Integrand size = 21, antiderivative size = 98 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \, dx=2 a b x+\frac {2 a b \cot (c+d x)}{d}+\frac {\left (a^2-b^2\right ) \cot ^2(c+d x)}{2 d}-\frac {2 a b \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^4(c+d x)}{4 d}+\frac {\left (a^2-b^2\right ) \log (\sin (c+d x))}{d} \] Output:
2*a*b*x+2*a*b*cot(d*x+c)/d+1/2*(a^2-b^2)*cot(d*x+c)^2/d-2/3*a*b*cot(d*x+c) ^3/d-1/4*a^2*cot(d*x+c)^4/d+(a^2-b^2)*ln(sin(d*x+c))/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.18 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {a^2 \csc ^2(c+d x)}{d}-\frac {b^2 \csc ^2(c+d x)}{2 d}-\frac {a^2 \csc ^4(c+d x)}{4 d}-\frac {2 a b \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(c+d x)\right )}{3 d}+\frac {a^2 \log (\sin (c+d x))}{d}-\frac {b^2 \log (\sin (c+d x))}{d} \] Input:
Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2,x]
Output:
(a^2*Csc[c + d*x]^2)/d - (b^2*Csc[c + d*x]^2)/(2*d) - (a^2*Csc[c + d*x]^4) /(4*d) - (2*a*b*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d *x]^2])/(3*d) + (a^2*Log[Sin[c + d*x]])/d - (b^2*Log[Sin[c + d*x]])/d
Time = 0.74 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.02, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 4025, 3042, 4012, 25, 3042, 4012, 3042, 4012, 25, 3042, 4014, 3042, 25, 3956}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^2}{\tan (c+d x)^5}dx\) |
| \(\Big \downarrow \) 4025 |
| \(\displaystyle \int \cot ^4(c+d x) \left (2 a b-\left (a^2-b^2\right ) \tan (c+d x)\right )dx-\frac {a^2 \cot ^4(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\tan (c+d x)^4}dx-\frac {a^2 \cot ^4(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \int -\cot ^3(c+d x) \left (a^2+2 b \tan (c+d x) a-b^2\right )dx-\frac {a^2 \cot ^4(c+d x)}{4 d}-\frac {2 a b \cot ^3(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \cot ^3(c+d x) \left (a^2+2 b \tan (c+d x) a-b^2\right )dx-\frac {a^2 \cot ^4(c+d x)}{4 d}-\frac {2 a b \cot ^3(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {a^2+2 b \tan (c+d x) a-b^2}{\tan (c+d x)^3}dx-\frac {a^2 \cot ^4(c+d x)}{4 d}-\frac {2 a b \cot ^3(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle -\int \cot ^2(c+d x) \left (2 a b-\left (a^2-b^2\right ) \tan (c+d x)\right )dx+\frac {\left (a^2-b^2\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 \cot ^4(c+d x)}{4 d}-\frac {2 a b \cot ^3(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\tan (c+d x)^2}dx+\frac {\left (a^2-b^2\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 \cot ^4(c+d x)}{4 d}-\frac {2 a b \cot ^3(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle -\int -\cot (c+d x) \left (a^2+2 b \tan (c+d x) a-b^2\right )dx+\frac {\left (a^2-b^2\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 \cot ^4(c+d x)}{4 d}-\frac {2 a b \cot ^3(c+d x)}{3 d}+\frac {2 a b \cot (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \cot (c+d x) \left (a^2+2 b \tan (c+d x) a-b^2\right )dx+\frac {\left (a^2-b^2\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 \cot ^4(c+d x)}{4 d}-\frac {2 a b \cot ^3(c+d x)}{3 d}+\frac {2 a b \cot (c+d x)}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a^2+2 b \tan (c+d x) a-b^2}{\tan (c+d x)}dx+\frac {\left (a^2-b^2\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 \cot ^4(c+d x)}{4 d}-\frac {2 a b \cot ^3(c+d x)}{3 d}+\frac {2 a b \cot (c+d x)}{d}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \left (a^2-b^2\right ) \int \cot (c+d x)dx+\frac {\left (a^2-b^2\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 \cot ^4(c+d x)}{4 d}-\frac {2 a b \cot ^3(c+d x)}{3 d}+\frac {2 a b \cot (c+d x)}{d}+2 a b x\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \left (a^2-b^2\right ) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\left (a^2-b^2\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 \cot ^4(c+d x)}{4 d}-\frac {2 a b \cot ^3(c+d x)}{3 d}+\frac {2 a b \cot (c+d x)}{d}+2 a b x\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\left (a^2-b^2\right ) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx+\frac {\left (a^2-b^2\right ) \cot ^2(c+d x)}{2 d}-\frac {a^2 \cot ^4(c+d x)}{4 d}-\frac {2 a b \cot ^3(c+d x)}{3 d}+\frac {2 a b \cot (c+d x)}{d}+2 a b x\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \cot ^2(c+d x)}{2 d}+\frac {\left (a^2-b^2\right ) \log (-\sin (c+d x))}{d}-\frac {a^2 \cot ^4(c+d x)}{4 d}-\frac {2 a b \cot ^3(c+d x)}{3 d}+\frac {2 a b \cot (c+d x)}{d}+2 a b x\) |
Input:
Int[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2,x]
Output:
2*a*b*x + (2*a*b*Cot[c + d*x])/d + ((a^2 - b^2)*Cot[c + d*x]^2)/(2*d) - (2 *a*b*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]^4)/(4*d) + ((a^2 - b^2)*Log [-Sin[c + d*x]])/d
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
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] + Simp[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 ]
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] + Simp[(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] && N eQ[a*c + b*d, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Time = 1.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.89
| method | result | size |
| derivativedivides | \(\frac {b^{2} \left (-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+2 a b \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(87\) |
| default | \(\frac {b^{2} \left (-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+2 a b \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(87\) |
| parallelrisch | \(\frac {6 \left (-a^{2}+b^{2}\right ) \ln \left (\sec \left (d x +c \right )^{2}\right )+12 \left (a^{2}-b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )-3 \cot \left (d x +c \right )^{4} a^{2}-8 \cot \left (d x +c \right )^{3} a b +6 \left (a^{2}-b^{2}\right ) \cot \left (d x +c \right )^{2}+24 \cot \left (d x +c \right ) a b +24 a b d x}{12 d}\) | \(105\) |
| norman | \(\frac {-\frac {a^{2}}{4 d}+\frac {\left (a^{2}-b^{2}\right ) \tan \left (d x +c \right )^{2}}{2 d}+2 a b x \tan \left (d x +c \right )^{4}-\frac {2 a b \tan \left (d x +c \right )}{3 d}+\frac {2 a b \tan \left (d x +c \right )^{3}}{d}}{\tan \left (d x +c \right )^{4}}+\frac {\left (a^{2}-b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d}\) | \(128\) |
| risch | \(2 a b x -i a^{2} x +i b^{2} x -\frac {2 i a^{2} c}{d}+\frac {2 i b^{2} c}{d}-\frac {2 \left (-12 i a b \,{\mathrm e}^{6 i \left (d x +c \right )}+6 a^{2} {\mathrm e}^{6 i \left (d x +c \right )}-3 b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+24 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}-6 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+6 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-20 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-3 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+8 i a b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{d}\) | \(230\) |
Input:
int(cot(d*x+c)^5*(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(b^2*(-1/2*cot(d*x+c)^2-ln(sin(d*x+c)))+2*a*b*(-1/3*cot(d*x+c)^3+cot(d *x+c)+d*x+c)+a^2*(-1/4*cot(d*x+c)^4+1/2*cot(d*x+c)^2+ln(sin(d*x+c))))
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.31 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {6 \, {\left (a^{2} - b^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{4} + 24 \, a b \tan \left (d x + c\right )^{3} + 3 \, {\left (8 \, a b d x + 3 \, a^{2} - 2 \, b^{2}\right )} \tan \left (d x + c\right )^{4} - 8 \, a b \tan \left (d x + c\right ) + 6 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} - 3 \, a^{2}}{12 \, d \tan \left (d x + c\right )^{4}} \] Input:
integrate(cot(d*x+c)^5*(a+b*tan(d*x+c))^2,x, algorithm="fricas")
Output:
1/12*(6*(a^2 - b^2)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^ 4 + 24*a*b*tan(d*x + c)^3 + 3*(8*a*b*d*x + 3*a^2 - 2*b^2)*tan(d*x + c)^4 - 8*a*b*tan(d*x + c) + 6*(a^2 - b^2)*tan(d*x + c)^2 - 3*a^2)/(d*tan(d*x + c )^4)
Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (88) = 176\).
