Integrand size = 34, antiderivative size = 185 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=-\frac {(2 B+i A n) \cot (c+d x) (a+i a \tan (c+d x))^n}{2 d}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}-\frac {(A-i B) \operatorname {Hypergeometric2F1}\left (1,n,1+n,\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^n}{2 d n}-\frac {\left (2 i B n-A \left (2-n+n^2\right )\right ) \operatorname {Hypergeometric2F1}(1,n,1+n,1+i \tan (c+d x)) (a+i a \tan (c+d x))^n}{2 d n} \]
-1/2*(2*B+I*A*n)*cot(d*x+c)*(a+I*a*tan(d*x+c))^n/d-1/2*A*cot(d*x+c)^2*(a+I *a*tan(d*x+c))^n/d-1/2*(A-I*B)*hypergeom([1, n],[1+n],1/2+1/2*I*tan(d*x+c) )*(a+I*a*tan(d*x+c))^n/d/n-1/2*(2*I*B*n-A*(n^2-n+2))*hypergeom([1, n],[1+n ],1+I*tan(d*x+c))*(a+I*a*tan(d*x+c))^n/d/n
Time = 3.17 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.19 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\frac {(a+i a \tan (c+d x))^n \left (-i (A-i B) n \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {1}{2} (1+i \tan (c+d x))\right ) (-i+\tan (c+d x))+2 (A+i B+A n+i B n+2 A n \operatorname {Hypergeometric2F1}(3,1+n,2+n,1+i \tan (c+d x))+2 A n \operatorname {Hypergeometric2F1}(1,1+n,2+n,1+i \tan (c+d x)) (1+i \tan (c+d x))+2 i A n \operatorname {Hypergeometric2F1}(3,1+n,2+n,1+i \tan (c+d x)) \tan (c+d x)-2 B n \operatorname {Hypergeometric2F1}(2,1+n,2+n,1+i \tan (c+d x)) (-i+\tan (c+d x)))\right )}{4 d n (1+n)} \]
((a + I*a*Tan[c + d*x])^n*((-I)*(A - I*B)*n*Hypergeometric2F1[1, 1 + n, 2 + n, (1 + I*Tan[c + d*x])/2]*(-I + Tan[c + d*x]) + 2*(A + I*B + A*n + I*B* n + 2*A*n*Hypergeometric2F1[3, 1 + n, 2 + n, 1 + I*Tan[c + d*x]] + 2*A*n*H ypergeometric2F1[1, 1 + n, 2 + n, 1 + I*Tan[c + d*x]]*(1 + I*Tan[c + d*x]) + (2*I)*A*n*Hypergeometric2F1[3, 1 + n, 2 + n, 1 + I*Tan[c + d*x]]*Tan[c + d*x] - 2*B*n*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + I*Tan[c + d*x]]*(-I + Tan[c + d*x]))))/(4*d*n*(1 + n))
Time = 1.14 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {3042, 4081, 3042, 4081, 3042, 4083, 3042, 3962, 78, 4082, 75}
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.
| \(\displaystyle \int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^n (A+B \tan (c+d x))}{\tan (c+d x)^3}dx\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \frac {\int \cot ^2(c+d x) (i \tan (c+d x) a+a)^n (a (2 B+i A n)-a A (2-n) \tan (c+d x))dx}{2 a}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(i \tan (c+d x) a+a)^n (a (2 B+i A n)-a A (2-n) \tan (c+d x))}{\tan (c+d x)^2}dx}{2 a}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \frac {\frac {\int \cot (c+d x) (i \tan (c+d x) a+a)^n \left (a^2 \left (2 i B n-A \left (n^2-n+2\right )\right )-a^2 (1-n) (2 B+i A n) \tan (c+d x)\right )dx}{a}-\frac {a (2 B+i A n) \cot (c+d x) (a+i a \tan (c+d x))^n}{d}}{2 a}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {(i \tan (c+d x) a+a)^n \left (a^2 \left (2 i B n-A \left (n^2-n+2\right )\right )-a^2 (1-n) (2 B+i A n) \tan (c+d x)\right )}{\tan (c+d x)}dx}{a}-\frac {a (2 B+i A n) \cot (c+d x) (a+i a \tan (c+d x))^n}{d}}{2 a}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}\) |
| \(\Big \downarrow \) 4083 |
| \(\displaystyle \frac {\frac {a \left (-A \left (n^2-n+2\right )+2 i B n\right ) \int \cot (c+d x) (a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^ndx-2 a^2 (B+i A) \int (i \tan (c+d x) a+a)^ndx}{a}-\frac {a (2 B+i A n) \cot (c+d x) (a+i a \tan (c+d x))^n}{d}}{2 a}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \left (-A \left (n^2-n+2\right )+2 i B n\right ) \int \frac {(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^n}{\tan (c+d x)}dx-2 a^2 (B+i A) \int (i \tan (c+d x) a+a)^ndx}{a}-\frac {a (2 B+i A n) \cot (c+d x) (a+i a \tan (c+d x))^n}{d}}{2 a}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}\) |
| \(\Big \downarrow \) 3962 |
