Integrand size = 21, antiderivative size = 274 \[ \int \frac {\sin ^5(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {a^5 b \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2} d}+\frac {a^3 b^2 \cos (c+d x)}{\left (a^2+b^2\right )^3 d}+\frac {a b^2 \cos (c+d x)}{\left (a^2+b^2\right )^2 d}-\frac {a \cos (c+d x)}{\left (a^2+b^2\right ) d}-\frac {a b^2 \cos ^3(c+d x)}{3 \left (a^2+b^2\right )^2 d}+\frac {2 a \cos ^3(c+d x)}{3 \left (a^2+b^2\right ) d}-\frac {a \cos ^5(c+d x)}{5 \left (a^2+b^2\right ) d}+\frac {a^4 b \sin (c+d x)}{\left (a^2+b^2\right )^3 d}+\frac {a^2 b \sin ^3(c+d x)}{3 \left (a^2+b^2\right )^2 d}+\frac {b \sin ^5(c+d x)}{5 \left (a^2+b^2\right ) d} \] Output:
a^5*b*arctanh((b*cos(d*x+c)-a*sin(d*x+c))/(a^2+b^2)^(1/2))/(a^2+b^2)^(7/2) /d+a^3*b^2*cos(d*x+c)/(a^2+b^2)^3/d+a*b^2*cos(d*x+c)/(a^2+b^2)^2/d-a*cos(d *x+c)/(a^2+b^2)/d-1/3*a*b^2*cos(d*x+c)^3/(a^2+b^2)^2/d+2/3*a*cos(d*x+c)^3/ (a^2+b^2)/d-1/5*a*cos(d*x+c)^5/(a^2+b^2)/d+a^4*b*sin(d*x+c)/(a^2+b^2)^3/d+ 1/3*a^2*b*sin(d*x+c)^3/(a^2+b^2)^2/d+1/5*b*sin(d*x+c)^5/(a^2+b^2)/d
Time = 3.37 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.05 \[ \int \frac {\sin ^5(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {-480 a^5 b \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )+\sqrt {a^2+b^2} \left (-30 a \left (5 a^4-4 a^2 b^2-b^4\right ) \cos (c+d x)+5 a \left (5 a^4+6 a^2 b^2+b^4\right ) \cos (3 (c+d x))-3 a^5 \cos (5 (c+d x))-6 a^3 b^2 \cos (5 (c+d x))-3 a b^4 \cos (5 (c+d x))+330 a^4 b \sin (c+d x)+120 a^2 b^3 \sin (c+d x)+30 b^5 \sin (c+d x)-35 a^4 b \sin (3 (c+d x))-50 a^2 b^3 \sin (3 (c+d x))-15 b^5 \sin (3 (c+d x))+3 a^4 b \sin (5 (c+d x))+6 a^2 b^3 \sin (5 (c+d x))+3 b^5 \sin (5 (c+d x))\right )}{240 \left (a^2+b^2\right )^{7/2} d} \] Input:
Integrate[Sin[c + d*x]^5/(a + b*Tan[c + d*x]),x]
Output:
(-480*a^5*b*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]] + Sqrt[a^2 + b^2]*(-30*a*(5*a^4 - 4*a^2*b^2 - b^4)*Cos[c + d*x] + 5*a*(5*a^4 + 6*a^2* b^2 + b^4)*Cos[3*(c + d*x)] - 3*a^5*Cos[5*(c + d*x)] - 6*a^3*b^2*Cos[5*(c + d*x)] - 3*a*b^4*Cos[5*(c + d*x)] + 330*a^4*b*Sin[c + d*x] + 120*a^2*b^3* Sin[c + d*x] + 30*b^5*Sin[c + d*x] - 35*a^4*b*Sin[3*(c + d*x)] - 50*a^2*b^ 3*Sin[3*(c + d*x)] - 15*b^5*Sin[3*(c + d*x)] + 3*a^4*b*Sin[5*(c + d*x)] + 6*a^2*b^3*Sin[5*(c + d*x)] + 3*b^5*Sin[5*(c + d*x)]))/(240*(a^2 + b^2)^(7/ 2)*d)
Time = 1.28 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.92, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 4001, 3042, 3588, 3042, 3044, 15, 3113, 2009, 3578, 3042, 3113, 2009, 3578, 3042, 3118, 3553, 219}
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.
| \(\displaystyle \int \frac {\sin ^5(c+d x)}{a+b \tan (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^5}{a+b \tan (c+d x)}dx\) |
| \(\Big \downarrow \) 4001 |
| \(\displaystyle \int \frac {\sin ^5(c+d x) \cos (c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^5 \cos (c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle \frac {a \int \sin ^5(c+d x)dx}{a^2+b^2}+\frac {b \int \cos (c+d x) \sin ^4(c+d x)dx}{a^2+b^2}-\frac {a b \int \frac {\sin ^4(c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \sin (c+d x)^5dx}{a^2+b^2}+\frac {b \int \cos (c+d x) \sin (c+d x)^4dx}{a^2+b^2}-\frac {a b \int \frac {\sin (c+d x)^4}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {b \int \sin ^4(c+d x)d\sin (c+d x)}{d \left (a^2+b^2\right )}+\frac {a \int \sin (c+d x)^5dx}{a^2+b^2}-\frac {a b \int \frac {\sin (c+d x)^4}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {a \int \sin (c+d x)^5dx}{a^2+b^2}-\frac {a b \int \frac {\sin (c+d x)^4}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {b \sin ^5(c+d x)}{5 d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle -\frac {a \int \left (\cos ^4(c+d x)-2 \cos ^2(c+d x)+1\right )d\cos (c+d x)}{d \left (a^2+b^2\right )}-\frac {a b \int \frac {\sin (c+d x)^4}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {b \sin ^5(c+d x)}{5 d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a b \int \frac {\sin (c+d x)^4}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {b \sin ^5(c+d x)}{5 d \left (a^2+b^2\right )}-\frac {a \left (\frac {1}{5} \cos ^5(c+d x)-\frac {2}{3} \cos ^3(c+d x)+\cos (c+d x)\right )}{d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3578 |
| \(\displaystyle -\frac {a b \left (\frac {b \int \sin ^3(c+d x)dx}{a^2+b^2}+\frac {a^2 \int \frac {\sin ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}-\frac {a \sin ^3(c+d x)}{3 d \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \sin ^5(c+d x)}{5 d \left (a^2+b^2\right )}-\frac {a \left (\frac {1}{5} \cos ^5(c+d x)-\frac {2}{3} \cos ^3(c+d x)+\cos (c+d x)\right )}{d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \left (\frac {b \int \sin (c+d x)^3dx}{a^2+b^2}+\frac {a^2 \int \frac {\sin (c+d x)^2}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}-\frac {a \sin ^3(c+d x)}{3 d \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \sin ^5(c+d x)}{5 d \left (a^2+b^2\right )}-\frac {a \left (\frac {1}{5} \cos ^5(c+d x)-\frac {2}{3} \cos ^3(c+d x)+\cos (c+d x)\right )}{d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle -\frac {a b \left (-\frac {b \int \left (1-\cos ^2(c+d x)\right )d\cos (c+d x)}{d \left (a^2+b^2\right )}+\frac {a^2 \int \frac {\sin (c+d x)^2}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}-\frac {a \sin ^3(c+d x)}{3 d \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \sin ^5(c+d x)}{5 d \left (a^2+b^2\right )}-\frac {a \left (\frac {1}{5} \cos ^5(c+d x)-\frac {2}{3} \cos ^3(c+d x)+\cos (c+d x)\right )}{d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a b \left (\frac {a^2 \int \frac {\sin (c+d x)^2}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}-\frac {a \sin ^3(c+d x)}{3 d \left (a^2+b^2\right )}-\frac {b \left (\cos (c+d x)-\frac {1}{3} \cos ^3(c+d x)\right )}{d \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \sin ^5(c+d x)}{5 d \left (a^2+b^2\right )}-\frac {a \left (\frac {1}{5} \cos ^5(c+d x)-\frac {2}{3} \cos ^3(c+d x)+\cos (c+d x)\right )}{d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3578 |
