Integrand size = 25, antiderivative size = 387 \[ \int \frac {(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\frac {4 (a-b) b \sqrt {a+b} \left (41 a^2-3 b^2\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{105 a^3 d}+\frac {2 (a-b) \sqrt {a+b} \left (25 a^2-57 a b-6 b^2\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{105 a^2 d}+\frac {2 a \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {16 b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (25 a^2+3 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 a d \cos ^{\frac {3}{2}}(c+d x)} \] Output:
4/105*(a-b)*b*(a+b)^(1/2)*(41*a^2-3*b^2)*cot(d*x+c)*EllipticE((a+b*cos(d*x +c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d* x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a^3/d+2/105*(a-b)*(a+b)^ (1/2)*(25*a^2-57*a*b-6*b^2)*cot(d*x+c)*EllipticF((a+b*cos(d*x+c))^(1/2)/(a +b)^(1/2)/cos(d*x+c)^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^ (1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a^2/d+2/7*a*(a+b*cos(d*x+c))^(1/2)*si n(d*x+c)/d/cos(d*x+c)^(7/2)+16/35*b*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/d/co s(d*x+c)^(5/2)+2/105*(25*a^2+3*b^2)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/a/d/ cos(d*x+c)^(3/2)
Result contains complex when optimal does not.
Time = 6.62 (sec) , antiderivative size = 1302, normalized size of antiderivative = 3.36 \[ \int \frac {(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(9/2),x]
Output:
((-4*a*(25*a^4 - 31*a^2*b^2 + 6*b^4)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*C os[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[( c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a *(-82*a^3*b + 6*a*b^3)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[ -(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x]) *Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^ 4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)* Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2 ]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*El lipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/ Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[ a + b*Cos[c + d*x]])) + 2*(-82*a^2*b^2 + 6*b^4)*((I*Cos[(c + d*x)/2]*Sqrt[ a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x] ]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x] ]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d* x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + ...
Time = 1.71 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 3278, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3477, 3042, 3295, 3473}
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.
\(\displaystyle \int \frac {(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
\(\Big \downarrow \) 3278 |
\(\displaystyle \frac {2}{7} \int \frac {4 a b \cos ^2(c+d x)+\left (5 a^2+7 b^2\right ) \cos (c+d x)+8 a b}{2 \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {2 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \int \frac {4 a b \cos ^2(c+d x)+\left (5 a^2+7 b^2\right ) \cos (c+d x)+8 a b}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {2 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \int \frac {4 a b \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (5 a^2+7 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+8 a b}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3534 |
\(\displaystyle \frac {1}{7} \left (\frac {2 \int \frac {44 b \cos (c+d x) a^2+16 b^2 \cos ^2(c+d x) a+\left (25 a^2+3 b^2\right ) a}{2 \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {16 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \left (\frac {\int \frac {44 b \cos (c+d x) a^2+16 b^2 \cos ^2(c+d x) a+\left (25 a^2+3 b^2\right ) a}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {16 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {\int \frac {44 b \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+16 b^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+\left (25 a^2+3 b^2\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}+\frac {16 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3534 |
\(\displaystyle \frac {1}{7} \left (\frac {\frac {2 \int \frac {\left (25 a^2+51 b^2\right ) \cos (c+d x) a^2+2 b \left (41 a^2-3 b^2\right ) a}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 \left (25 a^2+3 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {16 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \left (\frac {\frac {\int \frac {\left (25 a^2+51 b^2\right ) \cos (c+d x) a^2+2 b \left (41 a^2-3 b^2\right ) a}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 \left (25 a^2+3 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {16 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {\frac {\int \frac {\left (25 a^2+51 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+2 b \left (41 a^2-3 b^2\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}+\frac {2 \left (25 a^2+3 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {16 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3477 |
\(\displaystyle \frac {1}{7} \left (\frac {\frac {2 a b \left (41 a^2-3 b^2\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+a (a-b) \left (25 a^2-57 a b-6 b^2\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 \left (25 a^2+3 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {16 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {\frac {a (a-b) \left (25 a^2-57 a b-6 b^2\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+2 a b \left (41 a^2-3 b^2\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}+\frac {2 \left (25 a^2+3 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {16 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3295 |
\(\displaystyle \frac {1}{7} \left (\frac {\frac {2 a b \left (41 a^2-3 b^2\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (a-b) \sqrt {a+b} \left (25 a^2-57 a b-6 b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{3 a}+\frac {2 \left (25 a^2+3 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {16 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3473 |
\(\displaystyle \frac {1}{7} \left (\frac {\frac {\frac {2 (a-b) \sqrt {a+b} \left (25 a^2-57 a b-6 b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}+\frac {4 b (a-b) \sqrt {a+b} \left (41 a^2-3 b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a d}}{3 a}+\frac {2 \left (25 a^2+3 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {16 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
Input:
Int[(a + b*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(9/2),x]
Output:
(2*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((1 6*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (((4 *(a - b)*b*Sqrt[a + b]*(41*a^2 - 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt [a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b)) ]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b) ])/(a*d) + (2*(a - b)*Sqrt[a + b]*(25*a^2 - 57*a*b - 6*b^2)*Cot[c + d*x]*E llipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])] , -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Se c[c + d*x]))/(a - b)])/d)/(3*a) + (2*(25*a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d *x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)))/(5*a))/7
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_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*(a + b*Si n[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)*Simp[c*(a*c - b*d)*(m + 1) + d*(b*c - a*d)*(n - 1) + (d*(a*c - b*d)*(m + 1) - c*(b*c - a*d)*(m + 2))*Sin[e + f*x] - d*(b*c - a *d)*(m + n + 1)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1 ] && LtQ[1, n, 2] && IntegersQ[2*m, 2*n]
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] ], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) ^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* (c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] )/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(c + d)/b]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e , f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Leaf count of result is larger than twice the leaf count of optimal. \(1141\) vs. \(2(343)=686\).
Time = 27.70 (sec) , antiderivative size = 1142, normalized size of antiderivative = 2.95
Input:
int((a+cos(d*x+c)*b)^(3/2)/cos(d*x+c)^(9/2),x,method=_RETURNVERBOSE)
Output:
2/105/d*(((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*(cos(d*x+c)/(cos(d* x+c)+1))^(1/2)*a^3*b*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2)) *(82*cos(d*x+c)^5+164*cos(d*x+c)^4+82*cos(d*x+c)^3)+((a+cos(d*x+c)*b)/(cos (d*x+c)+1)/(a+b))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^2*b^2*Elliptic E(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(82*cos(d*x+c)^5+164*cos(d*x +c)^4+82*cos(d*x+c)^3)+((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*(cos( d*x+c)/(cos(d*x+c)+1))^(1/2)*a*b^3*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b) /(a+b))^(1/2))*(-6*cos(d*x+c)^5-12*cos(d*x+c)^4-6*cos(d*x+c)^3)+((a+cos(d* x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*b^4* EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(-6*cos(d*x+c)^5-12* cos(d*x+c)^4-6*cos(d*x+c)^3)+((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2) *(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^4*EllipticF(cot(d*x+c)-csc(d*x+c),(-( a-b)/(a+b))^(1/2))*(-25*cos(d*x+c)^5-50*cos(d*x+c)^4-25*cos(d*x+c)^3)+((a+ cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2 )*a^3*b*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(-82*cos(d*x +c)^5-164*cos(d*x+c)^4-82*cos(d*x+c)^3)+((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/( a+b))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^2*b^2*EllipticF(cot(d*x+c) -csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(-51*cos(d*x+c)^5-102*cos(d*x+c)^4-51*co s(d*x+c)^3)+((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*(cos(d*x+c)/(cos (d*x+c)+1))^(1/2)*a*b^3*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^...
\[ \int \frac {(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(3/2)/cos(d*x+c)^(9/2),x, algorithm="fricas")
Output:
integral((b*cos(d*x + c) + a)^(3/2)/cos(d*x + c)^(9/2), x)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(3/2)/cos(d*x+c)**(9/2),x)
Output:
Timed out
\[ \int \frac {(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(3/2)/cos(d*x+c)^(9/2),x, algorithm="maxima")
Output:
integrate((b*cos(d*x + c) + a)^(3/2)/cos(d*x + c)^(9/2), x)
\[ \int \frac {(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(3/2)/cos(d*x+c)^(9/2),x, algorithm="giac")
Output:
integrate((b*cos(d*x + c) + a)^(3/2)/cos(d*x + c)^(9/2), x)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^{9/2}} \,d x \] Input:
int((a + b*cos(c + d*x))^(3/2)/cos(c + d*x)^(9/2),x)
Output:
int((a + b*cos(c + d*x))^(3/2)/cos(c + d*x)^(9/2), x)
\[ \int \frac {(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) a +\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) b \] Input:
int((a+b*cos(d*x+c))^(3/2)/cos(d*x+c)^(9/2),x)
Output:
int((sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)))/cos(c + d*x)**5,x)*a + i nt((sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)))/cos(c + d*x)**4,x)*b