Integrand size = 33, antiderivative size = 169 \[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx=\frac {(A+i B) \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},-\frac {b \tan (c+d x)}{a},-i \tan (c+d x)\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (\frac {a+b \tan (c+d x)}{a}\right )^{-n}}{d}+\frac {(A-i B) \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},-\frac {b \tan (c+d x)}{a},i \tan (c+d x)\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (\frac {a+b \tan (c+d x)}{a}\right )^{-n}}{d} \] Output:
(A+I*B)*AppellF1(1/2,1,-n,3/2,-I*tan(d*x+c),-b*tan(d*x+c)/a)*tan(d*x+c)^(1 /2)*(a+b*tan(d*x+c))^n/d/(((a+b*tan(d*x+c))/a)^n)+(A-I*B)*AppellF1(1/2,1,- n,3/2,I*tan(d*x+c),-b*tan(d*x+c)/a)*tan(d*x+c)^(1/2)*(a+b*tan(d*x+c))^n/d/ (((a+b*tan(d*x+c))/a)^n)
\[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx=\int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx \] Input:
Integrate[((a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]] ,x]
Output:
Integrate[((a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]] , x]
Time = 0.63 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3042, 4086, 3042, 4085, 148, 334, 333}
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.
\(\displaystyle \int \frac {(A+B \tan (c+d x)) (a+b \tan (c+d x))^n}{\sqrt {\tan (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(A+B \tan (c+d x)) (a+b \tan (c+d x))^n}{\sqrt {\tan (c+d x)}}dx\) |
\(\Big \downarrow \) 4086 |
\(\displaystyle \frac {1}{2} (A+i B) \int \frac {(1-i \tan (c+d x)) (a+b \tan (c+d x))^n}{\sqrt {\tan (c+d x)}}dx+\frac {1}{2} (A-i B) \int \frac {(i \tan (c+d x)+1) (a+b \tan (c+d x))^n}{\sqrt {\tan (c+d x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} (A+i B) \int \frac {(1-i \tan (c+d x)) (a+b \tan (c+d x))^n}{\sqrt {\tan (c+d x)}}dx+\frac {1}{2} (A-i B) \int \frac {(i \tan (c+d x)+1) (a+b \tan (c+d x))^n}{\sqrt {\tan (c+d x)}}dx\) |
\(\Big \downarrow \) 4085 |
\(\displaystyle \frac {(A-i B) \int \frac {(a+b \tan (c+d x))^n}{(1-i \tan (c+d x)) \sqrt {\tan (c+d x)}}d\tan (c+d x)}{2 d}+\frac {(A+i B) \int \frac {(a+b \tan (c+d x))^n}{(i \tan (c+d x)+1) \sqrt {\tan (c+d x)}}d\tan (c+d x)}{2 d}\) |
\(\Big \downarrow \) 148 |
\(\displaystyle \frac {(A-i B) \int \frac {(a+b \tan (c+d x))^n}{1-i \tan (c+d x)}d\sqrt {\tan (c+d x)}}{d}+\frac {(A+i B) \int \frac {(a+b \tan (c+d x))^n}{i \tan (c+d x)+1}d\sqrt {\tan (c+d x)}}{d}\) |
\(\Big \downarrow \) 334 |
\(\displaystyle \frac {(A-i B) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} \int \frac {\left (\frac {b \tan (c+d x)}{a}+1\right )^n}{1-i \tan (c+d x)}d\sqrt {\tan (c+d x)}}{d}+\frac {(A+i B) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} \int \frac {\left (\frac {b \tan (c+d x)}{a}+1\right )^n}{i \tan (c+d x)+1}d\sqrt {\tan (c+d x)}}{d}\) |
\(\Big \downarrow \) 333 |
\(\displaystyle \frac {(A+i B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},1,-n,\frac {3}{2},-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{d}+\frac {(A-i B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},1,-n,\frac {3}{2},i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{d}\) |
Input:
Int[((a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]],x]