| \(\displaystyle -\frac {a b \left (\frac {a^2 \left (\frac {b \int \sin (c+d x)dx}{a^2+b^2}+\frac {a^2 \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}-\frac {a \sin (c+d x)}{d \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a \sin ^3(c+d x)}{3 d \left (a^2+b^2\right )}-\frac {b \left (\cos (c+d x)-\frac {1}{3} \cos ^3(c+d x)\right )}{d \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \sin ^5(c+d x)}{5 d \left (a^2+b^2\right )}-\frac {a \left (\frac {1}{5} \cos ^5(c+d x)-\frac {2}{3} \cos ^3(c+d x)+\cos (c+d x)\right )}{d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \left (\frac {a^2 \left (\frac {b \int \sin (c+d x)dx}{a^2+b^2}+\frac {a^2 \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}-\frac {a \sin (c+d x)}{d \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a \sin ^3(c+d x)}{3 d \left (a^2+b^2\right )}-\frac {b \left (\cos (c+d x)-\frac {1}{3} \cos ^3(c+d x)\right )}{d \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \sin ^5(c+d x)}{5 d \left (a^2+b^2\right )}-\frac {a \left (\frac {1}{5} \cos ^5(c+d x)-\frac {2}{3} \cos ^3(c+d x)+\cos (c+d x)\right )}{d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle -\frac {a b \left (\frac {a^2 \left (\frac {a^2 \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}-\frac {a \sin (c+d x)}{d \left (a^2+b^2\right )}-\frac {b \cos (c+d x)}{d \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a \sin ^3(c+d x)}{3 d \left (a^2+b^2\right )}-\frac {b \left (\cos (c+d x)-\frac {1}{3} \cos ^3(c+d x)\right )}{d \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \sin ^5(c+d x)}{5 d \left (a^2+b^2\right )}-\frac {a \left (\frac {1}{5} \cos ^5(c+d x)-\frac {2}{3} \cos ^3(c+d x)+\cos (c+d x)\right )}{d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle -\frac {a b \left (\frac {a^2 \left (-\frac {a^2 \int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{d \left (a^2+b^2\right )}-\frac {a \sin (c+d x)}{d \left (a^2+b^2\right )}-\frac {b \cos (c+d x)}{d \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a \sin ^3(c+d x)}{3 d \left (a^2+b^2\right )}-\frac {b \left (\cos (c+d x)-\frac {1}{3} \cos ^3(c+d x)\right )}{d \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \sin ^5(c+d x)}{5 d \left (a^2+b^2\right )}-\frac {a \left (\frac {1}{5} \cos ^5(c+d x)-\frac {2}{3} \cos ^3(c+d x)+\cos (c+d x)\right )}{d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a b \left (\frac {a^2 \left (-\frac {a^2 \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{d \left (a^2+b^2\right )^{3/2}}-\frac {a \sin (c+d x)}{d \left (a^2+b^2\right )}-\frac {b \cos (c+d x)}{d \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a \sin ^3(c+d x)}{3 d \left (a^2+b^2\right )}-\frac {b \left (\cos (c+d x)-\frac {1}{3} \cos ^3(c+d x)\right )}{d \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \sin ^5(c+d x)}{5 d \left (a^2+b^2\right )}-\frac {a \left (\frac {1}{5} \cos ^5(c+d x)-\frac {2}{3} \cos ^3(c+d x)+\cos (c+d x)\right )}{d \left (a^2+b^2\right )}\) |
Input:
Int[Sin[c + d*x]^5/(a + b*Tan[c + d*x]),x]
