\(\int (a+a \sin (e+f x))^m \tan ^4(e+f x) \, dx\) [135]

Optimal result

Integrand size = 21, antiderivative size = 319 \[ \int (a+a \sin (e+f x))^m \tan ^4(e+f x) \, dx=-\frac {a \left (3+7 m+m^2\right ) \sec (e+f x) (a+a \sin (e+f x))^{-1+m}}{3 f m}+\frac {\left (9-5 m-m^2\right ) \sec (e+f x) (a-a \sin (e+f x)) (a+a \sin (e+f x))^{-1+m}}{3 f (3-2 m)}+\frac {2^{-\frac {1}{2}+m} \left (9-12 m-7 m^2+6 m^3+m^4\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right ) \sec (e+f x) (1+\sin (e+f x))^{\frac {3}{2}-m} (a-a \sin (e+f x)) (a+a \sin (e+f x))^{-1+m}}{3 f (3-2 m) m}+\frac {a^2 (3+m) \sin (e+f x) (a+a \sin (e+f x))^{-1+m} \tan (e+f x)}{3 f m (a-a \sin (e+f x))}-\frac {a^2 \sin ^2(e+f x) (a+a \sin (e+f x))^{-1+m} \tan (e+f x)}{f m (a-a \sin (e+f x))} \] Output:

-1/3*a*(m^2+7*m+3)*sec(f*x+e)*(a+a*sin(f*x+e))^(-1+m)/f/m+1/3*(-m^2-5*m+9) 
*sec(f*x+e)*(a-a*sin(f*x+e))*(a+a*sin(f*x+e))^(-1+m)/f/(3-2*m)+1/3*2^(-1/2 
+m)*(m^4+6*m^3-7*m^2-12*m+9)*hypergeom([1/2, 3/2-m],[3/2],1/2-1/2*sin(f*x+ 
e))*sec(f*x+e)*(1+sin(f*x+e))^(3/2-m)*(a-a*sin(f*x+e))*(a+a*sin(f*x+e))^(- 
1+m)/f/(3-2*m)/m+1/3*a^2*(3+m)*sin(f*x+e)*(a+a*sin(f*x+e))^(-1+m)*tan(f*x+ 
e)/f/m/(a-a*sin(f*x+e))-a^2*sin(f*x+e)^2*(a+a*sin(f*x+e))^(-1+m)*tan(f*x+e 
)/f/m/(a-a*sin(f*x+e))
 

Mathematica [F]

\[ \int (a+a \sin (e+f x))^m \tan ^4(e+f x) \, dx=\int (a+a \sin (e+f x))^m \tan ^4(e+f x) \, dx \] Input:

Integrate[(a + a*Sin[e + f*x])^m*Tan[e + f*x]^4,x]
 

Output:

Integrate[(a + a*Sin[e + f*x])^m*Tan[e + f*x]^4, x]
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 3198, 111, 25, 27, 170, 25, 27, 162, 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.

\(\displaystyle \int \tan ^4(e+f x) (a \sin (e+f x)+a)^m \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \tan (e+f x)^4 (a \sin (e+f x)+a)^mdx\)

\(\Big \downarrow \) 3198

\(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \int \frac {a^4 \sin ^4(e+f x) (\sin (e+f x) a+a)^{m-\frac {5}{2}}}{(a-a \sin (e+f x))^{5/2}}d(a \sin (e+f x))}{a f}\)

