Integrand size = 23, antiderivative size = 101 \[ \int \left (a+b \sin ^2(c+d x)\right )^p \tan ^4(c+d x) \, dx=\frac {\operatorname {AppellF1}\left (\frac {5}{2},\frac {5}{2},-p,\frac {7}{2},\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \sin ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (1+\frac {b \sin ^2(c+d x)}{a}\right )^{-p} \tan (c+d x)}{5 d} \]
1/5*AppellF1(5/2,5/2,-p,7/2,sin(d*x+c)^2,-b*sin(d*x+c)^2/a)*sin(d*x+c)^4*( a+b*sin(d*x+c)^2)^p*(cos(d*x+c)^2)^(1/2)*tan(d*x+c)/d/((1+b*sin(d*x+c)^2/a )^p)
Time = 4.62 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.01 \[ \int \left (a+b \sin ^2(c+d x)\right )^p \tan ^4(c+d x) \, dx=\frac {\operatorname {AppellF1}\left (\frac {5}{2},\frac {5}{2},-p,\frac {7}{2},\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \sin ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {a+b \sin ^2(c+d x)}{a}\right )^{-p} \tan (c+d x)}{5 d} \]
(AppellF1[5/2, 5/2, -p, 7/2, Sin[c + d*x]^2, -((b*Sin[c + d*x]^2)/a)]*Sqrt [Cos[c + d*x]^2]*Sin[c + d*x]^4*(a + b*Sin[c + d*x]^2)^p*Tan[c + d*x])/(5* d*((a + b*Sin[c + d*x]^2)/a)^p)
Time = 0.30 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3675, 395, 394}
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.
\(\displaystyle \int \tan ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (c+d x)^4 \left (a+b \sin (c+d x)^2\right )^pdx\) |
\(\Big \downarrow \) 3675 |
\(\displaystyle \frac {\sqrt {\cos ^2(c+d x)} \sec (c+d x) \int \frac {\sin ^4(c+d x) \left (b \sin ^2(c+d x)+a\right )^p}{\left (1-\sin ^2(c+d x)\right )^{5/2}}d\sin (c+d x)}{d}\) |
\(\Big \downarrow \) 395 |
\(\displaystyle \frac {\sqrt {\cos ^2(c+d x)} \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {b \sin ^2(c+d x)}{a}+1\right )^{-p} \int \frac {\sin ^4(c+d x) \left (\frac {b \sin ^2(c+d x)}{a}+1\right )^p}{\left (1-\sin ^2(c+d x)\right )^{5/2}}d\sin (c+d x)}{d}\) |
\(\Big \downarrow \) 394 |
\(\displaystyle \frac {\sin ^4(c+d x) \sqrt {\cos ^2(c+d x)} \tan (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {b \sin ^2(c+d x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{2},\frac {5}{2},-p,\frac {7}{2},\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right )}{5 d}\) |
(AppellF1[5/2, 5/2, -p, 7/2, Sin[c + d*x]^2, -((b*Sin[c + d*x]^2)/a)]*Sqrt [Cos[c + d*x]^2]*Sin[c + d*x]^4*(a + b*Sin[c + d*x]^2)^p*Tan[c + d*x])/(5* d*(1 + (b*Sin[c + d*x]^2)/a)^p)
3.6.48.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 , -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^ FracPart[p]) Int[(e*x)^m*(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ [{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && !(IntegerQ[p] || GtQ[a, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ (m_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff^(m + 1 )*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])) Subst[Int[x^m*((a + b*ff^2*x^2) ^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b , e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
\[\int {\left (a +\left (\sin ^{2}\left (d x +c \right )\right ) b \right )}^{p} \left (\tan ^{4}\left (d x +c \right )\right )d x\]
\[ \int \left (a+b \sin ^2(c+d x)\right )^p \tan ^4(c+d x) \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \tan \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int \left (a+b \sin ^2(c+d x)\right )^p \tan ^4(c+d x) \, dx=\text {Timed out} \]
\[ \int \left (a+b \sin ^2(c+d x)\right )^p \tan ^4(c+d x) \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \tan \left (d x + c\right )^{4} \,d x } \]
\[ \int \left (a+b \sin ^2(c+d x)\right )^p \tan ^4(c+d x) \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \tan \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int \left (a+b \sin ^2(c+d x)\right )^p \tan ^4(c+d x) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^4\,{\left (b\,{\sin \left (c+d\,x\right )}^2+a\right )}^p \,d x \]