Integrand size = 25, antiderivative size = 132 \[ \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx=\frac {5 d^3 \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{6 b c}+\frac {d (d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)}}{3 b c}+\frac {5 d^4 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)}}{12 b \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \]
1/3*d*(d*cos(b*x+a))^(5/2)*(c*sin(b*x+a))^(1/2)/b/c+5/6*d^3*(d*cos(b*x+a)) ^(1/2)*(c*sin(b*x+a))^(1/2)/b/c-5/12*d^4*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a +1/4*Pi+b*x)*EllipticF(cos(a+1/4*Pi+b*x),2^(1/2))*sin(2*b*x+2*a)^(1/2)/b/( 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 = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.53 \[ \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx=\frac {2 (d \cos (a+b x))^{7/2} \cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{4},\frac {5}{4},\sin ^2(a+b x)\right ) \sec ^5(a+b x) \sqrt {c \sin (a+b x)}}{b c} \]
(2*(d*Cos[a + b*x])^(7/2)*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-5/4, 1 /4, 5/4, Sin[a + b*x]^2]*Sec[a + b*x]^5*Sqrt[c*Sin[a + b*x]])/(b*c)
Time = 0.56 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3049, 3042, 3049, 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.
\(\displaystyle \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}}dx\) |
\(\Big \downarrow \) 3049 |
\(\displaystyle \frac {5}{6} d^2 \int \frac {(d \cos (a+b x))^{3/2}}{\sqrt {c \sin (a+b x)}}dx+\frac {d \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{5/2}}{3 b c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} d^2 \int \frac {(d \cos (a+b x))^{3/2}}{\sqrt {c \sin (a+b x)}}dx+\frac {d \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{5/2}}{3 b c}\) |
\(\Big \downarrow \) 3049 |
\(\displaystyle \frac {5}{6} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}dx+\frac {d \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{b c}\right )+\frac {d \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{5/2}}{3 b c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}dx+\frac {d \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{b c}\right )+\frac {d \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{5/2}}{3 b c}\) |
\(\Big \downarrow \) 3053 |
\(\displaystyle \frac {5}{6} d^2 \left (\frac {d^2 \sqrt {\sin (2 a+2 b x)} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx}{2 \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}+\frac {d \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{b c}\right )+\frac {d \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{5/2}}{3 b c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} d^2 \left (\frac {d^2 \sqrt {\sin (2 a+2 b x)} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx}{2 \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}+\frac {d \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{b c}\right )+\frac {d \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{5/2}}{3 b c}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {5}{6} d^2 \left (\frac {d^2 \sqrt {\sin (2 a+2 b x)} \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right )}{2 b \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}+\frac {d \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{b c}\right )+\frac {d \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{5/2}}{3 b c}\) |
(d*(d*Cos[a + b*x])^(5/2)*Sqrt[c*Sin[a + b*x]])/(3*b*c) + (5*d^2*((d*Sqrt[ d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])/(b*c) + (d^2*EllipticF[a - Pi/4 + b* x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(2*b*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x ]])))/6
3.3.91.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[a*(b*Sin[e + f*x])^(n + 1)*((a*Cos[e + f*x])^(m - 1)/ (b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Sin[e + f*x])^n*(a *Cos[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && 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]
Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 1727, normalized size of antiderivative = 13.08
1/48/b*2^(1/2)*(d*cos(b*x+a))^(1/2)*d^3/(c*sin(b*x+a))^(1/2)*(-6*I*(-cot(b *x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc( b*x+a))^(1/2)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2-1/2*I,1/2*2^ (1/2))+6*I*(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2 )*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2 ),1/2+1/2*I,1/2*2^(1/2))+6*I*(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)- csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticPi((-cot(b*x+a)+ csc(b*x+a)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*sec(b*x+a)-6*I*(-cot(b*x+a)+csc (b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^( 1/2)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*se c(b*x+a)+8*2^(1/2)*cos(b*x+a)^2*sin(b*x+a)-6*(-cot(b*x+a)+csc(b*x+a)+1)^(1 /2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*Elliptic Pi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))+32*(-cot(b*x+a) +csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a ))^(1/2)*EllipticF((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2*2^(1/2))-6*(-cot(b *x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc( b*x+a))^(1/2)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2-1/2*I,1/2*2^ (1/2))-6*(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)* (cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2), 1/2+1/2*I,1/2*2^(1/2))*sec(b*x+a)+32*(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(...
\[ \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx=\int { \frac {\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}}{\sqrt {c \sin \left (b x + a\right )}} \,d x } \]
Timed out. \[ \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx=\text {Timed out} \]
\[ \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx=\int { \frac {\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}}{\sqrt {c \sin \left (b x + a\right )}} \,d x } \]
\[ \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx=\int { \frac {\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}}{\sqrt {c \sin \left (b x + a\right )}} \,d x } \]
Timed out. \[ \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx=\int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^{7/2}}{\sqrt {c\,\sin \left (a+b\,x\right )}} \,d x \]