Integrand size = 25, antiderivative size = 132 \[ \int (d \cos (a+b x))^{9/2} \sqrt {c \sin (a+b x)} \, dx=\frac {7 d^3 (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{30 b c}+\frac {d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{5 b c}+\frac {7 d^4 \sqrt {d \cos (a+b x)} E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sin (a+b x)}}{20 b \sqrt {\sin (2 a+2 b x)}} \]
7/30*d^3*(d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^(3/2)/b/c+1/5*d*(d*cos(b*x+a) )^(7/2)*(c*sin(b*x+a))^(3/2)/b/c-7/20*d^4*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin( a+1/4*Pi+b*x)*EllipticE(cos(a+1/4*Pi+b*x),2^(1/2))*(d*cos(b*x+a))^(1/2)*(c *sin(b*x+a))^(1/2)/b/sin(2*b*x+2*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 \[ \int (d \cos (a+b x))^{9/2} \sqrt {c \sin (a+b x)} \, dx=\frac {2 d^4 \sqrt {d \cos (a+b x)} \sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {3}{4},\frac {7}{4},\sin ^2(a+b x)\right ) \sqrt {c \sin (a+b x)} \tan (a+b x)}{3 b} \]
(2*d^4*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[-7/4, 3/4, 7/4, Sin[a + b*x]^2]*Sqrt[c*Sin[a + b*x]]*Tan[a + b*x])/(3*b)
Time = 0.58 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.05, 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, 3052, 3042, 3119}
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.
| \(\displaystyle \int \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{9/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{9/2}dx\) |
| \(\Big \downarrow \) 3049 |
| \(\displaystyle \frac {7}{10} d^2 \int (d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)}dx+\frac {d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{5 b c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{10} d^2 \int (d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)}dx+\frac {d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{5 b c}\) |
| \(\Big \downarrow \) 3049 |
| \(\displaystyle \frac {7}{10} d^2 \left (\frac {1}{2} d^2 \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}dx+\frac {d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{3 b c}\right )+\frac {d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{5 b c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{10} d^2 \left (\frac {1}{2} d^2 \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}dx+\frac {d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{3 b c}\right )+\frac {d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{5 b c}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle \frac {7}{10} d^2 \left (\frac {d^2 \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)} \int \sqrt {\sin (2 a+2 b x)}dx}{2 \sqrt {\sin (2 a+2 b x)}}+\frac {d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{3 b c}\right )+\frac {d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{5 b c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{10} d^2 \left (\frac {d^2 \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)} \int \sqrt {\sin (2 a+2 b x)}dx}{2 \sqrt {\sin (2 a+2 b x)}}+\frac {d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{3 b c}\right )+\frac {d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{5 b c}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {7}{10} d^2 \left (\frac {d^2 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{2 b \sqrt {\sin (2 a+2 b x)}}+\frac {d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{3 b c}\right )+\frac {d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{5 b c}\) |
(d*(d*Cos[a + b*x])^(7/2)*(c*Sin[a + b*x])^(3/2))/(5*b*c) + (7*d^2*((d*(d* Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^(3/2))/(3*b*c) + (d^2*Sqrt[d*Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(2*b*Sqrt[Sin[2*a + 2*b*x]])))/10
3.3.57.3.1 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[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] , x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]) Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(422\) vs. \(2(137)=274\).
Time = 0.91 (sec) , antiderivative size = 423, normalized size of antiderivative = 3.20
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \sqrt {d \cos \left (b x +a \right )}\, \sqrt {c \sin \left (b x +a \right )}\, \left (12 \sqrt {2}\, \left (\cos ^{6}\left (b x +a \right )\right )+2 \sqrt {2}\, \left (\cos ^{4}\left (b x +a \right )\right )-21 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )+42 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )-21 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )+42 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )+7 \sqrt {2}\, \left (\cos ^{2}\left (b x +a \right )\right )-21 \sqrt {2}\, \cos \left (b x +a \right )\right ) d^{4} \sec \left (b x +a \right ) \csc \left (b x +a \right )}{120 b}\) | \(423\) |
-1/120/b*2^(1/2)*(d*cos(b*x+a))^(1/2)*(c*sin(b*x+a))^(1/2)*(12*2^(1/2)*cos (b*x+a)^6+2*2^(1/2)*cos(b*x+a)^4-21*(-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))*cos(b*x+a)+42*(-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)* EllipticE((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2*2^(1/2))*cos(b*x+a)-21*(-co t(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)-c sc(b*x+a))^(1/2)*EllipticF((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2*2^(1/2))+4 2*(-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)*EllipticE((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2*2^(1 /2))+7*2^(1/2)*cos(b*x+a)^2-21*2^(1/2)*cos(b*x+a))*d^4*sec(b*x+a)*csc(b*x+ a)
\[ \int (d \cos (a+b x))^{9/2} \sqrt {c \sin (a+b x)} \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \sqrt {c \sin \left (b x + a\right )} \,d x } \]
Timed out. \[ \int (d \cos (a+b x))^{9/2} \sqrt {c \sin (a+b x)} \, dx=\text {Timed out} \]
\[ \int (d \cos (a+b x))^{9/2} \sqrt {c \sin (a+b x)} \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \sqrt {c \sin \left (b x + a\right )} \,d x } \]
\[ \int (d \cos (a+b x))^{9/2} \sqrt {c \sin (a+b x)} \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \sqrt {c \sin \left (b x + a\right )} \,d x } \]
Timed out. \[ \int (d \cos (a+b x))^{9/2} \sqrt {c \sin (a+b x)} \, dx=\int {\left (d\,\cos \left (a+b\,x\right )\right )}^{9/2}\,\sqrt {c\,\sin \left (a+b\,x\right )} \,d x \]