Integrand size = 25, antiderivative size = 309 \[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{3/2}} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{\sqrt {2} a^{3/2} b^{5/2} f}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{\sqrt {2} a^{3/2} b^{5/2} f}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)} \left (\sqrt {a}+\sqrt {a} \tan (e+f x)\right )}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{\sqrt {2} a^{3/2} b^{5/2} f}-\frac {2}{a b f \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)}} \] Output:
1/2*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)/a^(3/2)/b^(5/2)/f- 1/2*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)/a^(3/2)/b^(5/2)/f+ 1/2*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 )/a^(3/2)/b^(5/2)/f-2/a/b/f/(b*sec(f*x+e))^(1/2)/(a*sin(f*x+e))^(1/2)
Time = 2.05 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.47 \[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{3/2}} \, dx=-\frac {4+\sqrt {2} \arctan \left (\frac {-1+\sqrt {\tan ^2(e+f x)}}{\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}\right ) \sqrt [4]{\tan ^2(e+f x)}-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}{1+\sqrt {\tan ^2(e+f x)}}\right ) \sqrt [4]{\tan ^2(e+f x)}}{2 a b f \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)}} \] Input:
Integrate[1/((b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(3/2)),x]
Output:
-1/2*(4 + 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) - 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))/(a*b*f*Sq rt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]])
Time = 1.22 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.13, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 3061, 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.
| \(\displaystyle \int \frac {1}{(a \sin (e+f x))^{3/2} (b \sec (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (e+f x))^{3/2} (b \sec (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3061 |
| \(\displaystyle -\frac {\int \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)}dx}{a^2 b^2}-\frac {2}{a b f \sqrt {a \sin (e+f x)} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)}dx}{a^2 b^2}-\frac {2}{a b f \sqrt {a \sin (e+f x)} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3065 |
| \(\displaystyle -\frac {\sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}dx}{a^2 b^2}-\frac {2}{a b f \sqrt {a \sin (e+f x)} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}dx}{a^2 b^2}-\frac {2}{a b f \sqrt {a \sin (e+f x)} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3054 |
| \(\displaystyle -\frac {2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {a \tan (e+f x)}{b \left (\tan ^2(e+f x) a^2+a^2\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{a b f}-\frac {2}{a b f \sqrt {a \sin (e+f x)} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\int \frac {\tan (e+f x) a+a}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{a b f}-\frac {2}{a b f \sqrt {a \sin (e+f x)} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\int \frac {1}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}+\frac {\int \frac {1}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{a b f}-\frac {2}{a b f \sqrt {a \sin (e+f x)} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\int \frac {1}{-\frac {a \tan (e+f x)}{b}-1}d\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\int \frac {1}{-\frac {a \tan (e+f x)}{b}-1}d\left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{a b f}-\frac {2}{a b f \sqrt {a \sin (e+f x)} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{a b f}-\frac {2}{a b f \sqrt {a \sin (e+f x)} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}\right )}{a b f}-\frac {2}{a b f \sqrt {a \sin (e+f x)} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}\right )}{a b f}-\frac {2}{a b f \sqrt {a \sin (e+f x)} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} b}+\frac {\int \frac {\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {a} b}}{2 b}\right )}{a b f}-\frac {2}{a b f \sqrt {a \sin (e+f x)} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+a \tan (e+f x)+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+a \tan (e+f x)+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}\right )}{a b f}-\frac {2}{a b f \sqrt {a \sin (e+f x)} \sqrt {b \sec (e+f x)}}\) |
Input:
Int[1/((b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(3/2)),x]
Output:
(-2*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]])/(a*b*f) - 2/(a*b*f*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]])
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*Sin[e + f*x])^(m + 1)*((b*Sec[e + f*x])^(n + 1)/(a *b*f*(m + 1))), x] - Simp[(n + 1)/(a^2*b^2*(m + 1)) Int[(a*Sin[e + f*x])^ (m + 2)*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && LtQ[n , -1] && LtQ[m, -1] && 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 = 3.21 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.48
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (\ln \left (\frac {2 \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-\cos \left (f x +e \right ) \cot \left (f x +e \right )+\sin \left (f x +e \right )+2 \cos \left (f x +e \right )-\csc \left (f x +e \right )+2 \cot \left (f x +e \right )-2}{-1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )+2 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{-1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )-\ln \left (-\frac {\cos \left (f x +e \right ) \cot \left (f x +e \right )-2 \cot \left (f x +e \right )+2 \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\csc \left (f x +e \right )-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )+2}{-1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )+2 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-\cos \left (f x +e \right )+1}{-1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )+4 \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right )+4 \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\right )}{4 f \left (\cos \left (f x +e \right )+1\right ) a \sqrt {a \sin \left (f x +e \right )}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, b \sqrt {b \sec \left (f x +e \right )}}\) | \(457\) |
Input:
int(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/4/f*2^(1/2)*(ln((2*(-2*sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*si n(f*x+e)-cos(f*x+e)*cot(f*x+e)+sin(f*x+e)+2*cos(f*x+e)-csc(f*x+e)+2*cot(f* x+e)-2)/(-1+cos(f*x+e)))*sin(f*x+e)+2*arctan(((-2*sin(f*x+e)*cos(f*x+e)/(c os(f*x+e)+1)^2)^(1/2)*sin(f*x+e)+cos(f*x+e)-1)/(-1+cos(f*x+e)))*sin(f*x+e) -ln(-(cos(f*x+e)*cot(f*x+e)-2*cot(f*x+e)+2*(-2*sin(f*x+e)*cos(f*x+e)/(cos( f*x+e)+1)^2)^(1/2)*sin(f*x+e)+csc(f*x+e)-2*cos(f*x+e)-sin(f*x+e)+2)/(-1+co s(f*x+e)))*sin(f*x+e)+2*arctan(((-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)/(-1+cos(f*x+e)))*sin(f*x+e)+4*(-2*sin(f*x +e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e)+4*(-2*sin(f*x+e)*cos(f*x +e)/(cos(f*x+e)+1)^2)^(1/2))/(cos(f*x+e)+1)/a/(a*sin(f*x+e))^(1/2)/(-sin(f *x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)/b/(b*sec(f*x+e))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (245) = 490\).
Time = 0.15 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.76 \[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{3/2}} \, dx=\frac {2 \, \sqrt {2} a b \sqrt {\frac {1}{a b}} \arctan \left (-\frac {\sqrt {2} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sqrt {\frac {1}{a b}} \cos \left (f x + e\right )}{\cos \left (f x + e\right ) - \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + \sqrt {2} a b \sqrt {\frac {1}{a b}} \arctan \left (\frac {\sqrt {2} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sqrt {\frac {1}{a b}} + 2 \, \cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2}{2 \, {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 1\right )}}\right ) \sin \left (f x + e\right ) + \sqrt {2} a b \sqrt {\frac {1}{a b}} \arctan \left (\frac {\sqrt {2} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sqrt {\frac {1}{a b}} - 2 \, \cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 2}{2 \, {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 1\right )}}\right ) \sin \left (f x + e\right ) + \sqrt {2} a b \sqrt {\frac {1}{a b}} \log \left (2 \, \sqrt {2} {\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 )}} \sqrt {\frac {1}{a b}} + 4 \, \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) - \sqrt {2} a b \sqrt {\frac {1}{a b}} \log \left (-2 \, \sqrt {2} {\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 )}} \sqrt {\frac {1}{a b}} + 4 \, \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) - 16 \, \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{8 \, a^{2} b^{2} f \sin \left (f x + e\right )} \] Input:
integrate(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(3/2),x, algorithm="fricas ")
Output:
1/8*(2*sqrt(2)*a*b*sqrt(1/(a*b))*arctan(-sqrt(2)*sqrt(a*sin(f*x + e))*sqrt (b/cos(f*x + e))*sqrt(1/(a*b))*cos(f*x + e)/(cos(f*x + e) - sin(f*x + e))) *sin(f*x + e) + sqrt(2)*a*b*sqrt(1/(a*b))*arctan(1/2*(sqrt(2)*sqrt(a*sin(f *x + e))*sqrt(b/cos(f*x + e))*sqrt(1/(a*b)) + 2*cos(f*x + e)^2 - 2*cos(f*x + e)*sin(f*x + e) - 2)/(cos(f*x + e)^2 + cos(f*x + e)*sin(f*x + e) - 1))* sin(f*x + e) + sqrt(2)*a*b*sqrt(1/(a*b))*arctan(1/2*(sqrt(2)*sqrt(a*sin(f* x + e))*sqrt(b/cos(f*x + e))*sqrt(1/(a*b)) - 2*cos(f*x + e)^2 + 2*cos(f*x + e)*sin(f*x + e) + 2)/(cos(f*x + e)^2 + cos(f*x + e)*sin(f*x + e) - 1))*s in(f*x + e) + sqrt(2)*a*b*sqrt(1/(a*b))*log(2*sqrt(2)*(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))*sqrt(1/ (a*b)) + 4*cos(f*x + e)*sin(f*x + e) + 1)*sin(f*x + e) - sqrt(2)*a*b*sqrt( 1/(a*b))*log(-2*sqrt(2)*(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))*sqrt(1/(a*b)) + 4*cos(f*x + e)*sin(f* x + e) + 1)*sin(f*x + e) - 16*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e))*co s(f*x + e))/(a^2*b^2*f*sin(f*x + e))
Timed out. \[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(b*sec(f*x+e))**(3/2)/(a*sin(f*x+e))**(3/2),x)
Output:
Timed out
\[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(3/2),x, algorithm="maxima ")
Output:
integrate(1/((b*sec(f*x + e))^(3/2)*(a*sin(f*x + e))^(3/2)), x)
\[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(3/2),x, algorithm="giac")
Output:
integrate(1/((b*sec(f*x + e))^(3/2)*(a*sin(f*x + e))^(3/2)), x)
Timed out. \[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{{\left (a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \] Input:
int(1/((a*sin(e + f*x))^(3/2)*(b/cos(e + f*x))^(3/2)),x)
Output:
int(1/((a*sin(e + f*x))^(3/2)*(b/cos(e + f*x))^(3/2)), x)
\[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{3/2}} \, dx=\frac {\sqrt {b}\, \sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\sec \left (f x +e \right )}}{\sec \left (f x +e \right )^{2} \sin \left (f x +e \right )^{2}}d x \right )}{a^{2} b^{2}} \] Input:
int(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(3/2),x)
Output:
(sqrt(b)*sqrt(a)*int((sqrt(sin(e + f*x))*sqrt(sec(e + f*x)))/(sec(e + f*x) **2*sin(e + f*x)**2),x))/(a**2*b**2)