Output:
-((a*(Cos[c + d*x] - (2*Cos[c + d*x]^3)/3 + Cos[c + d*x]^5/5))/((a^2 + b^2 )*d)) + (b*Sin[c + d*x]^5)/(5*(a^2 + b^2)*d) - (a*b*(-((b*(Cos[c + d*x] - Cos[c + d*x]^3/3))/((a^2 + b^2)*d)) - (a*Sin[c + d*x]^3)/(3*(a^2 + b^2)*d) + (a^2*(-((a^2*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]] )/((a^2 + b^2)^(3/2)*d)) - (b*Cos[c + d*x])/((a^2 + b^2)*d) - (a*Sin[c + d *x])/((a^2 + b^2)*d)))/(a^2 + b^2)))/(a^2 + b^2)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x _Symbol] :> Simp[-d^(-1) Subst[Int[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
Int[sin[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin [(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[(-a)*(Sin[c + d*x]^(m - 1)/(d*(a^2 + b^2)*(m - 1))), x] + (Simp[a^2/(a^2 + b^2) Int[Sin[c + d*x]^(m - 2)/(a *Cos[c + d*x] + b*Sin[c + d*x]), x], x] + Simp[b/(a^2 + b^2) Int[Sin[c + d*x]^(m - 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ [m, 1]
Int[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_. ) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b /(a^2 + b^2) Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x], x] + (Simp[a/(a ^2 + b^2) Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x], x] - Simp[a*(b/(a^ 2 + b^2)) Int[Cos[c + d*x]^(m - 1)*(Sin[c + d*x]^(n - 1)/(a*Cos[c + d*x] + b*Sin[c + d*x])), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n _.), x_Symbol] :> Int[Sin[e + f*x]^m*((a*Cos[e + f*x] + b*Sin[e + f*x])^n/C os[e + f*x]^n), x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && ILtQ [n, 0] && ((LtQ[m, 5] && GtQ[n, -4]) || (EqQ[m, 5] && EqQ[n, -1]))
Time = 16.87 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.31
| method | result | size |
| derivativedivides | \(\frac {-\frac {64 a^{5} b \,\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (32 a^{6}+96 a^{4} b^{2}+96 a^{2} b^{4}+32 b^{6}\right ) \sqrt {a^{2}+b^{2}}}+\frac {2 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+2 a^{3} b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+2 \left (\frac {16}{3} a^{4} b +\frac {4}{3} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+2 \left (6 a^{3} b^{2}+2 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+2 \left (\frac {178}{15} a^{4} b +\frac {136}{15} a^{2} b^{3}+\frac {16}{5} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+2 \left (-\frac {16}{3} a^{5}-\frac {2}{3} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+2 \left (\frac {16}{3} a^{4} b +\frac {4}{3} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+2 \left (2 a^{3} b^{2}-\frac {8}{3} a^{5}+\frac {2}{3} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {16 a^{5}}{15}+\frac {6 a^{3} b^{2}}{5}+\frac {4 a \,b^{4}}{15}}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}}{d}\) | \(358\) |
