(2*b*d*tan(c + d*x)**3 - 2*I*b*d*tan(c
+ d*x)**2) - 4*B*log(tan(c + d*x))*tan(c + d*x)**2/(2*b*d*tan(c + d*x)**3 - 2*I*b*d*tan(c + d*x)**2) - 3*B*ta
n(c + d*x)**2/(2*b*d*tan(c + d*x)**3 - 2*I*b*d*tan(c + d*x)**2) + I*B*tan(c + d*x)/(2*b*d*tan(c + d*x)**3 - 2*
I*b*d*tan(c + d*x)**2) - B/(2*b*d*tan(c + d*x)**3 - 2*I*b*d*tan(c + d*x)**2) - 3*I*C*d*x*tan(c + d*x)**3/(2*b*
d*tan(c + d*x)**3 - 2*I*b*d*tan(c + d*x)**2) - 3*C*d*x*tan(c + d*x)**2/(2*b*d*tan(c + d*x)**3 - 2*I*b*d*tan(c
+ d*x)**2) - C*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(2*b*d*tan(c + d*x)**3 - 2*I*b*d*tan(c + d*x)**2) + I*
C*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(2*b*d*tan(c + d*x)**3 - 2*I*b*d*tan(c + d*x)**2) + 2*C*log(tan(c +
d*x))*tan(c + d*x)**3/(2*b*d*tan(c + d*x)**3 - 2*I*b*d*tan(c + d*x)**2) - 2*I*C*log(tan(c + d*x))*tan(c + d*x
)**2/(2*b*d*tan(c + d*x)**3 - 2*I*b*d*tan(c + d*x)**2) - 3*I*C*tan(c + d*x)**2/(2*b*d*tan(c + d*x)**3 - 2*I*b*
d*tan(c + d*x)**2) - 2*C*tan(c + d*x)/(2*b*d*tan(c + d*x)**3 - 2*I*b*d*tan(c + d*x)**2), Eq(a, -I*b)), (-3*B*d
*x*tan(c + d*x)**3/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - 3*I*B*d*x*tan(c + d*x)**2/(2*b*d*tan(c
+ d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - 2*I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(2*b*d*tan(c + d*x)**3 +
2*I*b*d*tan(c + d*x)**2) + 2*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(
c + d*x)**2) + 4*I*B*log(tan(c + d*x))*tan(c + d*x)**3/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - 4*B
*log(tan(c + d*x))*tan(c + d*x)**2/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - 3*B*tan(c + d*x)**2/(2*
b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - I*B*tan(c + d*x)/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x
)**2) - B/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) + 3*I*C*d*x*tan(c + d*x)**3/(2*b*d*tan(c + d*x)**3
+ 2*I*b*d*tan(c + d*x)**2) - 3*C*d*x*tan(c + d*x)**2/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - C*lo
g(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - I*C*log(tan(c + d*x
)**2 + 1)*tan(c + d*x)**2/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) + 2*C*log(tan(c + d*x))*tan(c + d*
x)**3/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) + 2*I*C*log(tan(c + d*x))*tan(c + d*x)**2/(2*b*d*tan(c
+ d*x)**3 + 2*I*b*d*tan(c + d*x)**2) + 3*I*C*tan(c + d*x)**2/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2
) - 2*C*tan(c + d*x)/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2), Eq(a, I*b)), (nan, Eq(c, -d*x)), (x*(B
*tan(c) + C*tan(c)**2)*cot(c)**4/(a + b*tan(c)), Eq(d, 0)), (B*a**4*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(
2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) - 2*B*a**4*log(tan(c + d*x))*tan(c + d*x)**2/(2*a**5
*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) - B*a**4/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c +
d*x)**2) + 2*B*a**3*b*d*x*tan(c + d*x)**2/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) + 2*B*a*
*3*b*tan(c + d*x)/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) - B*a**2*b**2/(2*a**5*d*tan(c + d
*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) + 2*B*a*b**3*tan(c + d*x)/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*ta
