3.14.9 \(\int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx\) [1309]

3.14.9.1 Optimal result

Integrand size = 25, antiderivative size = 223 \[ \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a+b) (c-i d) f (1+m)}-\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a+i b) (i c-d) f (1+m)}+\frac {d^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (1+m)} \]

output

1/2*hypergeom([1, 1+m],[2+m],(a+b*tan(f*x+e))/(a-I*b))*(a+b*tan(f*x+e))^(1 
+m)/(I*a+b)/(c-I*d)/f/(1+m)-1/2*hypergeom([1, 1+m],[2+m],(a+b*tan(f*x+e))/ 
(a+I*b))*(a+b*tan(f*x+e))^(1+m)/(I*a-b)/(c+I*d)/f/(1+m)+d^2*hypergeom([1, 
1+m],[2+m],-d*(a+b*tan(f*x+e))/(-a*d+b*c))*(a+b*tan(f*x+e))^(1+m)/(-a*d+b* 
c)/(c^2+d^2)/f/(1+m)
 

3.14.9.2 Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\frac {\left (\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right )}{(i a+b) (c-i d)}+\frac {i \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right )}{(a+i b) (c+i d)}-\frac {2 d^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {d (a+b \tan (e+f x))}{-b c+a d}\right )}{(-b c+a d) \left (c^2+d^2\right )}\right ) (a+b \tan (e+f x))^{1+m}}{2 f (1+m)} \]

input

Integrate[(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x]),x]
 

output

((Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]/((I*a 
 + b)*(c - I*d)) + (I*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f* 
x])/(a + I*b)])/((a + I*b)*(c + I*d)) - (2*d^2*Hypergeometric2F1[1, 1 + m, 
 2 + m, (d*(a + b*Tan[e + f*x]))/(-(b*c) + a*d)])/((-(b*c) + a*d)*(c^2 + d 
^2)))*(a + b*Tan[e + f*x])^(1 + m))/(2*f*(1 + m))
 

3.14.9.3 Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 4057, 3042, 4022, 3042, 4020, 25, 78, 4117, 78}

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.

\(\displaystyle \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)}dx\)

\(\Big \downarrow \) 4057

\(\displaystyle \frac {d^2 \int \frac {(a+b \tan (e+f x))^m \left (\tan ^2(e+f x)+1\right )}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {\int (a+b \tan (e+f x))^m (c-d \tan (e+f x))dx}{c^2+d^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d^2 \int \frac {(a+b \tan (e+f x))^m \left (\tan (e+f x)^2+1\right )}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {\int (a+b \tan (e+f x))^m (c-d \tan (e+f x))dx}{c^2+d^2}\)

\(\Big \downarrow \) 4022

\(\displaystyle \frac {d^2 \int \frac {(a+b \tan (e+f x))^m \left (\tan (e+f x)^2+1\right )}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {\frac {1}{2} (c-i d) \int (1-i \tan (e+f x)) (a+b \tan (e+f x))^mdx+\frac {1}{2} (c+i d) \int (i \tan (e+f x)+1) (a+b \tan (e+f x))^mdx}{c^2+d^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d^2 \int \frac {(a+b \tan (e+f x))^m \left (\tan (e+f x)^2+1\right )}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {\frac {1}{2} (c-i d) \int (1-i \tan (e+f x)) (a+b \tan (e+f x))^mdx+\frac {1}{2} (c+i d) \int (i \tan (e+f x)+1) (a+b \tan (e+f x))^mdx}{c^2+d^2}\)

\(\Big \downarrow \) 4020

\(\displaystyle \frac {d^2 \int \frac {(a+b \tan (e+f x))^m \left (\tan (e+f x)^2+1\right )}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {\frac {i (c+i d) \int -\frac {(a+b \tan (e+f x))^m}{1-i \tan (e+f x)}d(i \tan (e+f x))}{2 f}-\frac {i (c-i d) \int -\frac {(a+b \tan (e+f x))^m}{i \tan (e+f x)+1}d(-i \tan (e+f x))}{2 f}}{c^2+d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d^2 \int \frac {(a+b \tan (e+f x))^m \left (\tan (e+f x)^2+1\right )}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {\frac {i (c-i d) \int \frac {(a+b \tan (e+f x))^m}{i \tan (e+f x)+1}d(-i \tan (e+f x))}{2 f}-\frac {i (c+i d) \int \frac {(a+b \tan (e+f x))^m}{1-i \tan (e+f x)}d(i \tan (e+f x))}{2 f}}{c^2+d^2}\)

