\(\int (a+b \sec (c+d x)) (e \sin (c+d x))^m \, dx\) [259]

Optimal result

Integrand size = 21, antiderivative size = 119 \[ \int (a+b \sec (c+d x)) (e \sin (c+d x))^m \, dx=\frac {a \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) (e \sin (c+d x))^{1+m}}{d e (1+m) \sqrt {\cos ^2(c+d x)}}+\frac {b \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) (e \sin (c+d x))^{1+m}}{d e (1+m)} \] Output:

a*cos(d*x+c)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],sin(d*x+c)^2)*(e*sin(d 
*x+c))^(1+m)/d/e/(1+m)/(cos(d*x+c)^2)^(1/2)+b*hypergeom([1, 1/2+1/2*m],[3/ 
2+1/2*m],sin(d*x+c)^2)*(e*sin(d*x+c))^(1+m)/d/e/(1+m)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.82 \[ \int (a+b \sec (c+d x)) (e \sin (c+d x))^m \, dx=\frac {\left (a \sqrt {\cos ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right )+b \cos (c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right )\right ) (e \sin (c+d x))^m \tan (c+d x)}{d (1+m)} \] Input:

Integrate[(a + b*Sec[c + d*x])*(e*Sin[c + d*x])^m,x]
 

Output:

((a*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[ 
c + d*x]^2] + b*Cos[c + d*x]*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Si 
n[c + d*x]^2])*(e*Sin[c + d*x])^m*Tan[c + d*x])/(d*(1 + m))
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 4360, 25, 25, 3042, 3317, 3042, 3044, 27, 278, 3122}

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.

\(\displaystyle \int (a+b \sec (c+d x)) (e \sin (c+d x))^m \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right ) \left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^mdx\)

\(\Big \downarrow \) 4360

\(\displaystyle \int -\left (\sec (c+d x) (-a \cos (c+d x)-b) (e \sin (c+d x))^m\right )dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int -\left ((b+a \cos (c+d x)) \sec (c+d x) (e \sin (c+d x))^m\right )dx\)

\(\Big \downarrow \) 25

\(\displaystyle \int \sec (c+d x) (a \cos (c+d x)+b) (e \sin (c+d x))^mdx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+b\right ) \left (-e \cos \left (c+d x+\frac {\pi }{2}\right )\right )^m}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 3317

\(\displaystyle a \int (e \sin (c+d x))^mdx+b \int \sec (c+d x) (e \sin (c+d x))^mdx\)

\(\Big \downarrow \) 3042

\(\displaystyle a \int (e \sin (c+d x))^mdx+b \int \frac {(e \sin (c+d x))^m}{\cos (c+d x)}dx\)

\(\Big \downarrow \) 3044

\(\displaystyle a \int (e \sin (c+d x))^mdx+\frac {b \int \frac {e^2 (e \sin (c+d x))^m}{e^2-e^2 \sin ^2(c+d x)}d(e \sin (c+d x))}{d e}\)

\(\Big \downarrow \) 27

\(\displaystyle a \int (e \sin (c+d x))^mdx+\frac {b e \int \frac {(e \sin (c+d x))^m}{e^2-e^2 \sin ^2(c+d x)}d(e \sin (c+d x))}{d}\)

\(\Big \downarrow \) 278

\(\displaystyle a \int (e \sin (c+d x))^mdx+\frac {b (e \sin (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},\sin ^2(c+d x)\right )}{d e (m+1)}\)

\(\Big \downarrow \) 3122

\(\displaystyle \frac {a \cos (c+d x) (e \sin (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(c+d x)\right )}{d e (m+1) \sqrt {\cos ^2(c+d x)}}+\frac {b (e \sin (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},\sin ^2(c+d x)\right )}{d e (m+1)}\)

Input:

Int[(a + b*Sec[c + d*x])*(e*Sin[c + d*x])^m,x]
 

Output:

(a*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^ 
2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (b*Hyper 
geometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 
+ m))/(d*e*(1 + m))
 

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 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 3042

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

rule 3044

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ 
Symbol] :> Simp[1/(a*f)   Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a 
*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&  !(I 
ntegerQ[(m - 1)/2] && LtQ[0, m, n])
 

rule 3122

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( 
b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2 
F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] 
 &&  !IntegerQ[2*n]
 

rule 3317

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a   Int[(g*Co 
s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d   Int[(g*Cos[e + f*x])^ 
p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
 

rule 4360

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si 
n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
 

Maple [F]

\[\int \left (a +b \sec \left (d x +c \right )\right ) \left (e \sin \left (d x +c \right )\right )^{m}d x\]

Input:

int((a+b*sec(d*x+c))*(e*sin(d*x+c))^m,x)
 

Output:

int((a+b*sec(d*x+c))*(e*sin(d*x+c))^m,x)
 

Fricas [F]

\[ \int (a+b \sec (c+d x)) (e \sin (c+d x))^m \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{m} \,d x } \] Input:

integrate((a+b*sec(d*x+c))*(e*sin(d*x+c))^m,x, algorithm="fricas")
 

Output:

integral((b*sec(d*x + c) + a)*(e*sin(d*x + c))^m, x)
 

Sympy [F]

\[ \int (a+b \sec (c+d x)) (e \sin (c+d x))^m \, dx=\int \left (e \sin {\left (c + d x \right )}\right )^{m} \left (a + b \sec {\left (c + d x \right )}\right )\, dx \] Input:

integrate((a+b*sec(d*x+c))*(e*sin(d*x+c))**m,x)
 

Output:

Integral((e*sin(c + d*x))**m*(a + b*sec(c + d*x)), x)
 

Maxima [F]

\[ \int (a+b \sec (c+d x)) (e \sin (c+d x))^m \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{m} \,d x } \] Input:

integrate((a+b*sec(d*x+c))*(e*sin(d*x+c))^m,x, algorithm="maxima")
 

Output:

integrate((b*sec(d*x + c) + a)*(e*sin(d*x + c))^m, x)
                                                                                    
                                                                                    
 

Giac [F]

\[ \int (a+b \sec (c+d x)) (e \sin (c+d x))^m \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{m} \,d x } \] Input:

integrate((a+b*sec(d*x+c))*(e*sin(d*x+c))^m,x, algorithm="giac")
 

Output:

integrate((b*sec(d*x + c) + a)*(e*sin(d*x + c))^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+b \sec (c+d x)) (e \sin (c+d x))^m \, dx=\int {\left (e\,\sin \left (c+d\,x\right )\right )}^m\,\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right ) \,d x \] Input:

int((e*sin(c + d*x))^m*(a + b/cos(c + d*x)),x)
 

Output:

int((e*sin(c + d*x))^m*(a + b/cos(c + d*x)), x)
 

Reduce [F]

\[ \int (a+b \sec (c+d x)) (e \sin (c+d x))^m \, dx=e^{m} \left (\left (\int \sin \left (d x +c \right )^{m}d x \right ) a +\left (\int \sin \left (d x +c \right )^{m} \sec \left (d x +c \right )d x \right ) b \right ) \] Input:

int((a+b*sec(d*x+c))*(e*sin(d*x+c))^m,x)
 

Output:

e**m*(int(sin(c + d*x)**m,x)*a + int(sin(c + d*x)**m*sec(c + d*x),x)*b)