Integrand size = 21, antiderivative size = 119 \[ \int (a+a \sec (c+d x)) (e \sin (c+d x))^m \, dx=\frac {a \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) (e \sin (c+d x))^{1+m}}{d e (1+m) \sqrt {\cos ^2(c+d x)}}+\frac {a \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) (e \sin (c+d x))^{1+m}}{d e (1+m)} \]
a*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)+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)
Time = 0.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.82 \[ \int (a+a \sec (c+d x)) (e \sin (c+d x))^m \, dx=\frac {a (e \sin (c+d x))^m \left (\operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) \sin (c+d x)+\sqrt {\cos ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) \tan (c+d x)\right )}{d (1+m)} \]
(a*(e*Sin[c + d*x])^m*(Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x] + Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[1/2, (1 + m) /2, (3 + m)/2, Sin[c + d*x]^2]*Tan[c + d*x]))/(d*(1 + m))
Time = 0.49 (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.
\(\displaystyle \int (a \sec (c+d x)+a) (e \sin (c+d x))^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right ) \left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^mdx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int -\left (\sec (c+d x) (a (-\cos (c+d x))-a) (e \sin (c+d x))^m\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\left ((\cos (c+d x) a+a) \sec (c+d x) (e \sin (c+d x))^m\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \sec (c+d x) (a \cos (c+d x)+a) (e \sin (c+d x))^mdx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right ) \left (-e \cos \left (c+d x+\frac {\pi }{2}\right )\right )^m}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3317 |
\(\displaystyle a \int (e \sin (c+d x))^mdx+a \int \sec (c+d x) (e \sin (c+d x))^mdx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int (e \sin (c+d x))^mdx+a \int \frac {(e \sin (c+d x))^m}{\cos (c+d x)}dx\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle \frac {a \int \frac {e^2 (e \sin (c+d x))^m}{e^2-e^2 \sin ^2(c+d x)}d(e \sin (c+d x))}{d e}+a \int (e \sin (c+d x))^mdx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a e \int \frac {(e \sin (c+d x))^m}{e^2-e^2 \sin ^2(c+d x)}d(e \sin (c+d x))}{d}+a \int (e \sin (c+d x))^mdx\) |
\(\Big \downarrow \) 278 |
\(\displaystyle a \int (e \sin (c+d x))^mdx+\frac {a (e \sin (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},\sin ^2(c+d x)\right )}{d e (m+1)}\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle \frac {a (e \sin (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},\sin ^2(c+d x)\right )}{d e (m+1)}+\frac {a \cos (c+d x) (e \sin (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(c+d x)\right )}{d e (m+1) \sqrt {\cos ^2(c+d x)}}\) |
(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]) + (a*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))
3.2.36.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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])
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]
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]
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]
\[\int \left (a +a \sec \left (d x +c \right )\right ) \left (e \sin \left (d x +c \right )\right )^{m}d x\]
\[ \int (a+a \sec (c+d x)) (e \sin (c+d x))^m \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{m} \,d x } \]
\[ \int (a+a \sec (c+d x)) (e \sin (c+d x))^m \, dx=a \left (\int \left (e \sin {\left (c + d x \right )}\right )^{m}\, dx + \int \left (e \sin {\left (c + d x \right )}\right )^{m} \sec {\left (c + d x \right )}\, dx\right ) \]
\[ \int (a+a \sec (c+d x)) (e \sin (c+d x))^m \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{m} \,d x } \]
\[ \int (a+a \sec (c+d x)) (e \sin (c+d x))^m \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{m} \,d x } \]
Timed out. \[ \int (a+a \sec (c+d x)) (e \sin (c+d x))^m \, dx=\int {\left (e\,\sin \left (c+d\,x\right )\right )}^m\,\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right ) \,d x \]