Integrand size = 41, antiderivative size = 150 \[ \int (a+i a \tan (e+f x))^m (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx=\frac {(i A+B) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{2 f n}-\frac {2^{-1+n} (B (m-n)+i A (m+n)) \operatorname {Hypergeometric2F1}\left (m,-n,1+m,\frac {1}{2} (1+i \tan (e+f x))\right ) (1-i \tan (e+f x))^{-n} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{f m n} \]
1/2*(I*A+B)*(a+I*a*tan(f*x+e))^m*(c-I*c*tan(f*x+e))^n/f/n-2^(-1+n)*(B*(-n+ m)+I*A*(n+m))*hypergeom([m, -n],[1+m],1/2+1/2*I*tan(f*x+e))*(a+I*a*tan(f*x +e))^m*(c-I*c*tan(f*x+e))^n/f/m/n/((1-I*tan(f*x+e))^n)
Time = 1.13 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.81 \[ \int (a+i a \tan (e+f x))^m (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx=\frac {2^{-1+n} \left ((-i A-B) \operatorname {Hypergeometric2F1}\left (m,1-n,1+m,\frac {1}{2} (1+i \tan (e+f x))\right )+2 B \operatorname {Hypergeometric2F1}\left (m,-n,1+m,\frac {1}{2} (1+i \tan (e+f x))\right )\right ) (1-i \tan (e+f x))^{-n} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{f m} \]
(2^(-1 + n)*(((-I)*A - B)*Hypergeometric2F1[m, 1 - n, 1 + m, (1 + I*Tan[e + f*x])/2] + 2*B*Hypergeometric2F1[m, -n, 1 + m, (1 + I*Tan[e + f*x])/2])* (a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^n)/(f*m*(1 - I*Tan[e + f*x ])^n)
Time = 0.39 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {3042, 4071, 88, 80, 79}
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.
\(\displaystyle \int (a+i a \tan (e+f x))^m (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a+i a \tan (e+f x))^m (A+B \tan (e+f x)) (c-i c \tan (e+f x))^ndx\) |
\(\Big \downarrow \) 4071 |
\(\displaystyle \frac {a c \int (i \tan (e+f x) a+a)^{m-1} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{n-1}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 88 |
\(\displaystyle \frac {a c \left (\frac {(B+i A) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{2 a c n}-\frac {(-A (m+n)+i B (m-n)) \int (i \tan (e+f x) a+a)^{m-1} (c-i c \tan (e+f x))^nd\tan (e+f x)}{2 c n}\right )}{f}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {a c \left (\frac {(B+i A) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{2 a c n}-\frac {2^{n-1} (-A (m+n)+i B (m-n)) (1-i \tan (e+f x))^{-n} (c-i c \tan (e+f x))^n \int \left (\frac {1}{2}-\frac {1}{2} i \tan (e+f x)\right )^n (i \tan (e+f x) a+a)^{m-1}d\tan (e+f x)}{c n}\right )}{f}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {a c \left (\frac {i 2^{n-1} (-A (m+n)+i B (m-n)) (1-i \tan (e+f x))^{-n} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n \operatorname {Hypergeometric2F1}\left (m,-n,m+1,\frac {1}{2} (i \tan (e+f x)+1)\right )}{a c m n}+\frac {(B+i A) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{2 a c n}\right )}{f}\) |
(a*c*(((I*A + B)*(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^n)/(2*a*c *n) + (I*2^(-1 + n)*(I*B*(m - n) - A*(m + n))*Hypergeometric2F1[m, -n, 1 + m, (1 + I*Tan[e + f*x])/2]*(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x] )^n)/(a*c*m*n*(1 - I*Tan[e + f*x])^n)))/f
3.9.51.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && !RationalQ[p] && SumSimpl erQ[p, 1]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si mp[a*(c/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(A + B*x), x], x , Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]
\[\int \left (a +i a \tan \left (f x +e \right )\right )^{m} \left (A +B \tan \left (f x +e \right )\right ) \left (c -i c \tan \left (f x +e \right )\right )^{n}d x\]
\[ \int (a+i a \tan (e+f x))^m (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx=\int { {\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{n} \,d x } \]
integral(((A - I*B)*e^(2*I*f*x + 2*I*e) + A + I*B)*(2*c/(e^(2*I*f*x + 2*I* e) + 1))^n*e^(2*I*f*m*x + 2*I*e*m + m*log(a/c) + m*log(2*c/(e^(2*I*f*x + 2 *I*e) + 1)))/(e^(2*I*f*x + 2*I*e) + 1), x)
\[ \int (a+i a \tan (e+f x))^m (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx=\int \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m} \left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{n} \left (A + B \tan {\left (e + f x \right )}\right )\, dx \]
\[ \int (a+i a \tan (e+f x))^m (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx=\int { {\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{n} \,d x } \]
\[ \int (a+i a \tan (e+f x))^m (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx=\int { {\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{n} \,d x } \]
Timed out. \[ \int (a+i a \tan (e+f x))^m (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx=\int \left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \]