Integrand size = 25, antiderivative size = 350 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=-\frac {21 a^{9/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{32 \sqrt {2} \sqrt {b} f}+\frac {21 a^{9/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{32 \sqrt {2} \sqrt {b} f}-\frac {21 a^{9/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)} \left (\sqrt {a}+\sqrt {a} \tan (e+f x)\right )}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{32 \sqrt {2} \sqrt {b} f}-\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}} \] Output:
-21/64*a^(9/2)*arctan(1-2^(1/2)*b^(1/2)*(a*sin(f*x+e))^(1/2)/a^(1/2)/(b*co s(f*x+e))^(1/2))*(b*cos(f*x+e))^(1/2)*(b*sec(f*x+e))^(1/2)*2^(1/2)/b^(1/2) /f+21/64*a^(9/2)*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)*2^(1/2)/b^(1/ 2)/f-21/64*a^(9/2)*arctanh(2^(1/2)*b^(1/2)*(a*sin(f*x+e))^(1/2)/(b*cos(f*x +e))^(1/2)/(a^(1/2)+a^(1/2)*tan(f*x+e)))*(b*cos(f*x+e))^(1/2)*(b*sec(f*x+e ))^(1/2)*2^(1/2)/b^(1/2)/f-7/16*a^3*b*(a*sin(f*x+e))^(3/2)/f/(b*sec(f*x+e) )^(1/2)-1/4*a*b*(a*sin(f*x+e))^(7/2)/f/(b*sec(f*x+e))^(1/2)
Time = 3.29 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.48 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\frac {a^4 \cot (e+f x) \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)} \left (4 (-9+2 \cos (2 (e+f x))) \sin ^2(e+f x)+21 \sqrt {2} \arctan \left (\frac {-1+\sqrt {\tan ^2(e+f x)}}{\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}\right ) \sqrt [4]{\tan ^2(e+f x)}-21 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}{1+\sqrt {\tan ^2(e+f x)}}\right ) \sqrt [4]{\tan ^2(e+f x)}\right )}{64 f} \] Input:
Integrate[Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(9/2),x]
Output:
(a^4*Cot[e + f*x]*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]]*(4*(-9 + 2*Cos [2*(e + f*x)])*Sin[e + f*x]^2 + 21*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) - 21*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)))/(64*f)
Time = 1.44 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.11, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 3063, 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.
| \(\displaystyle \int (a \sin (e+f x))^{9/2} \sqrt {b \sec (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x))^{9/2} \sqrt {b \sec (e+f x)}dx\) |
| \(\Big \downarrow \) 3063 |
| \(\displaystyle \frac {7}{8} a^2 \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2}dx-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{8} a^2 \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2}dx-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3063 |
| \(\displaystyle \frac {7}{8} a^2 \left (\frac {3}{4} a^2 \int \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)}dx-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{8} a^2 \left (\frac {3}{4} a^2 \int \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)}dx-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3065 |
| \(\displaystyle \frac {7}{8} a^2 \left (\frac {3}{4} a^2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}dx-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{8} a^2 \left (\frac {3}{4} a^2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}dx-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3054 |
| \(\displaystyle \frac {7}{8} a^2 \left (\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {a \tan (e+f x)}{b \left (\tan ^2(e+f x) a^2+a^2\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {7}{8} a^2 \left (\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\int \frac {\tan (e+f x) a+a}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {7}{8} a^2 \left (\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\int \frac {1}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}+\frac {\int \frac {1}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {7}{8} a^2 \left (\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\int \frac {1}{-\frac {a \tan (e+f x)}{b}-1}d\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\int \frac {1}{-\frac {a \tan (e+f x)}{b}-1}d\left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {7}{8} a^2 \left (\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {7}{8} a^2 \left (\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {7}{8} a^2 \left (\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{8} a^2 \left (\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} b}+\frac {\int \frac {\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {a} b}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {7}{8} a^2 \left (\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+a \tan (e+f x)+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+a \tan (e+f x)+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}\) |
Input:
Int[Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(9/2),x]
Output:
-1/4*(a*b*(a*Sin[e + f*x])^(7/2))/(f*Sqrt[b*Sec[e + f*x]]) + (7*a^2*((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])) + 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]))/(2*b) - (-1/2*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]]/(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))*Sqr t[b*Sec[e + f*x]])/(2*f) - (a*b*(a*Sin[e + f*x])^(3/2))/(2*f*Sqrt[b*Sec[e + f*x]])))/8
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 = 51.64 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.31
| method | result | size |
| default | \(\frac {\sqrt {2}\, a^{4} \left (-21 \ln \left (-\frac {\cos \left (f x +e \right ) \cot \left (f x +e \right )+2 \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-2 \cot \left (f x +e \right )-\sin \left (f x +e \right )-2 \cos \left (f x +e \right )+\csc \left (f x +e \right )+2}{\cos \left (f x +e \right )-1}\right )+21 \ln \left (\frac {-\cos \left (f x +e \right ) \cot \left (f x +e \right )+2 \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+2 \cot \left (f x +e \right )+\sin \left (f x +e \right )+2 \cos \left (f x +e \right )-\csc \left (f x +e \right )-2}{\cos \left (f x +e \right )-1}\right )+42 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )-1}\right )+42 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{\cos \left (f x +e \right )-1}\right )+\left (16 \cos \left (f x +e \right )^{3}+16 \cos \left (f x +e \right )^{2}-44 \cos \left (f x +e \right )-44\right ) \sin \left (f x +e \right ) \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\right ) \sqrt {a \sin \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} \sqrt {b \sec \left (f x +e \right )}\, \sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{8} \csc \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{16384 f \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\) | \(458\) |
Input:
int((b*sec(f*x+e))^(1/2)*(a*sin(f*x+e))^(9/2),x,method=_RETURNVERBOSE)
Output:
1/16384/f*2^(1/2)*a^4*(-21*ln(-(cos(f*x+e)*cot(f*x+e)+2*(-2*sin(f*x+e)/(co s(f*x+e)+1)^2*cos(f*x+e))^(1/2)*sin(f*x+e)-2*cot(f*x+e)-sin(f*x+e)-2*cos(f *x+e)+csc(f*x+e)+2)/(cos(f*x+e)-1))+21*ln((-cos(f*x+e)*cot(f*x+e)+2*(-2*si n(f*x+e)/(cos(f*x+e)+1)^2*cos(f*x+e))^(1/2)*sin(f*x+e)+2*cot(f*x+e)+sin(f* x+e)+2*cos(f*x+e)-csc(f*x+e)-2)/(cos(f*x+e)-1))+42*arctan(((-2*sin(f*x+e)/ (cos(f*x+e)+1)^2*cos(f*x+e))^(1/2)*sin(f*x+e)-cos(f*x+e)+1)/(cos(f*x+e)-1) )+42*arctan(((-2*sin(f*x+e)/(cos(f*x+e)+1)^2*cos(f*x+e))^(1/2)*sin(f*x+e)+ cos(f*x+e)-1)/(cos(f*x+e)-1))+(16*cos(f*x+e)^3+16*cos(f*x+e)^2-44*cos(f*x+ e)-44)*sin(f*x+e)*(-2*sin(f*x+e)/(cos(f*x+e)+1)^2*cos(f*x+e))^(1/2))*(a*si n(f*x+e))^(1/2)*cos(f*x+e)*sin(f*x+e)^6*(b*sec(f*x+e))^(1/2)/(-sin(f*x+e)/ (cos(f*x+e)+1)^2*cos(f*x+e))^(1/2)*sec(1/2*f*x+1/2*e)^8*csc(1/2*f*x+1/2*e) ^6
Time = 0.17 (sec) , antiderivative size = 538, normalized size of antiderivative = 1.54 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=-\frac {42 \, \sqrt {2} \sqrt {a b} a^{4} \arctan \left (-\frac {\sqrt {2} \sqrt {a b} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a b \cos \left (f x + e\right ) - a b \sin \left (f x + e\right )}\right ) + 21 \, \sqrt {2} \sqrt {a b} a^{4} \arctan \left (\frac {2 \, a b \cos \left (f x + e\right )^{2} - 2 \, a b \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a b + \sqrt {2} \sqrt {a b} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{2 \, {\left (a b \cos \left (f x + e\right )^{2} + a b \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a b\right )}}\right ) + 21 \, \sqrt {2} \sqrt {a b} a^{4} \arctan \left (-\frac {2 \, a b \cos \left (f x + e\right )^{2} - 2 \, a b \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a b - \sqrt {2} \sqrt {a b} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{2 \, {\left (a b \cos \left (f x + e\right )^{2} + a b \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a b\right )}}\right ) + 21 \, \sqrt {2} \sqrt {a b} a^{4} \log \left (4 \, a b \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 2 \, \sqrt {2} \sqrt {a b} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right ) \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} + a b\right ) - 21 \, \sqrt {2} \sqrt {a b} a^{4} \log \left (4 \, a b \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, \sqrt {2} \sqrt {a b} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right ) \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} + a b\right ) - 16 \, {\left (4 \, a^{4} \cos \left (f x + e\right )^{3} - 11 \, a^{4} \cos \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{256 \, f} \] Input:
integrate((b*sec(f*x+e))^(1/2)*(a*sin(f*x+e))^(9/2),x, algorithm="fricas")
Output:
-1/256*(42*sqrt(2)*sqrt(a*b)*a^4*arctan(-sqrt(2)*sqrt(a*b)*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e))*cos(f*x + e)/(a*b*cos(f*x + e) - a*b*sin(f*x + e))) + 21*sqrt(2)*sqrt(a*b)*a^4*arctan(1/2*(2*a*b*cos(f*x + e)^2 - 2*a*b*c os(f*x + e)*sin(f*x + e) - 2*a*b + sqrt(2)*sqrt(a*b)*sqrt(a*sin(f*x + e))* sqrt(b/cos(f*x + e)))/(a*b*cos(f*x + e)^2 + a*b*cos(f*x + e)*sin(f*x + e) - a*b)) + 21*sqrt(2)*sqrt(a*b)*a^4*arctan(-1/2*(2*a*b*cos(f*x + e)^2 - 2*a *b*cos(f*x + e)*sin(f*x + e) - 2*a*b - sqrt(2)*sqrt(a*b)*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e)))/(a*b*cos(f*x + e)^2 + a*b*cos(f*x + e)*sin(f*x + e) - a*b)) + 21*sqrt(2)*sqrt(a*b)*a^4*log(4*a*b*cos(f*x + e)*sin(f*x + e) + 2*sqrt(2)*sqrt(a*b)*(cos(f*x + e)^2 + cos(f*x + e)*sin(f*x + e))*sqrt(a *sin(f*x + e))*sqrt(b/cos(f*x + e)) + a*b) - 21*sqrt(2)*sqrt(a*b)*a^4*log( 4*a*b*cos(f*x + e)*sin(f*x + e) - 2*sqrt(2)*sqrt(a*b)*(cos(f*x + e)^2 + co s(f*x + e)*sin(f*x + e))*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e)) + a*b) - 16*(4*a^4*cos(f*x + e)^3 - 11*a^4*cos(f*x + e))*sqrt(a*sin(f*x + e))*sqr t(b/cos(f*x + e))*sin(f*x + e))/f
Timed out. \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\text {Timed out} \] Input:
integrate((b*sec(f*x+e))**(1/2)*(a*sin(f*x+e))**(9/2),x)
Output:
Timed out
\[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {9}{2}} \,d x } \] Input:
integrate((b*sec(f*x+e))^(1/2)*(a*sin(f*x+e))^(9/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*sec(f*x + e))*(a*sin(f*x + e))^(9/2), x)
\[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {9}{2}} \,d x } \] Input:
integrate((b*sec(f*x+e))^(1/2)*(a*sin(f*x+e))^(9/2),x, algorithm="giac")
Output:
integrate(sqrt(b*sec(f*x + e))*(a*sin(f*x + e))^(9/2), x)
Timed out. \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\int {\left (a\,\sin \left (e+f\,x\right )\right )}^{9/2}\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x \] Input:
int((a*sin(e + f*x))^(9/2)*(b/cos(e + f*x))^(1/2),x)
Output:
int((a*sin(e + f*x))^(9/2)*(b/cos(e + f*x))^(1/2), x)
\[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\sqrt {b}\, \sqrt {a}\, \left (\int \sqrt {\sin \left (f x +e \right )}\, \sqrt {\sec \left (f x +e \right )}\, \sin \left (f x +e \right )^{4}d x \right ) a^{4} \] Input:
int((b*sec(f*x+e))^(1/2)*(a*sin(f*x+e))^(9/2),x)
Output:
sqrt(b)*sqrt(a)*int(sqrt(sin(e + f*x))*sqrt(sec(e + f*x))*sin(e + f*x)**4, x)*a**4