Integrand size = 21, antiderivative size = 59 \[ \int \left (a+b \sin ^2(c+d x)\right )^p \tan (c+d x) \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {a+b \sin ^2(c+d x)}{a+b}\right ) \left (a+b \sin ^2(c+d x)\right )^{1+p}}{2 (a+b) d (1+p)} \] Output:
1/2*hypergeom([1, p+1],[2+p],(a+b*sin(d*x+c)^2)/(a+b))*(a+b*sin(d*x+c)^2)^ (p+1)/(a+b)/d/(p+1)
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.03 \[ \int \left (a+b \sin ^2(c+d x)\right )^p \tan (c+d x) \, dx=\frac {\left (a+b-b \cos ^2(c+d x)\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1-\frac {b \cos ^2(c+d x)}{a+b}\right )}{2 (a+b) d (1+p)} \] Input:
Integrate[(a + b*Sin[c + d*x]^2)^p*Tan[c + d*x],x]
Output:
((a + b - b*Cos[c + d*x]^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (b*Cos[c + d*x]^2)/(a + b)])/(2*(a + b)*d*(1 + p))
Time = 0.23 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3673, 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.
\(\displaystyle \int \tan (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (c+d x) \left (a+b \sin (c+d x)^2\right )^pdx\) |
\(\Big \downarrow \) 3673 |
\(\displaystyle \frac {\int \frac {\left (b \sin ^2(c+d x)+a\right )^p}{1-\sin ^2(c+d x)}d\sin ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 78 |
\(\displaystyle \frac {\left (a+b \sin ^2(c+d x)\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b \sin ^2(c+d x)+a}{a+b}\right )}{2 d (p+1) (a+b)}\) |
Input:
Int[(a + b*Sin[c + d*x]^2)^p*Tan[c + d*x],x]
Output:
(Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sin[c + d*x]^2)/(a + b)]*(a + b *Sin[c + d*x]^2)^(1 + p))/(2*(a + b)*d*(1 + p))
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_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ (m_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, Simp[ff^((m + 1)/2)/(2*f) Subst[Int[x^((m - 1)/2)*((a + b*ff*x)^p/(1 - ff*x)^((m + 1 )/2)), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && Integ erQ[(m - 1)/2]
\[\int \left (a +b \sin \left (d x +c \right )^{2}\right )^{p} \tan \left (d x +c \right )d x\]
Input:
int((a+b*sin(d*x+c)^2)^p*tan(d*x+c),x)
Output:
int((a+b*sin(d*x+c)^2)^p*tan(d*x+c),x)
\[ \int \left (a+b \sin ^2(c+d x)\right )^p \tan (c+d x) \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \tan \left (d x + c\right ) \,d x } \] Input:
integrate((a+b*sin(d*x+c)^2)^p*tan(d*x+c),x, algorithm="fricas")
Output:
integral((-b*cos(d*x + c)^2 + a + b)^p*tan(d*x + c), x)
\[ \int \left (a+b \sin ^2(c+d x)\right )^p \tan (c+d x) \, dx=\int \left (a + b \sin ^{2}{\left (c + d x \right )}\right )^{p} \tan {\left (c + d x \right )}\, dx \] Input:
integrate((a+b*sin(d*x+c)**2)**p*tan(d*x+c),x)
Output:
Integral((a + b*sin(c + d*x)**2)**p*tan(c + d*x), x)
\[ \int \left (a+b \sin ^2(c+d x)\right )^p \tan (c+d x) \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \tan \left (d x + c\right ) \,d x } \] Input:
integrate((a+b*sin(d*x+c)^2)^p*tan(d*x+c),x, algorithm="maxima")
Output:
integrate((b*sin(d*x + c)^2 + a)^p*tan(d*x + c), x)
\[ \int \left (a+b \sin ^2(c+d x)\right )^p \tan (c+d x) \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \tan \left (d x + c\right ) \,d x } \] Input:
integrate((a+b*sin(d*x+c)^2)^p*tan(d*x+c),x, algorithm="giac")
Output:
integrate((b*sin(d*x + c)^2 + a)^p*tan(d*x + c), x)
Timed out. \[ \int \left (a+b \sin ^2(c+d x)\right )^p \tan (c+d x) \, dx=\int \mathrm {tan}\left (c+d\,x\right )\,{\left (b\,{\sin \left (c+d\,x\right )}^2+a\right )}^p \,d x \] Input:
int(tan(c + d*x)*(a + b*sin(c + d*x)^2)^p,x)
Output:
int(tan(c + d*x)*(a + b*sin(c + d*x)^2)^p, x)
\[ \int \left (a+b \sin ^2(c+d x)\right )^p \tan (c+d x) \, dx=\int \left (\sin \left (d x +c \right )^{2} b +a \right )^{p} \tan \left (d x +c \right )d x \] Input:
int((a+b*sin(d*x+c)^2)^p*tan(d*x+c),x)
Output:
int((sin(c + d*x)**2*b + a)**p*tan(c + d*x),x)