| default | \(\frac {-\frac {64 a^{5} b \,\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (32 a^{6}+96 a^{4} b^{2}+96 a^{2} b^{4}+32 b^{6}\right ) \sqrt {a^{2}+b^{2}}}+\frac {2 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+2 a^{3} b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+2 \left (\frac {16}{3} a^{4} b +\frac {4}{3} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+2 \left (6 a^{3} b^{2}+2 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+2 \left (\frac {178}{15} a^{4} b +\frac {136}{15} a^{2} b^{3}+\frac {16}{5} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+2 \left (-\frac {16}{3} a^{5}-\frac {2}{3} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+2 \left (\frac {16}{3} a^{4} b +\frac {4}{3} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+2 \left (2 a^{3} b^{2}-\frac {8}{3} a^{5}+\frac {2}{3} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {16 a^{5}}{15}+\frac {6 a^{3} b^{2}}{5}+\frac {4 a \,b^{4}}{15}}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}}{d}\) | \(358\) |
| risch | \(-\frac {i {\mathrm e}^{3 i \left (d x +c \right )} b}{32 \left (-2 i a b +a^{2}-b^{2}\right ) d}+\frac {5 \,{\mathrm e}^{3 i \left (d x +c \right )} a}{96 \left (-2 i a b +a^{2}-b^{2}\right ) d}+\frac {i {\mathrm e}^{i \left (d x +c \right )} a b}{4 \left (-3 i a^{2} b +i b^{3}+a^{3}-3 a \,b^{2}\right ) d}-\frac {5 \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{16 \left (-3 i a^{2} b +i b^{3}+a^{3}-3 a \,b^{2}\right ) d}+\frac {{\mathrm e}^{i \left (d x +c \right )} b^{2}}{16 \left (-3 i a^{2} b +i b^{3}+a^{3}-3 a \,b^{2}\right ) d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} a b}{4 \left (i b +a \right )^{3} d}-\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{16 \left (i b +a \right )^{3} d}+\frac {{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{16 \left (i b +a \right )^{3} d}-\frac {5 \,{\mathrm e}^{-3 i \left (d x +c \right )} a}{96 \left (-i a +b \right )^{2} d}-\frac {i {\mathrm e}^{-3 i \left (d x +c \right )} b}{32 \left (-i a +b \right )^{2} d}+\frac {i a^{5} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i b +a}{\sqrt {-a^{2}-b^{2}}}\right )}{\sqrt {-a^{2}-b^{2}}\, \left (a^{2}+b^{2}\right )^{3} d}-\frac {i a^{5} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i b +a}{\sqrt {-a^{2}-b^{2}}}\right )}{\sqrt {-a^{2}-b^{2}}\, \left (a^{2}+b^{2}\right )^{3} d}+\frac {a \cos \left (5 d x +5 c \right )}{80 d \left (-a^{2}-b^{2}\right )}-\frac {b \sin \left (5 d x +5 c \right )}{80 d \left (-a^{2}-b^{2}\right )}\) | \(496\) |
Input:
int(sin(d*x+c)^5/(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(-64*a^5*b/(32*a^6+96*a^4*b^2+96*a^2*b^4+32*b^6)/(a^2+b^2)^(1/2)*arcta nh(1/2*(2*a*tan(1/2*d*x+1/2*c)-2*b)/(a^2+b^2)^(1/2))+2/(a^4+2*a^2*b^2+b^4) /(a^2+b^2)*(a^4*b*tan(1/2*d*x+1/2*c)^9+a^3*b^2*tan(1/2*d*x+1/2*c)^8+(16/3* a^4*b+4/3*a^2*b^3)*tan(1/2*d*x+1/2*c)^7+(6*a^3*b^2+2*a*b^4)*tan(1/2*d*x+1/ 2*c)^6+(178/15*a^4*b+136/15*a^2*b^3+16/5*b^5)*tan(1/2*d*x+1/2*c)^5+(-16/3* a^5-2/3*a*b^4)*tan(1/2*d*x+1/2*c)^4+(16/3*a^4*b+4/3*a^2*b^3)*tan(1/2*d*x+1 /2*c)^3+(2*a^3*b^2-8/3*a^5+2/3*a*b^4)*tan(1/2*d*x+1/2*c)^2+a^4*b*tan(1/2*d *x+1/2*c)-8/15*a^5+3/5*a^3*b^2+2/15*a*b^4)/(1+tan(1/2*d*x+1/2*c)^2)^5)
Time = 0.12 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.35 \[ \int \frac {\sin ^5(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {15 \, \sqrt {a^{2} + b^{2}} a^{5} b \log \left (\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) - 6 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (2 \, a^{7} + 5 \, a^{5} b^{2} + 4 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{3} - 30 \, {\left (a^{7} + a^{5} b^{2}\right )} \cos \left (d x + c\right ) + 2 \, {\left (23 \, a^{6} b + 34 \, a^{4} b^{3} + 14 \, a^{2} b^{5} + 3 \, b^{7} + 3 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{4} - {\left (11 \, a^{6} b + 28 \, a^{4} b^{3} + 23 \, a^{2} b^{5} + 6 \, b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} d} \] Input:
integrate(sin(d*x+c)^5/(a+b*tan(d*x+c)),x, algorithm="fricas")
Output:
1/30*(15*sqrt(a^2 + b^2)*a^5*b*log((2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 - 2*a^2 - b^2 - 2*sqrt(a^2 + b^2)*(b*cos(d*x + c) - a*sin(d*x + c)))/(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2)) - 6*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cos(d*x + c)^5 + 1 0*(2*a^7 + 5*a^5*b^2 + 4*a^3*b^4 + a*b^6)*cos(d*x + c)^3 - 30*(a^7 + a^5*b ^2)*cos(d*x + c) + 2*(23*a^6*b + 34*a^4*b^3 + 14*a^2*b^5 + 3*b^7 + 3*(a^6* b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cos(d*x + c)^4 - (11*a^6*b + 28*a^4*b^3 + 23*a^2*b^5 + 6*b^7)*cos(d*x + c)^2)*sin(d*x + c))/((a^8 + 4*a^6*b^2 + 6*a ^4*b^4 + 4*a^2*b^6 + b^8)*d)
Timed out. \[ \int \frac {\sin ^5(c+d x)}{a+b \tan (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(sin(d*x+c)**5/(a+b*tan(d*x+c)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 658 vs. \(2 (260) = 520\).
Time = 0.14 (sec) , antiderivative size = 658, normalized size of antiderivative = 2.40 \[ \int \frac {\sin ^5(c+d x)}{a+b \tan (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(sin(d*x+c)^5/(a+b*tan(d*x+c)),x, algorithm="maxima")
Output:
1/15*(15*a^5*b*log((b - a*sin(d*x + c)/(cos(d*x + c) + 1) + sqrt(a^2 + b^2 ))/(b - a*sin(d*x + c)/(cos(d*x + c) + 1) - sqrt(a^2 + b^2)))/((a^6 + 3*a^ 4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) - 2*(8*a^5 - 9*a^3*b^2 - 2*a*b^4 - 15*a^4*b*sin(d*x + c)/(cos(d*x + c) + 1) - 15*a^3*b^2*sin(d*x + c)^8/(c os(d*x + c) + 1)^8 - 15*a^4*b*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 10*(4* a^5 - 3*a^3*b^2 - a*b^4)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 20*(4*a^4*b + a^2*b^3)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 10*(8*a^5 + a*b^4)*sin(d *x + c)^4/(cos(d*x + c) + 1)^4 - 2*(89*a^4*b + 68*a^2*b^3 + 24*b^5)*sin(d* x + c)^5/(cos(d*x + c) + 1)^5 - 30*(3*a^3*b^2 + a*b^4)*sin(d*x + c)^6/(cos (d*x + c) + 1)^6 - 20*(4*a^4*b + a^2*b^3)*sin(d*x + c)^7/(cos(d*x + c) + 1 )^7)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + 5*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*(a^6 + 3*a^4*b^2 + 3*a^2*b^ 4 + b^6)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 10*(a^6 + 3*a^4*b^2 + 3*a^2 *b^4 + b^6)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5*(a^6 + 3*a^4*b^2 + 3*a ^2*b^4 + b^6)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + (a^6 + 3*a^4*b^2 + 3*a ^2*b^4 + b^6)*sin(d*x + c)^10/(cos(d*x + c) + 1)^10))/d
Time = 0.23 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.69 \[ \int \frac {\sin ^5(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\frac {15 \, a^{5} b \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (15 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 15 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 80 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 20 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 90 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 30 