\(\Big \downarrow \) 111

\(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (-\frac {\int -\frac {a^3 \sin ^2(e+f x) (\sin (e+f x) a+a)^{m-\frac {5}{2}} (m \sin (e+f x) a+3 a)}{(a-a \sin (e+f x))^{5/2}}d(a \sin (e+f x))}{m}-\frac {a^3 \sin ^3(e+f x) (a \sin (e+f x)+a)^{m-\frac {3}{2}}}{m (a-a \sin (e+f x))^{3/2}}\right )}{a f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (\frac {\int \frac {a^3 \sin ^2(e+f x) (\sin (e+f x) a+a)^{m-\frac {5}{2}} (m \sin (e+f x) a+3 a)}{(a-a \sin (e+f x))^{5/2}}d(a \sin (e+f x))}{m}-\frac {a^3 \sin ^3(e+f x) (a \sin (e+f x)+a)^{m-\frac {3}{2}}}{m (a-a \sin (e+f x))^{3/2}}\right )}{a f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (\frac {a \int \frac {a^2 \sin ^2(e+f x) (\sin (e+f x) a+a)^{m-\frac {5}{2}} (m \sin (e+f x) a+3 a)}{(a-a \sin (e+f x))^{5/2}}d(a \sin (e+f x))}{m}-\frac {a^3 \sin ^3(e+f x) (a \sin (e+f x)+a)^{m-\frac {3}{2}}}{m (a-a \sin (e+f x))^{3/2}}\right )}{a f}\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (\frac {a \left (\frac {\int -\frac {a^2 \sin (e+f x) (\sin (e+f x) a+a)^{m-\frac {5}{2}} \left (2 a m-a \left (-m^2-3 m+3\right ) \sin (e+f x)\right )}{(a-a \sin (e+f x))^{5/2}}d(a \sin (e+f x))}{1-m}+\frac {a^2 m \sin ^2(e+f x) (a \sin (e+f x)+a)^{m-\frac {3}{2}}}{(1-m) (a-a \sin (e+f x))^{3/2}}\right )}{m}-\frac {a^3 \sin ^3(e+f x) (a \sin (e+f x)+a)^{m-\frac {3}{2}}}{m (a-a \sin (e+f x))^{3/2}}\right )}{a f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (\frac {a \left (\frac {a^2 m \sin ^2(e+f x) (a \sin (e+f x)+a)^{m-\frac {3}{2}}}{(1-m) (a-a \sin (e+f x))^{3/2}}-\frac {\int \frac {a^2 \sin (e+f x) (\sin (e+f x) a+a)^{m-\frac {5}{2}} \left (2 a m-a \left (-m^2-3 m+3\right ) \sin (e+f x)\right )}{(a-a \sin (e+f x))^{5/2}}d(a \sin (e+f x))}{1-m}\right )}{m}-\frac {a^3 \sin ^3(e+f x) (a \sin (e+f x)+a)^{m-\frac {3}{2}}}{m (a-a \sin (e+f x))^{3/2}}\right )}{a f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (\frac {a \left (\frac {a^2 m \sin ^2(e+f x) (a \sin (e+f x)+a)^{m-\frac {3}{2}}}{(1-m) (a-a \sin (e+f x))^{3/2}}-\frac {a \int \frac {a \sin (e+f x) (\sin (e+f x) a+a)^{m-\frac {5}{2}} \left (2 a m-a \left (-m^2-3 m+3\right ) \sin (e+f x)\right )}{(a-a \sin (e+f x))^{5/2}}d(a \sin (e+f x))}{1-m}\right )}{m}-\frac {a^3 \sin ^3(e+f x) (a \sin (e+f x)+a)^{m-\frac {3}{2}}}{m (a-a \sin (e+f x))^{3/2}}\right )}{a f}\)