Output:
((A + I*B)*AppellF1[1/2, 1, -n, 3/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x]) /a)]*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^n)/(d*(1 + (b*Tan[c + d*x])/a )^n) + ((A - I*B)*AppellF1[1/2, 1, -n, 3/2, I*Tan[c + d*x], -((b*Tan[c + d *x])/a)]*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^n)/(d*(1 + (b*Tan[c + d*x ])/a)^n)
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))^(p_.), x_] :> With[{k = Denominator[m]}, Simp[k/b Subst[Int[x^(k*(m + 1) - 1)*(c + d*(x^k/b))^n*(e + f*(x^k/b))^p, x], x, (b*x)^(1/k)], x]] /; FreeQ[{b, c, d, e, f, n, p}, x] && FractionQ[m] && IntegerQ[p]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[ (1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && !(IntegerQ[p] || GtQ[a, 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^2/f Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), 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] && NeQ[a^2 + b^2, 0] && !IntegerQ[m] && !IntegerQ[n] && !IntegersQ[2*m, 2*n ] && 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] :> Si mp[(A + I*B)/2 Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 - I*T an[e + f*x]), x], x] + Simp[(A - I*B)/2 Int[(a + b*Tan[e + f*x])^m*(c + d *Tan[e + f*x])^n*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A , B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && !IntegerQ[m] & & !IntegerQ[n] && !IntegersQ[2*m, 2*n] && NeQ[A^2 + B^2, 0]
\[\int \frac {\left (a +b \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )\right )}{\sqrt {\tan \left (d x +c \right )}}d x\]
Input:
int((a+b*tan(d*x+c))^n*(A+B*tan(d*x+c))/tan(d*x+c)^(1/2),x)
Output:
int((a+b*tan(d*x+c))^n*(A+B*tan(d*x+c))/tan(d*x+c)^(1/2),x)
\[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\tan \left (d x + c\right )}} \,d x } \] Input:
integrate((a+b*tan(d*x+c))^n*(A+B*tan(d*x+c))/tan(d*x+c)^(1/2),x, algorith m="fricas")
Output:
integral((B*tan(d*x + c) + A)*(b*tan(d*x + c) + a)^n/sqrt(tan(d*x + c)), x )
\[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{n}}{\sqrt {\tan {\left (c + d x \right )}}}\, dx \] Input:
integrate((a+b*tan(d*x+c))**n*(A+B*tan(d*x+c))/tan(d*x+c)**(1/2),x)
Output:
Integral((A + B*tan(c + d*x))*(a + b*tan(c + d*x))**n/sqrt(tan(c + d*x)), x)
Timed out. \[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx=\text {Timed out} \] Input:
integrate((a+b*tan(d*x+c))^n*(A+B*tan(d*x+c))/tan(d*x+c)^(1/2),x, algorith m="maxima")
Output:
Timed out
Timed out. \[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx=\text {Timed out} \] Input:
integrate((a+b*tan(d*x+c))^n*(A+B*tan(d*x+c))/tan(d*x+c)^(1/2),x, algorith m="giac")
Output:
Timed out
Timed out. \[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx=\int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}} \,d x \] Input:
int(((A + B*tan(c + d*x))*(a + b*tan(c + d*x))^n)/tan(c + d*x)^(1/2),x)
Output:
int(((A + B*tan(c + d*x))*(a + b*tan(c + d*x))^n)/tan(c + d*x)^(1/2), x)
