Integrand size = 25, antiderivative size = 177 \[ \int \sqrt {a+b \sin ^2(e+f x)} \tan ^5(e+f x) \, dx=\frac {\left (8 a^2+24 a b+15 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a+b}}\right )}{8 (a+b)^{3/2} f}-\frac {\left (8 a^2+24 a b+15 b^2\right ) \sqrt {a+b \sin ^2(e+f x)}}{8 (a+b)^2 f}-\frac {(8 a+7 b) \sec ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{8 (a+b)^2 f}+\frac {\sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 (a+b) f} \]
1/8*(8*a^2+24*a*b+15*b^2)*arctanh((a+b*sin(f*x+e)^2)^(1/2)/(a+b)^(1/2))/(a +b)^(3/2)/f-1/8*(8*a+7*b)*sec(f*x+e)^2*(a+b*sin(f*x+e)^2)^(3/2)/(a+b)^2/f+ 1/4*sec(f*x+e)^4*(a+b*sin(f*x+e)^2)^(3/2)/(a+b)/f-1/8*(8*a^2+24*a*b+15*b^2 )*(a+b*sin(f*x+e)^2)^(1/2)/(a+b)^2/f
Time = 0.61 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.81 \[ \int \sqrt {a+b \sin ^2(e+f x)} \tan ^5(e+f x) \, dx=\frac {-\left ((8 a+7 b) \sec ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}\right )+2 (a+b) \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}+\left (8 a^2+24 a b+15 b^2\right ) \left (\sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a+b}}\right )-\sqrt {a+b \sin ^2(e+f x)}\right )}{8 (a+b)^2 f} \]
(-((8*a + 7*b)*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2)) + 2*(a + b)*Se c[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2) + (8*a^2 + 24*a*b + 15*b^2)*(Sqr t[a + b]*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]] - Sqrt[a + b*Sin[ e + f*x]^2]))/(8*(a + b)^2*f)
Time = 0.35 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3673, 100, 27, 87, 60, 73, 221}
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.
\(\displaystyle \int \tan ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (e+f x)^5 \sqrt {a+b \sin (e+f x)^2}dx\) |
\(\Big \downarrow \) 3673 |
\(\displaystyle \frac {\int \frac {\sin ^4(e+f x) \sqrt {b \sin ^2(e+f x)+a}}{\left (1-\sin ^2(e+f x)\right )^3}d\sin ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 (a+b) \left (1-\sin ^2(e+f x)\right )^2}-\frac {\int \frac {\sqrt {b \sin ^2(e+f x)+a} \left (4 (a+b) \sin ^2(e+f x)+4 a+3 b\right )}{2 \left (1-\sin ^2(e+f x)\right )^2}d\sin ^2(e+f x)}{2 (a+b)}}{2 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 (a+b) \left (1-\sin ^2(e+f x)\right )^2}-\frac {\int \frac {\sqrt {b \sin ^2(e+f x)+a} \left (4 (a+b) \sin ^2(e+f x)+4 a+3 b\right )}{\left (1-\sin ^2(e+f x)\right )^2}d\sin ^2(e+f x)}{4 (a+b)}}{2 f}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 (a+b) \left (1-\sin ^2(e+f x)\right )^2}-\frac {\frac {(8 a+7 b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{(a+b) \left (1-\sin ^2(e+f x)\right )}-\frac {\left (8 a^2+24 a b+15 b^2\right ) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{1-\sin ^2(e+f x)}d\sin ^2(e+f x)}{2 (a+b)}}{4 (a+b)}}{2 f}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 (a+b) \left (1-\sin ^2(e+f x)\right )^2}-\frac {\frac {(8 a+7 b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{(a+b) \left (1-\sin ^2(e+f x)\right )}-\frac {\left (8 a^2+24 a b+15 b^2\right ) \left ((a+b) \int \frac {1}{\left (1-\sin ^2(e+f x)\right ) \sqrt {b \sin ^2(e+f x)+a}}d\sin ^2(e+f x)-2 \sqrt {a+b \sin ^2(e+f x)}\right )}{2 (a+b)}}{4 (a+b)}}{2 f}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 (a+b) \left (1-\sin ^2(e+f x)\right )^2}-\frac {\frac {(8 a+7 b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{(a+b) \left (1-\sin ^2(e+f x)\right )}-\frac {\left (8 a^2+24 a b+15 b^2\right ) \left (\frac {2 (a+b) \int \frac {1}{\frac {a+b}{b}-\frac {\sin ^4(e+f x)}{b}}d\sqrt {b \sin ^2(e+f x)+a}}{b}-2 \sqrt {a+b \sin ^2(e+f x)}\right )}{2 (a+b)}}{4 (a+b)}}{2 f}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 (a+b) \left (1-\sin ^2(e+f x)\right )^2}-\frac {\frac {(8 a+7 b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{(a+b) \left (1-\sin ^2(e+f x)\right )}-\frac {\left (8 a^2+24 a b+15 b^2\right ) \left (2 \sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a+b}}\right )-2 \sqrt {a+b \sin ^2(e+f x)}\right )}{2 (a+b)}}{4 (a+b)}}{2 f}\) |
((a + b*Sin[e + f*x]^2)^(3/2)/(2*(a + b)*(1 - Sin[e + f*x]^2)^2) - (((8*a + 7*b)*(a + b*Sin[e + f*x]^2)^(3/2))/((a + b)*(1 - Sin[e + f*x]^2)) - ((8* a^2 + 24*a*b + 15*b^2)*(2*Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/S qrt[a + b]] - 2*Sqrt[a + b*Sin[e + f*x]^2]))/(2*(a + b)))/(4*(a + b)))/(2* f)
3.5.88.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(720\) vs. \(2(157)=314\).