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 178 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 136 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 80 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 10 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 80 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 20 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 30 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a^{5} + 9 \, a^{3} b^{2} + 2 \, a b^{4}\right )}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{15 \, d} \] Input:
integrate(sin(d*x+c)^5/(a+b*tan(d*x+c)),x, algorithm="giac")
Output:
1/15*(15*a^5*b*log(abs(2*a*tan(1/2*d*x + 1/2*c) - 2*b - 2*sqrt(a^2 + b^2)) /abs(2*a*tan(1/2*d*x + 1/2*c) - 2*b + 2*sqrt(a^2 + b^2)))/((a^6 + 3*a^4*b^ 2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) + 2*(15*a^4*b*tan(1/2*d*x + 1/2*c)^9 + 15*a^3*b^2*tan(1/2*d*x + 1/2*c)^8 + 80*a^4*b*tan(1/2*d*x + 1/2*c)^7 + 2 0*a^2*b^3*tan(1/2*d*x + 1/2*c)^7 + 90*a^3*b^2*tan(1/2*d*x + 1/2*c)^6 + 30* a*b^4*tan(1/2*d*x + 1/2*c)^6 + 178*a^4*b*tan(1/2*d*x + 1/2*c)^5 + 136*a^2* b^3*tan(1/2*d*x + 1/2*c)^5 + 48*b^5*tan(1/2*d*x + 1/2*c)^5 - 80*a^5*tan(1/ 2*d*x + 1/2*c)^4 - 10*a*b^4*tan(1/2*d*x + 1/2*c)^4 + 80*a^4*b*tan(1/2*d*x + 1/2*c)^3 + 20*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 - 40*a^5*tan(1/2*d*x + 1/2* c)^2 + 30*a^3*b^2*tan(1/2*d*x + 1/2*c)^2 + 10*a*b^4*tan(1/2*d*x + 1/2*c)^2 + 15*a^4*b*tan(1/2*d*x + 1/2*c) - 8*a^5 + 9*a^3*b^2 + 2*a*b^4)/((a^6 + 3* a^4*b^2 + 3*a^2*b^4 + b^6)*(tan(1/2*d*x + 1/2*c)^2 + 1)^5))/d
Time = 4.07 (sec) , antiderivative size = 683, normalized size of antiderivative = 2.49 \[ \int \frac {\sin ^5(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\frac {2\,\left (-8\,a^5+9\,a^3\,b^2+2\,a\,b^4\right )}{15\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (4\,a^4\,b+a^2\,b^3\right )}{3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (3\,a^3\,b^2+a\,b^4\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-4\,a^5+3\,a^3\,b^2+a\,b^4\right )}{3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (89\,a^4\,b+68\,a^2\,b^3+24\,b^5\right )}{15\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (8\,a^5+a\,b^4\right )}{3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {2\,a^3\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {8\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (4\,a^4+a^2\,b^2\right )}{3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {2\,a^4\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {2\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {2\,a^5\,b\,\mathrm {atanh}\left (\frac {2\,a^6\,b+2\,b^7+6\,a^2\,b^5+6\,a^4\,b^3-2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}{2\,{\left (a^2+b^2\right )}^{7/2}}\right )}{d\,{\left (a^2+b^2\right )}^{7/2}} \] Input:
int(sin(c + d*x)^5/(a + b*tan(c + d*x)),x)
Output:
((2*(2*a*b^4 - 8*a^5 + 9*a^3*b^2))/(15*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) ) + (8*tan(c/2 + (d*x)/2)^3*(4*a^4*b + a^2*b^3))/(3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (4*tan(c/2 + (d*x)/2)^6*(a*b^4 + 3*a^3*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (4*tan(c/2 + (d*x)/2)^2*(a*b^4 - 4*a^5 + 3*a^3*b ^2))/(3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (4*tan(c/2 + (d*x)/2)^5*(89 *a^4*b + 24*b^5 + 68*a^2*b^3))/(15*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (4*tan(c/2 + (d*x)/2)^4*(a*b^4 + 8*a^5))/(3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4 *b^2)) + (2*a^3*b^2*tan(c/2 + (d*x)/2)^8)/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b ^2) + (8*b*tan(c/2 + (d*x)/2)^7*(4*a^4 + a^2*b^2))/(3*(a^6 + b^6 + 3*a^2*b ^4 + 3*a^4*b^2)) + (2*a^4*b*tan(c/2 + (d*x)/2))/(a^6 + b^6 + 3*a^2*b^4 + 3 *a^4*b^2) + (2*a^4*b*tan(c/2 + (d*x)/2)^9)/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4* b^2))/(d*(5*tan(c/2 + (d*x)/2)^2 + 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 + 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 + 1)) + (2*a^5 *b*atanh((2*a^6*b + 2*b^7 + 6*a^2*b^5 + 6*a^4*b^3 - 2*a*tan(c/2 + (d*x)/2) *(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))/(2*(a^2 + b^2)^(7/2))))/(d*(a^2 + b^ 2)^(7/2))
Time = 0.17 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.66 \[ \int \frac {\sin ^5(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\cos \left (d x +c \right ) a^{5} b^{2}+11 \cos \left (d x +c \right ) a^{3} b^{4}+15 \sin \left (d x +c \right ) a^{4} b^{3}-3 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{7}-4 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{7}+3 \sin \left (d x +c \right )^{5} a^{6} b +9 \sin \left (d x +c \right )^{5} a^{4} b^{3}+9 \sin \left (d x +c \right )^{5} a^{2} b^{5}+5 \sin \left (d x +c \right )^{3} a^{6} b +10 \sin \left (d x +c \right )^{3} a^{4} b^{3}+5 \sin \left (d x +c \right )^{3} a^{2} b^{5}+2 \cos \left (d x +c \right ) a \,b^{6}+15 \sin \left (d x +c \right ) a^{6} b -9 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{5} b^{2}-9 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{3} b^{4}-3 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a \,b^{6}-7 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{5} b^{2}-2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{3} b^{4}+\cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a \,b^{6}+30 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a i -b i}{\sqrt {a^{2}+b^{2}}}\right ) a^{5} b i +3 \sin \left (d x +c \right )^{5} b^{7}-5 a^{5} b^{2}+5 a^{3} b^{4}+2 a \,b^{6}-8 \cos \left (d x +c \right ) a^{7}-8 a^{7}}{15 d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )} \] Input:
int(sin(d*x+c)^5/(a+b*tan(d*x+c)),x)
Output:
(30*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2)) *a**5*b*i - 3*cos(c + d*x)*sin(c + d*x)**4*a**7 - 9*cos(c + d*x)*sin(c + d *x)**4*a**5*b**2 - 9*cos(c + d*x)*sin(c + d*x)**4*a**3*b**4 - 3*cos(c + d* x)*sin(c + d*x)**4*a*b**6 - 4*cos(c + d*x)*sin(c + d*x)**2*a**7 - 7*cos(c + d*x)*sin(c + d*x)**2*a**5*b**2 - 2*cos(c + d*x)*sin(c + d*x)**2*a**3*b** 4 + cos(c + d*x)*sin(c + d*x)**2*a*b**6 - 8*cos(c + d*x)*a**7 + cos(c + d* x)*a**5*b**2 + 11*cos(c + d*x)*a**3*b**4 + 2*cos(c + d*x)*a*b**6 + 3*sin(c + d*x)**5*a**6*b + 9*sin(c + d*x)**5*a**4*b**3 + 9*sin(c + d*x)**5*a**2*b **5 + 3*sin(c + d*x)**5*b**7 + 5*sin(c + d*x)**3*a**6*b + 10*sin(c + d*x)* *3*a**4*b**3 + 5*sin(c + d*x)**3*a**2*b**5 + 15*sin(c + d*x)*a**6*b + 15*s in(c + d*x)*a**4*b**3 - 8*a**7 - 5*a**5*b**2 + 5*a**3*b**4 + 2*a*b**6)/(15 *d*(a**8 + 4*a**6*b**2 + 6*a**4*b**4 + 4*a**2*b**6 + b**8))