n(c + d*x)**2) - 2*B*b**4*log(a/b + tan(c + d*x))*tan(c + d*x)**2/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*ta
n(c + d*x)**2) + 2*B*b**4*log(tan(c + d*x))*tan(c + d*x)**2/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c +
d*x)**2) - 2*C*a**4*d*x*tan(c + d*x)**2/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) - 2*C*a**4*
tan(c + d*x)/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) + C*a**3*b*log(tan(c + d*x)**2 + 1)*ta
n(c + d*x)**2/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) - 2*C*a**3*b*log(tan(c + d*x))*tan(c
+ d*x)**2/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) - 2*C*a**2*b**2*tan(c + d*x)/(2*a**5*d*ta
n(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) + 2*C*a*b**3*log(a/b + tan(c + d*x))*tan(c + d*x)**2/(2*a**5*d*
tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) - 2*C*a*b**3*log(tan(c + d*x))*tan(c + d*x)**2/(2*a**5*d*tan(
c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2), True))
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Giac [A]
time = 1.47, size = 214, normalized size = 1.56 \begin {gather*} -\frac {\frac {2 \, {\left (C a - B b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} - \frac {{\left (B a + C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, {\left (C a b^{4} - B b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{5} b + a^{3} b^{3}} + \frac {2 \, {\left (B a^{2} + C a b - B b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac {3 \, B a^{2} \tan \left (d x + c\right )^{2} + 3 \, C a b \tan \left (d x + c\right )^{2} - 3 \, B b^{2} \tan \left (d x + c\right )^{2} - 2 \, C a^{2} \tan \left (d x + c\right ) + 2 \, B a b \tan \left (d x + c\right ) - B a^{2}}{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)^4*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c)),x, algorithm="giac")
[Out]
-1/2*(2*(C*a - B*b)*(d*x + c)/(a^2 + b^2) - (B*a + C*b)*log(tan(d*x + c)^2 + 1)/(a^2 + b^2) - 2*(C*a*b^4 - B*b
^5)*log(abs(b*tan(d*x + c) + a))/(a^5*b + a^3*b^3) + 2*(B*a^2 + C*a*b - B*b^2)*log(abs(tan(d*x + c)))/a^3 - (3
*B*a^2*tan(d*x + c)^2 + 3*C*a*b*tan(d*x + c)^2 - 3*B*b^2*tan(d*x + c)^2 - 2*C*a^2*tan(d*x + c) + 2*B*a*b*tan(d
*x + c) - B*a^2)/(a^3*tan(d*x + c)^2))/d
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Mupad [B]
time = 10.93, size = 175, normalized size = 1.28 \begin {gather*} -\frac {{\mathrm {cot}\left (c+d\,x\right )}^2\,\left (\frac {B}{2\,a}-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (B\,b-C\,a\right )}{a^2}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (-b+a\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,a^2+C\,a\,b-B\,b^2\right )}{a^3\,d}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,b^4-C\,a\,b^3\right )}{d\,\left (a^5+a^3\,b^2\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )}{2\,d\,\left (a-b\,1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((cot(c + d*x)^4*(B*tan(c + d*x) + C*tan(c + d*x)^2))/(a + b*tan(c + d*x)),x)
[Out]
(log(tan(c + d*x) - 1i)*(B*1i - C))/(2*d*(a*1i - b)) - (cot(c + d*x)^2*(B/(2*a) - (tan(c + d*x)*(B*b - C*a))/a
^2))/d - (log(tan(c + d*x))*(B*a^2 - B*b^2 + C*a*b))/(a^3*d) - (log(a + b*tan(c + d*x))*(B*b^4 - C*a*b^3))/(d*
(a^5 + a^3*b^2)) + (log(tan(c + d*x) + 1i)*(B - C*1i))/(2*d*(a - b*1i))
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