\(\Big \downarrow \) 78

\(\displaystyle \frac {d^2 \int \frac {(a+b \tan (e+f x))^m \left (\tan (e+f x)^2+1\right )}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {\frac {i (c-i d) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b)}-\frac {i (c+i d) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (a-i b)}}{c^2+d^2}\)

\(\Big \downarrow \) 4117

\(\displaystyle \frac {d^2 \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)}d\tan (e+f x)}{f \left (c^2+d^2\right )}+\frac {\frac {i (c-i d) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b)}-\frac {i (c+i d) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (a-i b)}}{c^2+d^2}\)

\(\Big \downarrow \) 78

\(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right )}{f (m+1) \left (c^2+d^2\right ) (b c-a d)}+\frac {\frac {i (c-i d) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b)}-\frac {i (c+i d) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (a-i b)}}{c^2+d^2}\)

input

Int[(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x]),x]
 

output

(d^2*Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - 
a*d))]*(a + b*Tan[e + f*x])^(1 + m))/((b*c - a*d)*(c^2 + d^2)*f*(1 + m)) + 
 (((-1/2*I)*(c + I*d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f* 
x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/((a - I*b)*f*(1 + m)) + ((I/2 
)*(c - I*d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I 
*b)]*(a + b*Tan[e + f*x])^(1 + m))/((a + I*b)*f*(1 + m)))/(c^2 + d^2)
 

3.14.9.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 78

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4020

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f)   Subst[Int[(a + (b/d)*x)^m/(d^2 + 
c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ 
b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
 

rule 4022

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2   Int[(a + b*Tan[e + f*x])^m*( 
1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2   Int[(a + b*Tan[e + f*x])^m 
*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c 
 - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]
 

rule 4057

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)/((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)]), x_Symbol] :> Simp[1/(c^2 + d^2)   Int[(a + b*Tan[e + f*x])^m 
*(c - d*Tan[e + f*x]), x], x] + Simp[d^2/(c^2 + d^2)   Int[(a + b*Tan[e + f 
*x])^m*((1 + Tan[e + f*x]^2)/(c + d*Tan[e + f*x])), x], x] /; FreeQ[{a, b, 
c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d 
^2, 0] &&  !IntegerQ[m]
 

rule 4117

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) 
+ (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
 Simp[A/f   Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; 
FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
 

3.14.9.4 Maple [F]

\[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{m}}{c +d \tan \left (f x +e \right )}d x\]

input

int((a+b*tan(f*x+e))^m/(c+d*tan(f*x+e)),x)
 

output

int((a+b*tan(f*x+e))^m/(c+d*tan(f*x+e)),x)
 

3.14.9.5 Fricas [F]

\[ \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{d \tan \left (f x + e\right ) + c} \,d x } \]

input

integrate((a+b*tan(f*x+e))^m/(c+d*tan(f*x+e)),x, algorithm="fricas")
 

output

integral((b*tan(f*x + e) + a)^m/(d*tan(f*x + e) + c), x)
 

3.14.9.6 Sympy [F]

\[ \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{m}}{c + d \tan {\left (e + f x \right )}}\, dx \]

input

integrate((a+b*tan(f*x+e))**m/(c+d*tan(f*x+e)),x)
 

output

Integral((a + b*tan(e + f*x))**m/(c + d*tan(e + f*x)), x)
 

3.14.9.7 Maxima [F]

\[ \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{d \tan \left (f x + e\right ) + c} \,d x } \]

input

integrate((a+b*tan(f*x+e))^m/(c+d*tan(f*x+e)),x, algorithm="maxima")
 

output

integrate((b*tan(f*x + e) + a)^m/(d*tan(f*x + e) + c), x)
 

3.14.9.8 Giac [F]

\[ \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{d \tan \left (f x + e\right ) + c} \,d x } \]

input

integrate((a+b*tan(f*x+e))^m/(c+d*tan(f*x+e)),x, algorithm="giac")
 

output

integrate((b*tan(f*x + e) + a)^m/(d*tan(f*x + e) + c), x)
 

3.14.9.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^m}{c+d\,\mathrm {tan}\left (e+f\,x\right )} \,d x \]

input

int((a + b*tan(e + f*x))^m/(c + d*tan(e + f*x)),x)
 

output

int((a + b*tan(e + f*x))^m/(c + d*tan(e + f*x)), x)