Time = 2.15 (sec) , antiderivative size = 721, normalized size of antiderivative = 4.07
method | result | size |
default | \(\frac {\left (-16 \left (a +b \right )^{\frac {3}{2}} \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\, a^{2}-48 \left (a +b \right )^{\frac {3}{2}} \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\, a b -30 b^{2} \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (a +b \right )^{\frac {3}{2}}+8 \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right ) a^{4}+40 \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right ) a^{3} b +71 \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right ) a^{2} b^{2}+54 \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right ) a \,b^{3}+15 \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right ) b^{4}+8 \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right ) a^{4}+40 \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right ) a^{3} b +71 \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right ) a^{2} b^{2}+54 \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right ) a \,b^{3}+15 \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right ) b^{4}\right ) \left (\cos ^{4}\left (f x +e \right )\right )-2 \left (a +b \right )^{\frac {3}{2}} {\left (a +b -b \left (\cos ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}} \left (8 a +7 b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+4 \left (a +b \right )^{\frac {3}{2}} {\left (a +b -b \left (\cos ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}} a +4 b {\left (a +b -b \left (\cos ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}} \left (a +b \right )^{\frac {3}{2}}}{16 \left (a +b \right )^{\frac {3}{2}} \cos \left (f x +e \right )^{4} \left (a^{2}+2 a b +b^{2}\right ) f}\) | \(721\) |
1/16*((-16*(a+b)^(3/2)*(a+b-b*cos(f*x+e)^2)^(1/2)*a^2-48*(a+b)^(3/2)*(a+b- b*cos(f*x+e)^2)^(1/2)*a*b-30*b^2*(a+b-b*cos(f*x+e)^2)^(1/2)*(a+b)^(3/2)+8* ln(2/(1+sin(f*x+e))*((a+b)^(1/2)*(a+b-b*cos(f*x+e)^2)^(1/2)-b*sin(f*x+e)+a ))*a^4+40*ln(2/(1+sin(f*x+e))*((a+b)^(1/2)*(a+b-b*cos(f*x+e)^2)^(1/2)-b*si n(f*x+e)+a))*a^3*b+71*ln(2/(1+sin(f*x+e))*((a+b)^(1/2)*(a+b-b*cos(f*x+e)^2 )^(1/2)-b*sin(f*x+e)+a))*a^2*b^2+54*ln(2/(1+sin(f*x+e))*((a+b)^(1/2)*(a+b- b*cos(f*x+e)^2)^(1/2)-b*sin(f*x+e)+a))*a*b^3+15*ln(2/(1+sin(f*x+e))*((a+b) ^(1/2)*(a+b-b*cos(f*x+e)^2)^(1/2)-b*sin(f*x+e)+a))*b^4+8*ln(2/(sin(f*x+e)- 1)*((a+b)^(1/2)*(a+b-b*cos(f*x+e)^2)^(1/2)+b*sin(f*x+e)+a))*a^4+40*ln(2/(s in(f*x+e)-1)*((a+b)^(1/2)*(a+b-b*cos(f*x+e)^2)^(1/2)+b*sin(f*x+e)+a))*a^3* b+71*ln(2/(sin(f*x+e)-1)*((a+b)^(1/2)*(a+b-b*cos(f*x+e)^2)^(1/2)+b*sin(f*x +e)+a))*a^2*b^2+54*ln(2/(sin(f*x+e)-1)*((a+b)^(1/2)*(a+b-b*cos(f*x+e)^2)^( 1/2)+b*sin(f*x+e)+a))*a*b^3+15*ln(2/(sin(f*x+e)-1)*((a+b)^(1/2)*(a+b-b*cos (f*x+e)^2)^(1/2)+b*sin(f*x+e)+a))*b^4)*cos(f*x+e)^4-2*(a+b)^(3/2)*(a+b-b*c os(f*x+e)^2)^(3/2)*(8*a+7*b)*cos(f*x+e)^2+4*(a+b)^(3/2)*(a+b-b*cos(f*x+e)^ 2)^(3/2)*a+4*b*(a+b-b*cos(f*x+e)^2)^(3/2)*(a+b)^(3/2))/(a+b)^(3/2)/cos(f*x +e)^4/(a^2+2*a*b+b^2)/f
Time = 0.84 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.00 \[ \int \sqrt {a+b \sin ^2(e+f x)} \tan ^5(e+f x) \, dx=\left [\frac {{\left (8 \, a^{2} + 24 \, a b + 15 \, b^{2}\right )} \sqrt {a + b} \cos \left (f x + e\right )^{4} \log \left (\frac {b \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a + b} - 2 \, a - 2 \, b}{\cos \left (f x + e\right )^{2}}\right ) - 2 \, {\left (8 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + {\left (8 \, a^{2} + 17 \, a b + 9 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, a^{2} - 4 \, a b - 2 \, b^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{16 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} f \cos \left (f x + e\right )^{4}}, -\frac {{\left (8 \, a^{2} + 24 \, a b + 15 \, b^{2}\right )} \sqrt {-a - b} \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a - b}}{a + b}\right ) \cos \left (f x + e\right )^{4} + {\left (8 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + {\left (8 \, a^{2} + 17 \, a b + 9 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, a^{2} - 4 \, a b - 2 \, b^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{8 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} f \cos \left (f x + e\right )^{4}}\right ] \]