| \(\displaystyle \frac {\frac {\frac {2 i a^3 (B+i A) \int \frac {(i \tan (c+d x) a+a)^{n-1}}{a-i a \tan (c+d x)}d(i a \tan (c+d x))}{d}+a \left (-A \left (n^2-n+2\right )+2 i B n\right ) \int \frac {(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^n}{\tan (c+d x)}dx}{a}-\frac {a (2 B+i A n) \cot (c+d x) (a+i a \tan (c+d x))^n}{d}}{2 a}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}\) |
| \(\Big \downarrow \) 78 |
| \(\displaystyle \frac {\frac {a \left (-A \left (n^2-n+2\right )+2 i B n\right ) \int \frac {(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^n}{\tan (c+d x)}dx+\frac {i a^2 (B+i A) (a+i a \tan (c+d x))^n \operatorname {Hypergeometric2F1}\left (1,n,n+1,\frac {i \tan (c+d x) a+a}{2 a}\right )}{d n}}{a}-\frac {a (2 B+i A n) \cot (c+d x) (a+i a \tan (c+d x))^n}{d}}{2 a}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}\) |
| \(\Big \downarrow \) 4082 |
| \(\displaystyle \frac {\frac {\frac {a^3 \left (-A \left (n^2-n+2\right )+2 i B n\right ) \int \cot (c+d x) (i \tan (c+d x) a+a)^{n-1}d\tan (c+d x)}{d}+\frac {i a^2 (B+i A) (a+i a \tan (c+d x))^n \operatorname {Hypergeometric2F1}\left (1,n,n+1,\frac {i \tan (c+d x) a+a}{2 a}\right )}{d n}}{a}-\frac {a (2 B+i A n) \cot (c+d x) (a+i a \tan (c+d x))^n}{d}}{2 a}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}\) |
| \(\Big \downarrow \) 75 |
| \(\displaystyle \frac {\frac {\frac {i a^2 (B+i A) (a+i a \tan (c+d x))^n \operatorname {Hypergeometric2F1}\left (1,n,n+1,\frac {i \tan (c+d x) a+a}{2 a}\right )}{d n}-\frac {a^2 \left (-A \left (n^2-n+2\right )+2 i B n\right ) (a+i a \tan (c+d x))^n \operatorname {Hypergeometric2F1}(1,n,n+1,i \tan (c+d x)+1)}{d n}}{a}-\frac {a (2 B+i A n) \cot (c+d x) (a+i a \tan (c+d x))^n}{d}}{2 a}-\frac {A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}\) |
-1/2*(A*Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^n)/d + (-((a*(2*B + I*A*n)*C ot[c + d*x]*(a + I*a*Tan[c + d*x])^n)/d) + (-((a^2*((2*I)*B*n - A*(2 - n + n^2))*Hypergeometric2F1[1, n, 1 + n, 1 + I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/(d*n)) + (I*a^2*(I*A + B)*Hypergeometric2F1[1, n, 1 + n, (a + I* a*Tan[c + d*x])/(2*a)]*(a + I*a*Tan[c + d*x])^n)/(d*n))/a)/(2*a)
3.3.25.3.1 Defintions of rubi rules used
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b *c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] && !IntegerQ[m] && IntegerQ[n]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[-b/d S ubst[Int[(a + x)^(n - 1)/(a - x), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b , c, d, n}, x] && EqQ[a^2 + b^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(A*d - B*c)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 + d^2))), x] - Simp[1/(a*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c* m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Fr eeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, -1]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(B/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[ a^2 + b^2, 0] && EqQ[A*b + a*B, 0]
Int[(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[( A*b + a*B)/(b*c + a*d) Int[(a + b*Tan[e + f*x])^m, x], x] - Simp[(B*c - A *d)/(b*c + a*d) Int[(a + b*Tan[e + f*x])^m*((a - b*Tan[e + f*x])/(c + d*T an[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
\[\int \left (\cot ^{3}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )\right )d x\]
\[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{3} \,d x } \]
integral(((-I*A - B)*e^(6*I*d*x + 6*I*c) + (-3*I*A - B)*e^(4*I*d*x + 4*I*c ) + (-3*I*A + B)*e^(2*I*d*x + 2*I*c) - I*A + B)*(2*a*e^(2*I*d*x + 2*I*c)/( e^(2*I*d*x + 2*I*c) + 1))^n/(e^(6*I*d*x + 6*I*c) - 3*e^(4*I*d*x + 4*I*c) + 3*e^(2*I*d*x + 2*I*c) - 1), x)
\[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n} \left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{3}{\left (c + d x \right )}\, dx \]
\[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{3} \,d x } \]
\[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{3} \,d x } \]
Timed out. \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^3\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \]