\(\int (a+a \sin (e+f x))^m \tan ^3(e+f x) \, dx\) [130]

Optimal result

Integrand size = 21, antiderivative size = 163 \[ \int (a+a \sin (e+f x))^m \tan ^3(e+f x) \, dx=\frac {a m (a+a \sin (e+f x))^{-1+m}}{4 f (1-m)}+\frac {a^2 (2+m) (a+a \sin (e+f x))^{-1+m}}{2 f m (a-a \sin (e+f x))}-\frac {a^2 \sin ^2(e+f x) (a+a \sin (e+f x))^{-1+m}}{f m (a-a \sin (e+f x))}-\frac {(4+m) \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^m}{8 f m} \] Output:

1/4*a*m*(a+a*sin(f*x+e))^(-1+m)/f/(1-m)+1/2*a^2*(2+m)*(a+a*sin(f*x+e))^(-1 
+m)/f/m/(a-a*sin(f*x+e))-a^2*sin(f*x+e)^2*(a+a*sin(f*x+e))^(-1+m)/f/m/(a-a 
*sin(f*x+e))-1/8*(4+m)*hypergeom([1, m],[1+m],1/2+1/2*sin(f*x+e))*(a+a*sin 
(f*x+e))^m/f/m
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.64 \[ \int (a+a \sin (e+f x))^m \tan ^3(e+f x) \, dx=\frac {a (a (1+\sin (e+f x)))^{-1+m} \left (-2 \left (-2+3 m+m^2\right )-m (4+m) \operatorname {Hypergeometric2F1}\left (1,-1+m,m,\frac {1}{2} (1+\sin (e+f x))\right ) (-1+\sin (e+f x))+4 m \sin (e+f x)+4 (-1+m) \sin ^2(e+f x)\right )}{4 f (-1+m) m (-1+\sin (e+f x))} \] Input:

Integrate[(a + a*Sin[e + f*x])^m*Tan[e + f*x]^3,x]
 

Output:

(a*(a*(1 + Sin[e + f*x]))^(-1 + m)*(-2*(-2 + 3*m + m^2) - m*(4 + m)*Hyperg 
eometric2F1[1, -1 + m, m, (1 + Sin[e + f*x])/2]*(-1 + Sin[e + f*x]) + 4*m* 
Sin[e + f*x] + 4*(-1 + m)*Sin[e + f*x]^2))/(4*f*(-1 + m)*m*(-1 + Sin[e + f 
*x]))
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3186, 111, 25, 27, 163, 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.

\(\displaystyle \int \tan ^3(e+f x) (a \sin (e+f x)+a)^m \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \tan (e+f x)^3 (a \sin (e+f x)+a)^mdx\)

\(\Big \downarrow \) 3186

\(\displaystyle \frac {\int \frac {a^3 \sin ^3(e+f x) (\sin (e+f x) a+a)^{m-2}}{(a-a \sin (e+f x))^2}d(a \sin (e+f x))}{f}\)

\(\Big \downarrow \) 111

\(\displaystyle \frac {-\frac {\int -\frac {a^2 \sin (e+f x) (\sin (e+f x) a+a)^{m-2} (m \sin (e+f x) a+2 a)}{(a-a \sin (e+f x))^2}d(a \sin (e+f x))}{m}-\frac {a^2 \sin ^2(e+f x) (a \sin (e+f x)+a)^{m-1}}{m (a-a \sin (e+f x))}}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {a^2 \sin (e+f x) (\sin (e+f x) a+a)^{m-2} (m \sin (e+f x) a+2 a)}{(a-a \sin (e+f x))^2}d(a \sin (e+f x))}{m}-\frac {a^2 \sin ^2(e+f x) (a \sin (e+f x)+a)^{m-1}}{m (a-a \sin (e+f x))}}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {a \int \frac {a \sin (e+f x) (\sin (e+f x) a+a)^{m-2} (m \sin (e+f x) a+2 a)}{(a-a \sin (e+f x))^2}d(a \sin (e+f x))}{m}-\frac {a^2 \sin ^2(e+f x) (a \sin (e+f x)+a)^{m-1}}{m (a-a \sin (e+f x))}}{f}\)

\(\Big \downarrow \) 163

\(\displaystyle \frac {\frac {a \left (\frac {(a \sin (e+f x)+a)^{m-1} \left (2 a m \sin (e+f x)+a \left (-m^2-3 m+2\right )\right )}{2 (1-m) (a-a \sin (e+f x))}-\frac {1}{2} a m (m+4) \int \frac {(\sin (e+f x) a+a)^{m-2}}{a-a \sin (e+f x)}d(a \sin (e+f x))\right )}{m}-\frac {a^2 \sin ^2(e+f x) (a \sin (e+f x)+a)^{m-1}}{m (a-a \sin (e+f x))}}{f}\)

\(\Big \downarrow \) 78

\(\displaystyle \frac {\frac {a \left (\frac {m (m+4) (a \sin (e+f x)+a)^{m-1} \operatorname {Hypergeometric2F1}\left (1,m-1,m,\frac {\sin (e+f x) a+a}{2 a}\right )}{4 (1-m)}+\frac {\left (2 a m \sin (e+f x)+a \left (-m^2-3 m+2\right )\right ) (a \sin (e+f x)+a)^{m-1}}{2 (1-m) (a-a \sin (e+f x))}\right )}{m}-\frac {a^2 \sin ^2(e+f x) (a \sin (e+f x)+a)^{m-1}}{m (a-a \sin (e+f x))}}{f}\)

Input:

Int[(a + a*Sin[e + f*x])^m*Tan[e + f*x]^3,x]
 

Output:

(-((a^2*Sin[e + f*x]^2*(a + a*Sin[e + f*x])^(-1 + m))/(m*(a - a*Sin[e + f* 
x]))) + (a*((m*(4 + m)*Hypergeometric2F1[1, -1 + m, m, (a + a*Sin[e + f*x] 
)/(2*a)]*(a + a*Sin[e + f*x])^(-1 + m))/(4*(1 - m)) + ((a + a*Sin[e + f*x] 
)^(-1 + m)*(a*(2 - 3*m - m^2) + 2*a*m*Sin[e + f*x]))/(2*(1 - m)*(a - a*Sin 
[e + f*x]))))/m)/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 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 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 163

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

rule 3042

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

rule 3186

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p 
_.), x_Symbol] :> Simp[1/f   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] && E 
qQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]
 

Maple [F]

\[\int \left (a +\sin \left (f x +e \right ) a \right )^{m} \tan \left (f x +e \right )^{3}d x\]

Input:

int((a+sin(f*x+e)*a)^m*tan(f*x+e)^3,x)
 

Output:

int((a+sin(f*x+e)*a)^m*tan(f*x+e)^3,x)
 

Fricas [F]

\[ \int (a+a \sin (e+f x))^m \tan ^3(e+f x) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \tan \left (f x + e\right )^{3} \,d x } \] Input:

integrate((a+a*sin(f*x+e))^m*tan(f*x+e)^3,x, algorithm="fricas")
 

Output:

integral((a*sin(f*x + e) + a)^m*tan(f*x + e)^3, x)
 

Sympy [F]

\[ \int (a+a \sin (e+f x))^m \tan ^3(e+f x) \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \tan ^{3}{\left (e + f x \right )}\, dx \] Input:

integrate((a+a*sin(f*x+e))**m*tan(f*x+e)**3,x)
 

Output:

Integral((a*(sin(e + f*x) + 1))**m*tan(e + f*x)**3, x)
 

Maxima [F]

\[ \int (a+a \sin (e+f x))^m \tan ^3(e+f x) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \tan \left (f x + e\right )^{3} \,d x } \] Input:

integrate((a+a*sin(f*x+e))^m*tan(f*x+e)^3,x, algorithm="maxima")
 

Output:

integrate((a*sin(f*x + e) + a)^m*tan(f*x + e)^3, x)
 

Giac [F]

\[ \int (a+a \sin (e+f x))^m \tan ^3(e+f x) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \tan \left (f x + e\right )^{3} \,d x } \] Input:

integrate((a+a*sin(f*x+e))^m*tan(f*x+e)^3,x, algorithm="giac")
 

Output:

integrate((a*sin(f*x + e) + a)^m*tan(f*x + e)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+a \sin (e+f x))^m \tan ^3(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^3\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \] Input:

int(tan(e + f*x)^3*(a + a*sin(e + f*x))^m,x)
 

Output:

int(tan(e + f*x)^3*(a + a*sin(e + f*x))^m, x)
 

Reduce [F]

\[ \int (a+a \sin (e+f x))^m \tan ^3(e+f x) \, dx=\int \left (a +a \sin \left (f x +e \right )\right )^{m} \tan \left (f x +e \right )^{3}d x \] Input:

int((a+a*sin(f*x+e))^m*tan(f*x+e)^3,x)
 

Output:

int((sin(e + f*x)*a + a)**m*tan(e + f*x)**3,x)