Time = 1.07 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.81 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \, dx=\begin {cases} \tilde {\infty } a^{2} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right )^{2} \cot ^{5}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } a^{2} x & \text {for}\: c = - d x \\- \frac {a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {a^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {a^{2}}{4 d \tan ^{4}{\left (c + d x \right )}} + 2 a b x + \frac {2 a b}{d \tan {\left (c + d x \right )}} - \frac {2 a b}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {b^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \] Input:
integrate(cot(d*x+c)**5*(a+b*tan(d*x+c))**2,x)
Output:
Piecewise((zoo*a**2*x, Eq(c, 0) & Eq(d, 0)), (x*(a + b*tan(c))**2*cot(c)** 5, Eq(d, 0)), (zoo*a**2*x, Eq(c, -d*x)), (-a**2*log(tan(c + d*x)**2 + 1)/( 2*d) + a**2*log(tan(c + d*x))/d + a**2/(2*d*tan(c + d*x)**2) - a**2/(4*d*t an(c + d*x)**4) + 2*a*b*x + 2*a*b/(d*tan(c + d*x)) - 2*a*b/(3*d*tan(c + d* x)**3) + b**2*log(tan(c + d*x)**2 + 1)/(2*d) - b**2*log(tan(c + d*x))/d - b**2/(2*d*tan(c + d*x)**2), True))
Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.13 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {24 \, {\left (d x + c\right )} a b - 6 \, {\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \, {\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {24 \, a b \tan \left (d x + c\right )^{3} - 8 \, a b \tan \left (d x + c\right ) + 6 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} - 3 \, a^{2}}{\tan \left (d x + c\right )^{4}}}{12 \, d} \] Input:
integrate(cot(d*x+c)^5*(a+b*tan(d*x+c))^2,x, algorithm="maxima")
Output:
1/12*(24*(d*x + c)*a*b - 6*(a^2 - b^2)*log(tan(d*x + c)^2 + 1) + 12*(a^2 - b^2)*log(tan(d*x + c)) + (24*a*b*tan(d*x + c)^3 - 8*a*b*tan(d*x + c) + 6* (a^2 - b^2)*tan(d*x + c)^2 - 3*a^2)/tan(d*x + c)^4)/d
Time = 0.33 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.21 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {2 \, {\left (d x + c\right )} a b}{d} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} + \frac {{\left (a^{2} - b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{d} + \frac {24 \, a b \tan \left (d x + c\right )^{3} - 8 \, a b \tan \left (d x + c\right ) + 6 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} - 3 \, a^{2}}{12 \, d \tan \left (d x + c\right )^{4}} \] Input:
integrate(cot(d*x+c)^5*(a+b*tan(d*x+c))^2,x, algorithm="giac")
Output:
2*(d*x + c)*a*b/d - 1/2*(a^2 - b^2)*log(tan(d*x + c)^2 + 1)/d + (a^2 - b^2 )*log(abs(tan(d*x + c)))/d + 1/12*(24*a*b*tan(d*x + c)^3 - 8*a*b*tan(d*x + c) + 6*(a^2 - b^2)*tan(d*x + c)^2 - 3*a^2)/(d*tan(d*x + c)^4)
Time = 1.24 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.32 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2-b^2\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^2}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^2}{2\,d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^4\,\left (\frac {a^2}{4}-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {a^2}{2}-\frac {b^2}{2}\right )+\frac {2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )}{3}-2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}{d} \] Input:
int(cot(c + d*x)^5*(a + b*tan(c + d*x))^2,x)
Output:
(log(tan(c + d*x) + 1i)*(a*1i + b)^2)/(2*d) - (log(tan(c + d*x) - 1i)*(a + b*1i)^2)/(2*d) + (log(tan(c + d*x))*(a^2 - b^2))/d - (cot(c + d*x)^4*(a^2 /4 - tan(c + d*x)^2*(a^2/2 - b^2/2) + (2*a*b*tan(c + d*x))/3 - 2*a*b*tan(c + d*x)^3))/d
Time = 0.27 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.23 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {256 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a b -64 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a b -96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{4} a^{2}+96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{4} b^{2}+96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4} a^{2}-96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4} b^{2}-39 \sin \left (d x +c \right )^{4} a^{2}+192 \sin \left (d x +c \right )^{4} a b d x +24 \sin \left (d x +c \right )^{4} b^{2}+96 \sin \left (d x +c \right )^{2} a^{2}-48 \sin \left (d x +c \right )^{2} b^{2}-24 a^{2}}{96 \sin \left (d x +c \right )^{4} d} \] Input:
int(cot(d*x+c)^5*(a+b*tan(d*x+c))^2,x)
Output:
(256*cos(c + d*x)*sin(c + d*x)**3*a*b - 64*cos(c + d*x)*sin(c + d*x)*a*b - 96*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**4*a**2 + 96*log(tan((c + d* x)/2)**2 + 1)*sin(c + d*x)**4*b**2 + 96*log(tan((c + d*x)/2))*sin(c + d*x) **4*a**2 - 96*log(tan((c + d*x)/2))*sin(c + d*x)**4*b**2 - 39*sin(c + d*x) **4*a**2 + 192*sin(c + d*x)**4*a*b*d*x + 24*sin(c + d*x)**4*b**2 + 96*sin( c + d*x)**2*a**2 - 48*sin(c + d*x)**2*b**2 - 24*a**2)/(96*sin(c + d*x)**4* d)