Integrand size = 19, antiderivative size = 72 \[ \int (a+a \sin (e+f x))^m \tan (e+f x) \, dx=-\frac {(a+a \sin (e+f x))^m}{2 f m}+\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^{1+m}}{4 a f (1+m)} \] Output:
-1/2*(a+a*sin(f*x+e))^m/f/m+1/4*hypergeom([1, 1+m],[2+m],1/2+1/2*sin(f*x+e ))*(a+a*sin(f*x+e))^(1+m)/a/f/(1+m)
Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.88 \[ \int (a+a \sin (e+f x))^m \tan (e+f x) \, dx=\frac {(a (1+\sin (e+f x)))^m \left (-2 (1+m)+m \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {1}{2} (1+\sin (e+f x))\right ) (1+\sin (e+f x))\right )}{4 f m (1+m)} \] Input:
Integrate[(a + a*Sin[e + f*x])^m*Tan[e + f*x],x]
Output:
((a*(1 + Sin[e + f*x]))^m*(-2*(1 + m) + m*Hypergeometric2F1[1, 1 + m, 2 + m, (1 + Sin[e + f*x])/2]*(1 + Sin[e + f*x])))/(4*f*m*(1 + m))
Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3186, 88, 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.
\(\displaystyle \int \tan (e+f x) (a \sin (e+f x)+a)^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (e+f x) (a \sin (e+f x)+a)^mdx\) |
\(\Big \downarrow \) 3186 |
\(\displaystyle \frac {\int \frac {a \sin (e+f x) (\sin (e+f x) a+a)^{m-1}}{a-a \sin (e+f x)}d(a \sin (e+f x))}{f}\) |
\(\Big \downarrow \) 88 |
\(\displaystyle \frac {\frac {1}{2} \int \frac {(\sin (e+f x) a+a)^m}{a-a \sin (e+f x)}d(a \sin (e+f x))-\frac {(a \sin (e+f x)+a)^m}{2 m}}{f}\) |
\(\Big \downarrow \) 78 |
\(\displaystyle \frac {\frac {(a \sin (e+f x)+a)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {\sin (e+f x) a+a}{2 a}\right )}{4 a (m+1)}-\frac {(a \sin (e+f x)+a)^m}{2 m}}{f}\) |
Input:
Int[(a + a*Sin[e + f*x])^m*Tan[e + f*x],x]
Output:
(-1/2*(a + a*Sin[e + f*x])^m/m + (Hypergeometric2F1[1, 1 + m, 2 + m, (a + a*Sin[e + f*x])/(2*a)]*(a + a*Sin[e + f*x])^(1 + m))/(4*a*(1 + m)))/f
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]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && !RationalQ[p] && SumSimpl erQ[p, 1]
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]
\[\int \left (a +\sin \left (f x +e \right ) a \right )^{m} \tan \left (f x +e \right )d x\]
Input:
int((a+sin(f*x+e)*a)^m*tan(f*x+e),x)
Output:
int((a+sin(f*x+e)*a)^m*tan(f*x+e),x)
\[ \int (a+a \sin (e+f x))^m \tan (e+f x) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \tan \left (f x + e\right ) \,d x } \] Input:
integrate((a+a*sin(f*x+e))^m*tan(f*x+e),x, algorithm="fricas")
Output:
integral((a*sin(f*x + e) + a)^m*tan(f*x + e), x)
\[ \int (a+a \sin (e+f x))^m \tan (e+f x) \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \tan {\left (e + f x \right )}\, dx \] Input:
integrate((a+a*sin(f*x+e))**m*tan(f*x+e),x)
Output:
Integral((a*(sin(e + f*x) + 1))**m*tan(e + f*x), x)
\[ \int (a+a \sin (e+f x))^m \tan (e+f x) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \tan \left (f x + e\right ) \,d x } \] Input:
integrate((a+a*sin(f*x+e))^m*tan(f*x+e),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^m*tan(f*x + e), x)
\[ \int (a+a \sin (e+f x))^m \tan (e+f x) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \tan \left (f x + e\right ) \,d x } \] Input:
integrate((a+a*sin(f*x+e))^m*tan(f*x+e),x, algorithm="giac")
Output:
integrate((a*sin(f*x + e) + a)^m*tan(f*x + e), x)
Timed out. \[ \int (a+a \sin (e+f x))^m \tan (e+f x) \, dx=\int \mathrm {tan}\left (e+f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \] Input:
int(tan(e + f*x)*(a + a*sin(e + f*x))^m,x)
Output:
int(tan(e + f*x)*(a + a*sin(e + f*x))^m, x)
\[ \int (a+a \sin (e+f x))^m \tan (e+f x) \, dx=\int \left (a +a \sin \left (f x +e \right )\right )^{m} \tan \left (f x +e \right )d x \] Input:
int((a+a*sin(f*x+e))^m*tan(f*x+e),x)
Output:
int((sin(e + f*x)*a + a)**m*tan(e + f*x),x)