Integrand size = 25, antiderivative size = 133 \[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{9/2}} \, dx=\frac {2 c \sqrt {c \sin (a+b x)}}{7 b d (d \cos (a+b x))^{7/2}}-\frac {2 c \sqrt {c \sin (a+b x)}}{21 b d^3 (d \cos (a+b x))^{3/2}}-\frac {2 c^2 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)}}{21 b d^4 \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \] Output:
2/7*c*(c*sin(b*x+a))^(1/2)/b/d/(d*cos(b*x+a))^(7/2)-2/21*c*(c*sin(b*x+a))^ (1/2)/b/d^3/(d*cos(b*x+a))^(3/2)-2/21*c^2*InverseJacobiAM(a-1/4*Pi+b*x,2^( 1/2))*sin(2*b*x+2*a)^(1/2)/b/d^4/(d*cos(b*x+a))^(1/2)/(c*sin(b*x+a))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.53 \[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{9/2}} \, dx=\frac {2 \cos ^2(a+b x)^{7/4} \cot (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {11}{4},\frac {9}{4},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{7/2}}{5 b c^2 (d \cos (a+b x))^{9/2}} \] Input:
Integrate[(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(9/2),x]
Output:
(2*(Cos[a + b*x]^2)^(7/4)*Cot[a + b*x]*Hypergeometric2F1[5/4, 11/4, 9/4, S in[a + b*x]^2]*(c*Sin[a + b*x])^(7/2))/(5*b*c^2*(d*Cos[a + b*x])^(9/2))
Time = 0.57 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3046, 3042, 3051, 3042, 3053, 3042, 3120}
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.
| \(\displaystyle \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3046 |
| \(\displaystyle \frac {2 c \sqrt {c \sin (a+b x)}}{7 b d (d \cos (a+b x))^{7/2}}-\frac {c^2 \int \frac {1}{(d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)}}dx}{7 d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \sqrt {c \sin (a+b x)}}{7 b d (d \cos (a+b x))^{7/2}}-\frac {c^2 \int \frac {1}{(d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)}}dx}{7 d^2}\) |
| \(\Big \downarrow \) 3051 |
| \(\displaystyle \frac {2 c \sqrt {c \sin (a+b x)}}{7 b d (d \cos (a+b x))^{7/2}}-\frac {c^2 \left (\frac {2 \int \frac {1}{\sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}dx}{3 d^2}+\frac {2 \sqrt {c \sin (a+b x)}}{3 b c d (d \cos (a+b x))^{3/2}}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \sqrt {c \sin (a+b x)}}{7 b d (d \cos (a+b x))^{7/2}}-\frac {c^2 \left (\frac {2 \int \frac {1}{\sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}dx}{3 d^2}+\frac {2 \sqrt {c \sin (a+b x)}}{3 b c d (d \cos (a+b x))^{3/2}}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 3053 |
| \(\displaystyle \frac {2 c \sqrt {c \sin (a+b x)}}{7 b d (d \cos (a+b x))^{7/2}}-\frac {c^2 \left (\frac {2 \sqrt {\sin (2 a+2 b x)} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx}{3 d^2 \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}+\frac {2 \sqrt {c \sin (a+b x)}}{3 b c d (d \cos (a+b x))^{3/2}}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \sqrt {c \sin (a+b x)}}{7 b d (d \cos (a+b x))^{7/2}}-\frac {c^2 \left (\frac {2 \sqrt {\sin (2 a+2 b x)} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx}{3 d^2 \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}+\frac {2 \sqrt {c \sin (a+b x)}}{3 b c d (d \cos (a+b x))^{3/2}}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {2 c \sqrt {c \sin (a+b x)}}{7 b d (d \cos (a+b x))^{7/2}}-\frac {c^2 \left (\frac {2 \sqrt {\sin (2 a+2 b x)} \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right )}{3 b d^2 \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}+\frac {2 \sqrt {c \sin (a+b x)}}{3 b c d (d \cos (a+b x))^{3/2}}\right )}{7 d^2}\) |
Input:
Int[(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(9/2),x]
Output:
(2*c*Sqrt[c*Sin[a + b*x]])/(7*b*d*(d*Cos[a + b*x])^(7/2)) - (c^2*((2*Sqrt[ c*Sin[a + b*x]])/(3*b*c*d*(d*Cos[a + b*x])^(3/2)) + (2*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(3*b*d^2*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin [a + b*x]])))/(7*d^2)
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(a*Sin[e + f*x])^(m - 1)*((b*Cos[e + f*x])^(n + 1)/(b*f*(n + 1))), x] + Simp[a^2*((m - 1)/(b^2*(n + 1))) Int[(a*Sin[e + f *x])^(m - 2)*(b*Cos[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(-(b*Sin[e + f*x])^(n + 1))*((a*Cos[e + f*x])^(m + 1) /(a*b*f*(m + 1))), x] + Simp[(m + n + 2)/(a^2*(m + 1)) Int[(b*Sin[e + f*x ])^n*(a*Cos[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m , -1] && IntegersQ[2*m, 2*n]
Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ )]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b *Cos[e + f*x]]) Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f }, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Time = 11.47 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(-\frac {2 \sqrt {c \sin \left (b x +a \right )}\, c \left (\sec \left (b x +a \right )-3 \sec \left (b x +a \right )^{3}+\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \operatorname {EllipticF}\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {2 \cot \left (b x +a \right )-2 \csc \left (b x +a \right )+2}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \left (\cot \left (b x +a \right )+\csc \left (b x +a \right )\right )\right )}{21 b \sqrt {d \cos \left (b x +a \right )}\, d^{4}}\) | \(140\) |
Input:
int((c*sin(b*x+a))^(3/2)/(d*cos(b*x+a))^(9/2),x,method=_RETURNVERBOSE)
Output:
-2/21/b*(c*sin(b*x+a))^(1/2)*c/(d*cos(b*x+a))^(1/2)/d^4*(sec(b*x+a)-3*sec( b*x+a)^3+(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*EllipticF((-cot(b*x+a)+csc(b*x+a )+1)^(1/2),1/2*2^(1/2))*(2*cot(b*x+a)-2*csc(b*x+a)+2)^(1/2)*(cot(b*x+a)-cs c(b*x+a))^(1/2)*(cot(b*x+a)+csc(b*x+a)))
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.91 \[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{9/2}} \, dx=\frac {2 \, {\left (\sqrt {i \, c d} c \cos \left (b x + a\right )^{4} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) + \sqrt {-i \, c d} c \cos \left (b x + a\right )^{4} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) - {\left (c \cos \left (b x + a\right )^{2} - 3 \, c\right )} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )}\right )}}{21 \, b d^{5} \cos \left (b x + a\right )^{4}} \] Input:
integrate((c*sin(b*x+a))^(3/2)/(d*cos(b*x+a))^(9/2),x, algorithm="fricas")
Output:
2/21*(sqrt(I*c*d)*c*cos(b*x + a)^4*elliptic_f(arcsin(cos(b*x + a) + I*sin( b*x + a)), -1) + sqrt(-I*c*d)*c*cos(b*x + a)^4*elliptic_f(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1) - (c*cos(b*x + a)^2 - 3*c)*sqrt(d*cos(b*x + a)) *sqrt(c*sin(b*x + a)))/(b*d^5*cos(b*x + a)^4)
Timed out. \[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((c*sin(b*x+a))**(3/2)/(d*cos(b*x+a))**(9/2),x)
Output:
Timed out
\[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{9/2}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((c*sin(b*x+a))^(3/2)/(d*cos(b*x+a))^(9/2),x, algorithm="maxima")
Output:
integrate((c*sin(b*x + a))^(3/2)/(d*cos(b*x + a))^(9/2), x)
\[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{9/2}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((c*sin(b*x+a))^(3/2)/(d*cos(b*x+a))^(9/2),x, algorithm="giac")
Output:
integrate((c*sin(b*x + a))^(3/2)/(d*cos(b*x + a))^(9/2), x)
Timed out. \[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{9/2}} \, dx=\int \frac {{\left (c\,\sin \left (a+b\,x\right )\right )}^{3/2}}{{\left (d\,\cos \left (a+b\,x\right )\right )}^{9/2}} \,d x \] Input:
int((c*sin(a + b*x))^(3/2)/(d*cos(a + b*x))^(9/2),x)
Output:
int((c*sin(a + b*x))^(3/2)/(d*cos(a + b*x))^(9/2), x)
\[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{9/2}} \, dx=\frac {\sqrt {d}\, \sqrt {c}\, c \left (-\cos \left (b x +a \right )^{4} \left (\int \frac {\sqrt {\sin \left (b x +a \right )}\, \sqrt {\cos \left (b x +a \right )}}{\cos \left (b x +a \right )^{3} \sin \left (b x +a \right )}d x \right ) b +2 \sqrt {\sin \left (b x +a \right )}\, \sqrt {\cos \left (b x +a \right )}\right )}{7 \cos \left (b x +a \right )^{4} b \,d^{5}} \] Input:
int((c*sin(b*x+a))^(3/2)/(d*cos(b*x+a))^(9/2),x)
Output:
(sqrt(d)*sqrt(c)*c*( - cos(a + b*x)**4*int((sqrt(sin(a + b*x))*sqrt(cos(a + b*x)))/(cos(a + b*x)**3*sin(a + b*x)),x)*b + 2*sqrt(sin(a + b*x))*sqrt(c os(a + b*x))))/(7*cos(a + b*x)**4*b*d**5)