\(\Big \downarrow \) 162

\(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (\frac {a \left (\frac {a^2 m \sin ^2(e+f x) (a \sin (e+f x)+a)^{m-\frac {3}{2}}}{(1-m) (a-a \sin (e+f x))^{3/2}}-\frac {a \left (\frac {1}{3} \left (m^4+6 m^3-7 m^2-12 m+9\right ) \int \frac {(\sin (e+f x) a+a)^{m-\frac {5}{2}}}{\sqrt {a-a \sin (e+f x)}}d(a \sin (e+f x))+\frac {\left (a \left (-m^3-7 m^2-m+6\right )-a \left (-m^3-8 m^2-6 m+9\right ) \sin (e+f x)\right ) (a \sin (e+f x)+a)^{m-\frac {3}{2}}}{3 (a-a \sin (e+f x))^{3/2}}\right )}{1-m}\right )}{m}-\frac {a^3 \sin ^3(e+f x) (a \sin (e+f x)+a)^{m-\frac {3}{2}}}{m (a-a \sin (e+f x))^{3/2}}\right )}{a f}\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (\frac {a \left (\frac {a^2 m \sin ^2(e+f x) (a \sin (e+f x)+a)^{m-\frac {3}{2}}}{(1-m) (a-a \sin (e+f x))^{3/2}}-\frac {a \left (\frac {2^{m-\frac {5}{2}} \left (m^4+6 m^3-7 m^2-12 m+9\right ) (a \sin (e+f x)+a)^{m-\frac {1}{2}} \left (\frac {a \sin (e+f x)+a}{a}\right )^{\frac {1}{2}-m} \int \frac {\left (\frac {1}{2} \sin (e+f x)+\frac {1}{2}\right )^{m-\frac {5}{2}}}{\sqrt {a-a \sin (e+f x)}}d(a \sin (e+f x))}{3 a^2}+\frac {(a \sin (e+f x)+a)^{m-\frac {3}{2}} \left (a \left (-m^3-7 m^2-m+6\right )-a \left (-m^3-8 m^2-6 m+9\right ) \sin (e+f x)\right )}{3 (a-a \sin (e+f x))^{3/2}}\right )}{1-m}\right )}{m}-\frac {a^3 \sin ^3(e+f x) (a \sin (e+f x)+a)^{m-\frac {3}{2}}}{m (a-a \sin (e+f x))^{3/2}}\right )}{a f}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (\frac {a \left (\frac {a^2 m \sin ^2(e+f x) (a \sin (e+f x)+a)^{m-\frac {3}{2}}}{(1-m) (a-a \sin (e+f x))^{3/2}}-\frac {a \left (\frac {(a \sin (e+f x)+a)^{m-\frac {3}{2}} \left (a \left (-m^3-7 m^2-m+6\right )-a \left (-m^3-8 m^2-6 m+9\right ) \sin (e+f x)\right )}{3 (a-a \sin (e+f x))^{3/2}}-\frac {2^{m-\frac {3}{2}} \left (m^4+6 m^3-7 m^2-12 m+9\right ) \sqrt {a-a \sin (e+f x)} (a \sin (e+f x)+a)^{m-\frac {1}{2}} \left (\frac {a \sin (e+f x)+a}{a}\right )^{\frac {1}{2}-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{2}-m,\frac {3}{2},\frac {a-a \sin (e+f x)}{2 a}\right )}{3 a^2}\right )}{1-m}\right )}{m}-\frac {a^3 \sin ^3(e+f x) (a \sin (e+f x)+a)^{m-\frac {3}{2}}}{m (a-a \sin (e+f x))^{3/2}}\right )}{a f}\)

Input:

Int[(a + a*Sin[e + f*x])^m*Tan[e + f*x]^4,x]
 

Output:

(Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]]*(-((a^3*Si 
n[e + f*x]^3*(a + a*Sin[e + f*x])^(-3/2 + m))/(m*(a - a*Sin[e + f*x])^(3/2 
))) + (a*((a^2*m*Sin[e + f*x]^2*(a + a*Sin[e + f*x])^(-3/2 + m))/((1 - m)* 
(a - a*Sin[e + f*x])^(3/2)) - (a*(-1/3*(2^(-3/2 + m)*(9 - 12*m - 7*m^2 + 6 
*m^3 + m^4)*Hypergeometric2F1[1/2, 5/2 - m, 3/2, (a - a*Sin[e + f*x])/(2*a 
)]*Sqrt[a - a*Sin[e + f*x]]*(a + a*Sin[e + f*x])^(-1/2 + m)*((a + a*Sin[e 
+ f*x])/a)^(1/2 - m))/a^2 + ((a + a*Sin[e + f*x])^(-3/2 + m)*(a*(6 - m - 7 
*m^2 - m^3) - a*(9 - 6*m - 8*m^2 - m^3)*Sin[e + f*x]))/(3*(a - a*Sin[e + f 
*x])^(3/2))))/(1 - m)))/m))/(a*f)
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 79

