Integrand size = 25, antiderivative size = 414 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2} \, dx=-\frac {3 a^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{4 \sqrt {2} \sqrt {b} f}+\frac {3 a^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{4 \sqrt {2} \sqrt {b} f}+\frac {3 a^{5/2} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{8 \sqrt {2} \sqrt {b} f}-\frac {3 a^{5/2} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{8 \sqrt {2} \sqrt {b} f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}} \]
-1/2*a*b*(a*sin(f*x+e))^(3/2)/f/(b*sec(f*x+e))^(1/2)-3/8*a^(5/2)*arctan(1- 2^(1/2)*b^(1/2)*(a*sin(f*x+e))^(1/2)/a^(1/2)/(b*cos(f*x+e))^(1/2))*(b*cos( f*x+e))^(1/2)*(b*sec(f*x+e))^(1/2)/f*2^(1/2)/b^(1/2)+3/8*a^(5/2)*arctan(1+ 2^(1/2)*b^(1/2)*(a*sin(f*x+e))^(1/2)/a^(1/2)/(b*cos(f*x+e))^(1/2))*(b*cos( f*x+e))^(1/2)*(b*sec(f*x+e))^(1/2)/f*2^(1/2)/b^(1/2)+3/16*a^(5/2)*ln(a^(1/ 2)-2^(1/2)*b^(1/2)*(a*sin(f*x+e))^(1/2)/(b*cos(f*x+e))^(1/2)+a^(1/2)*tan(f *x+e))*(b*cos(f*x+e))^(1/2)*(b*sec(f*x+e))^(1/2)/f*2^(1/2)/b^(1/2)-3/16*a^ (5/2)*ln(a^(1/2)+2^(1/2)*b^(1/2)*(a*sin(f*x+e))^(1/2)/(b*cos(f*x+e))^(1/2) +a^(1/2)*tan(f*x+e))*(b*cos(f*x+e))^(1/2)*(b*sec(f*x+e))^(1/2)/f*2^(1/2)/b ^(1/2)
Time = 1.32 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.38 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2} \, dx=-\frac {a^2 \cot (e+f x) \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)} \left (4 \sin ^2(e+f x)-3 \sqrt {2} \arctan \left (\frac {-1+\sqrt {\tan ^2(e+f x)}}{\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}\right ) \sqrt [4]{\tan ^2(e+f x)}+3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}{1+\sqrt {\tan ^2(e+f x)}}\right ) \sqrt [4]{\tan ^2(e+f x)}\right )}{8 f} \]
-1/8*(a^2*Cot[e + f*x]*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]]*(4*Sin[e + f*x]^2 - 3*Sqrt[2]*ArcTan[(-1 + Sqrt[Tan[e + f*x]^2])/(Sqrt[2]*(Tan[e + f*x]^2)^(1/4))]*(Tan[e + f*x]^2)^(1/4) + 3*Sqrt[2]*ArcTanh[(Sqrt[2]*(Tan[e + f*x]^2)^(1/4))/(1 + Sqrt[Tan[e + f*x]^2])]*(Tan[e + f*x]^2)^(1/4)))/f
Time = 0.70 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.84, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 3063, 3042, 3065, 3042, 3054, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int (a \sin (e+f x))^{5/2} \sqrt {b \sec (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x))^{5/2} \sqrt {b \sec (e+f x)}dx\) |
| \(\Big \downarrow \) 3063 |
| \(\displaystyle \frac {3}{4} a^2 \int \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)}dx-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} a^2 \int \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)}dx-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3065 |
| \(\displaystyle \frac {3}{4} a^2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}dx-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} a^2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}dx-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3054 |
| \(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {a \tan (e+f x)}{b \left (\tan ^2(e+f x) a^2+a^2\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\int \frac {\tan (e+f x) a+a}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\int \frac {1}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}+\frac {\int \frac {1}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\int \frac {1}{-\frac {a \tan (e+f x)}{b}-1}d\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\int \frac {1}{-\frac {a \tan (e+f x)}{b}-1}d\left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} b}+\frac {\int \frac {\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {a} b}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+a \tan (e+f x)+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+a \tan (e+f x)+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\) |
(3*a^3*b*Sqrt[b*Cos[e + f*x]]*((-(ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]/(Sqrt[2]*Sqrt[a]*Sqrt[b])) + Arc Tan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x ]])]/(Sqrt[2]*Sqrt[a]*Sqrt[b]))/(2*b) - (-1/2*Log[a - (Sqrt[2]*Sqrt[a]*Sqr t[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + a*Tan[e + f*x]]/(Sqrt[2] *Sqrt[a]*Sqrt[b]) + Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[a*Sin[e + f*x]]) /Sqrt[b*Cos[e + f*x]] + a*Tan[e + f*x]]/(2*Sqrt[2]*Sqrt[a]*Sqrt[b]))/(2*b) )*Sqrt[b*Sec[e + f*x]])/(2*f) - (a*b*(a*Sin[e + f*x])^(3/2))/(2*f*Sqrt[b*S ec[e + f*x]])