\[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx=\frac {2 \sqrt {\tan \left (d x +c \right )}\, \left (a +\tan \left (d x +c \right ) b \right )^{n} a -4 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \left (a +\tan \left (d x +c \right ) b \right )^{n} \tan \left (d x +c \right )^{2}}{2 \tan \left (d x +c \right ) b n +\tan \left (d x +c \right ) b +2 a n +a}d x \right ) a b d \,n^{2}-4 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \left (a +\tan \left (d x +c \right ) b \right )^{n} \tan \left (d x +c \right )^{2}}{2 \tan \left (d x +c \right ) b n +\tan \left (d x +c \right ) b +2 a n +a}d x \right ) a b d n -\left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \left (a +\tan \left (d x +c \right ) b \right )^{n} \tan \left (d x +c \right )^{2}}{2 \tan \left (d x +c \right ) b n +\tan \left (d x +c \right ) b +2 a n +a}d x \right ) a b d -2 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \left (a +\tan \left (d x +c \right ) b \right )^{n} \tan \left (d x +c \right )}{2 \tan \left (d x +c \right ) b n +\tan \left (d x +c \right ) b +2 a n +a}d x \right ) a^{2} d n -\left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \left (a +\tan \left (d x +c \right ) b \right )^{n} \tan \left (d x +c \right )}{2 \tan \left (d x +c \right ) b n +\tan \left (d x +c \right ) b +2 a n +a}d x \right ) a^{2} d +4 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \left (a +\tan \left (d x +c \right ) b \right )^{n}}{2 \tan \left (d x +c \right )^{2} b n +\tan \left (d x +c \right )^{2} b +2 \tan \left (d x +c \right ) a n +\tan \left (d x +c \right ) a}d x \right ) a^{2} d \,n^{2}+2 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \left (a +\tan \left (d x +c \right ) b \right )^{n}}{2 \tan \left (d x +c \right )^{2} b n +\tan \left (d x +c \right )^{2} b +2 \tan \left (d x +c \right ) a n +\tan \left (d x +c \right ) a}d x \right ) a^{2} d n +2 \left (\int \sqrt {\tan \left (d x +c \right )}\, \left (a +\tan \left (d x +c \right ) b \right )^{n}d x \right ) b d n +\left (\int \sqrt {\tan \left (d x +c \right )}\, \left (a +\tan \left (d x +c \right ) b \right )^{n}d x \right ) b d}{d \left (2 n +1\right )} \] Input:
int((a+b*tan(d*x+c))^n*(A+B*tan(d*x+c))/tan(d*x+c)^(1/2),x)
Output:
(2*sqrt(tan(c + d*x))*(tan(c + d*x)*b + a)**n*a - 4*int((sqrt(tan(c + d*x) )*(tan(c + d*x)*b + a)**n*tan(c + d*x)**2)/(2*tan(c + d*x)*b*n + tan(c + d *x)*b + 2*a*n + a),x)*a*b*d*n**2 - 4*int((sqrt(tan(c + d*x))*(tan(c + d*x) *b + a)**n*tan(c + d*x)**2)/(2*tan(c + d*x)*b*n + tan(c + d*x)*b + 2*a*n + a),x)*a*b*d*n - int((sqrt(tan(c + d*x))*(tan(c + d*x)*b + a)**n*tan(c + d *x)**2)/(2*tan(c + d*x)*b*n + tan(c + d*x)*b + 2*a*n + a),x)*a*b*d - 2*int ((sqrt(tan(c + d*x))*(tan(c + d*x)*b + a)**n*tan(c + d*x))/(2*tan(c + d*x) *b*n + tan(c + d*x)*b + 2*a*n + a),x)*a**2*d*n - int((sqrt(tan(c + d*x))*( tan(c + d*x)*b + a)**n*tan(c + d*x))/(2*tan(c + d*x)*b*n + tan(c + d*x)*b + 2*a*n + a),x)*a**2*d + 4*int((sqrt(tan(c + d*x))*(tan(c + d*x)*b + a)**n )/(2*tan(c + d*x)**2*b*n + tan(c + d*x)**2*b + 2*tan(c + d*x)*a*n + tan(c + d*x)*a),x)*a**2*d*n**2 + 2*int((sqrt(tan(c + d*x))*(tan(c + d*x)*b + a)* *n)/(2*tan(c + d*x)**2*b*n + tan(c + d*x)**2*b + 2*tan(c + d*x)*a*n + tan( c + d*x)*a),x)*a**2*d*n + 2*int(sqrt(tan(c + d*x))*(tan(c + d*x)*b + a)**n ,x)*b*d*n + int(sqrt(tan(c + d*x))*(tan(c + d*x)*b + a)**n,x)*b*d)/(d*(2*n + 1))