[1/16*((8*a^2 + 24*a*b + 15*b^2)*sqrt(a + b)*cos(f*x + e)^4*log((b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b) - 2*a - 2*b)/cos(f *x + e)^2) - 2*(8*(a^2 + 2*a*b + b^2)*cos(f*x + e)^4 + (8*a^2 + 17*a*b + 9 *b^2)*cos(f*x + e)^2 - 2*a^2 - 4*a*b - 2*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4), -1/8*((8*a^2 + 24*a*b + 15*b^ 2)*sqrt(-a - b)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(a + b ))*cos(f*x + e)^4 + (8*(a^2 + 2*a*b + b^2)*cos(f*x + e)^4 + (8*a^2 + 17*a* b + 9*b^2)*cos(f*x + e)^2 - 2*a^2 - 4*a*b - 2*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4)]
\[ \int \sqrt {a+b \sin ^2(e+f x)} \tan ^5(e+f x) \, dx=\int \sqrt {a + b \sin ^{2}{\left (e + f x \right )}} \tan ^{5}{\left (e + f x \right )}\, dx \]
Time = 0.30 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.30 \[ \int \sqrt {a+b \sin ^2(e+f x)} \tan ^5(e+f x) \, dx=-\frac {16 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} b^{3} + \frac {{\left (8 \, a^{2} b^{3} + 24 \, a b^{4} + 15 \, b^{5}\right )} \log \left (\frac {\sqrt {b \sin \left (f x + e\right )^{2} + a} - \sqrt {a + b}}{\sqrt {b \sin \left (f x + e\right )^{2} + a} + \sqrt {a + b}}\right )}{{\left (a + b\right )}^{\frac {3}{2}}} - \frac {2 \, {\left ({\left (8 \, a b^{4} + 9 \, b^{5}\right )} {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} - {\left (8 \, a^{2} b^{4} + 15 \, a b^{5} + 7 \, b^{6}\right )} \sqrt {b \sin \left (f x + e\right )^{2} + a}\right )}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{2} {\left (a + b\right )} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} - 2 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )} {\left (a^{2} + 2 \, a b + b^{2}\right )}}}{16 \, b^{3} f} \]
-1/16*(16*sqrt(b*sin(f*x + e)^2 + a)*b^3 + (8*a^2*b^3 + 24*a*b^4 + 15*b^5) *log((sqrt(b*sin(f*x + e)^2 + a) - sqrt(a + b))/(sqrt(b*sin(f*x + e)^2 + a ) + sqrt(a + b)))/(a + b)^(3/2) - 2*((8*a*b^4 + 9*b^5)*(b*sin(f*x + e)^2 + a)^(3/2) - (8*a^2*b^4 + 15*a*b^5 + 7*b^6)*sqrt(b*sin(f*x + e)^2 + a))/((b *sin(f*x + e)^2 + a)^2*(a + b) + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*(b*sin( f*x + e)^2 + a)*(a^2 + 2*a*b + b^2)))/(b^3*f)
Leaf count of result is larger than twice the leaf count of optimal. 2646 vs. \(2 (157) = 314\).
Time = 2.13 (sec) , antiderivative size = 2646, normalized size of antiderivative = 14.95 \[ \int \sqrt {a+b \sin ^2(e+f x)} \tan ^5(e+f x) \, dx=\text {Too large to display} \]
-1/4*((8*a^2 + 24*a*b + 15*b^2)*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^ 2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1 /2*f*x + 1/2*e)^2 + a) - sqrt(a))/sqrt(-a - b))/((a + b)*sqrt(-a - b)) - 1 6*((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*t an(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b - sqrt(a)*b)/(( sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1 /2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 + 2*(sqrt(a)*tan(1/ 2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e )^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) + a + 4*b) - 2*(8*(sqrt(a)* tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^7*a^2 + 16*(sqrt(a)*tan(1/2*f *x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^7*a*b + 7*(sqrt(a)*tan(1/2*f*x + 1/2*e )^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan (1/2*f*x + 1/2*e)^2 + a))^7*b^2 - 56*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqr t(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^6*a^(5/2) - 80*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*t an(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2 *e)^2 + a))^6*a^(3/2)*b - 17*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan( 1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2...
Timed out. \[ \int \sqrt {a+b \sin ^2(e+f x)} \tan ^5(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^5\,\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a} \,d x \]