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]))
 

rule 80

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])
 

rule 111

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

rule 162

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) 
 - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e 
*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + 
e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b 
*c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim 
p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d 
*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( 
b^2*(b*c - a*d)^2*(m + 1)*(m + 2)))   Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] 
, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + 
 n + 3, 0] &&  !LtQ[n, -2]))
 

rule 170

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegerQ[m]
 

rule 3042

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

rule 3198

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_ 
), x_Symbol] :> Simp[Sqrt[a + b*Sin[e + f*x]]*(Sqrt[a - b*Sin[e + f*x]]/(b* 
f*Cos[e + f*x]))   Subst[Int[x^p*((a + x)^(m - (p + 1)/2)/(a - x)^((p + 1)/ 
2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b 
^2, 0] &&  !IntegerQ[m] && IntegerQ[p/2]
 

Maple [F]

\[\int \left (a +\sin \left (f x +e \right ) a \right )^{m} \tan \left (f x +e \right )^{4}d x\]

Input:

int((a+sin(f*x+e)*a)^m*tan(f*x+e)^4,x)
 

Output:

int((a+sin(f*x+e)*a)^m*tan(f*x+e)^4,x)
 

Fricas [F]

\[ \int (a+a \sin (e+f x))^m \tan ^4(e+f x) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \tan \left (f x + e\right )^{4} \,d x } \] Input:

integrate((a+a*sin(f*x+e))^m*tan(f*x+e)^4,x, algorithm="fricas")
 

Output:

integral((a*sin(f*x + e) + a)^m*tan(f*x + e)^4, x)
 

Sympy [F]

\[ \int (a+a \sin (e+f x))^m \tan ^4(e+f x) \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \tan ^{4}{\left (e + f x \right )}\, dx \] Input:

integrate((a+a*sin(f*x+e))**m*tan(f*x+e)**4,x)
 

Output:

Integral((a*(sin(e + f*x) + 1))**m*tan(e + f*x)**4, x)
 

Maxima [F]

\[ \int (a+a \sin (e+f x))^m \tan ^4(e+f x) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \tan \left (f x + e\right )^{4} \,d x } \] Input:

integrate((a+a*sin(f*x+e))^m*tan(f*x+e)^4,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate((a*sin(f*x + e) + a)^m*tan(f*x + e)^4, x)
 

Giac [F]

\[ \int (a+a \sin (e+f x))^m \tan ^4(e+f x) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \tan \left (f x + e\right )^{4} \,d x } \] Input:

integrate((a+a*sin(f*x+e))^m*tan(f*x+e)^4,x, algorithm="giac")
 

Output:

integrate((a*sin(f*x + e) + a)^m*tan(f*x + e)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+a \sin (e+f x))^m \tan ^4(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^4\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \] Input:

int(tan(e + f*x)^4*(a + a*sin(e + f*x))^m,x)
 

Output:

int(tan(e + f*x)^4*(a + a*sin(e + f*x))^m, x)
 

Reduce [F]

\[ \int (a+a \sin (e+f x))^m \tan ^4(e+f x) \, dx=\int \left (a +a \sin \left (f x +e \right )\right )^{m} \tan \left (f x +e \right )^{4}d x \] Input:

int((a+a*sin(f*x+e))^m*tan(f*x+e)^4,x)
 

Output:

int((sin(e + f*x)*a + a)**m*tan(e + f*x)**4,x)