3.5.51.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> With[{k = Denominator[m]}, Simp[k*a*(b/f) Subst[Int[x^(k *(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Sin[e + f*x])^(1/k)/(b*Cos[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*b*(a*Sin[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n - 1)/(f*(m - n))), x] + Simp[a^2*((m - 1)/(m - n)) Int[(a*Sin[e + f*x])^( m - 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m - n, 0] && IntegersQ[2*m, 2*n]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(b*Cos[e + f*x])^n*(b*Sec[e + f*x])^n Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && Int egerQ[m - 1/2] && IntegerQ[n - 1/2]
Time = 4.46 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.08
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (4 \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {2}\, \sin \left (f x +e \right ) \cos \left (f x +e \right )+4 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+3 \ln \left (2 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cot \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \csc \left (f x +e \right )+2-2 \cot \left (f x +e \right )\right )-6 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )-1}\right )-6 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{\cos \left (f x +e \right )-1}\right )-3 \ln \left (-2 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cot \left (f x +e \right )-2 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \csc \left (f x +e \right )+2-2 \cot \left (f x +e \right )\right )\right ) \sqrt {a \sin \left (f x +e \right )}\, \sqrt {b \sec \left (f x +e \right )}\, a^{2} \cos \left (f x +e \right )}{16 f \left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\) | \(447\) |
-1/16/f*2^(1/2)*(4*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*2^(1/2) *sin(f*x+e)*cos(f*x+e)+4*2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2) ^(1/2)*sin(f*x+e)+3*ln(2*2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2) ^(1/2)*cot(f*x+e)+2*2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2 )*csc(f*x+e)+2-2*cot(f*x+e))-6*arctan((2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(co s(f*x+e)+1)^2)^(1/2)*sin(f*x+e)-cos(f*x+e)+1)/(cos(f*x+e)-1))-6*arctan((2^ (1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)+cos(f*x+e )-1)/(cos(f*x+e)-1))-3*ln(-2*2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1 )^2)^(1/2)*cot(f*x+e)-2*2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^ (1/2)*csc(f*x+e)+2-2*cot(f*x+e)))*(a*sin(f*x+e))^(1/2)*(b*sec(f*x+e))^(1/2 )*a^2*cos(f*x+e)/(cos(f*x+e)+1)/(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^ (1/2)
Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 1129, normalized size of antiderivative = 2.73 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2} \, dx=\text {Too large to display} \]
-1/32*(16*sqrt(a*sin(f*x + e))*a^2*sqrt(b/cos(f*x + e))*cos(f*x + e)*sin(f *x + e) + 3*(-a^10*b^2/f^4)^(1/4)*f*log(27/2*a^8*b^2*cos(f*x + e)*sin(f*x + e) + 27/2*((-a^10*b^2/f^4)^(1/4)*a^5*b*f*cos(f*x + e)*sin(f*x + e) - (-a ^10*b^2/f^4)^(3/4)*f^3*cos(f*x + e)^2)*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e)) - 27/4*sqrt(-a^10*b^2/f^4)*(2*a^3*b*f^2*cos(f*x + e)^2 - a^3*b*f^2) ) - 3*(-a^10*b^2/f^4)^(1/4)*f*log(27/2*a^8*b^2*cos(f*x + e)*sin(f*x + e) - 27/2*((-a^10*b^2/f^4)^(1/4)*a^5*b*f*cos(f*x + e)*sin(f*x + e) - (-a^10*b^ 2/f^4)^(3/4)*f^3*cos(f*x + e)^2)*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e)) - 27/4*sqrt(-a^10*b^2/f^4)*(2*a^3*b*f^2*cos(f*x + e)^2 - a^3*b*f^2)) - 3* I*(-a^10*b^2/f^4)^(1/4)*f*log(27/2*a^8*b^2*cos(f*x + e)*sin(f*x + e) - 27/ 2*(I*(-a^10*b^2/f^4)^(1/4)*a^5*b*f*cos(f*x + e)*sin(f*x + e) + I*(-a^10*b^ 2/f^4)^(3/4)*f^3*cos(f*x + e)^2)*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e)) + 27/4*sqrt(-a^10*b^2/f^4)*(2*a^3*b*f^2*cos(f*x + e)^2 - a^3*b*f^2)) + 3* I*(-a^10*b^2/f^4)^(1/4)*f*log(27/2*a^8*b^2*cos(f*x + e)*sin(f*x + e) - 27/ 2*(-I*(-a^10*b^2/f^4)^(1/4)*a^5*b*f*cos(f*x + e)*sin(f*x + e) - I*(-a^10*b ^2/f^4)^(3/4)*f^3*cos(f*x + e)^2)*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e) ) + 27/4*sqrt(-a^10*b^2/f^4)*(2*a^3*b*f^2*cos(f*x + e)^2 - a^3*b*f^2)) + 3 *(-a^10*b^2/f^4)^(1/4)*f*log(27*a^8*b^2 + 54*((-a^10*b^2/f^4)^(1/4)*a^5*b* f*cos(f*x + e)^2 - (-a^10*b^2/f^4)^(3/4)*f^3*cos(f*x + e)*sin(f*x + e))*sq rt(a*sin(f*x + e))*sqrt(b/cos(f*x + e))) - 3*(-a^10*b^2/f^4)^(1/4)*f*lo...
Timed out. \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]
\[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {5}{2}} \,d x } \]
\[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2} \, dx=\int {\left (a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x \]