Integrand size = 25, antiderivative size = 433 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \, dx=-\frac {5 (a-b) \sqrt {a+b} \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{4 d}+\frac {\sqrt {a+b} (5 a+2 b) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{4 d}-\frac {\sqrt {a+b} \left (3 a^2+4 b^2\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{4 b d}+\frac {3 a \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 d \sqrt {\cos (c+d x)}}+\frac {(a+b \cos (c+d x))^{3/2} \sin (c+d x)}{2 d \sqrt {\cos (c+d x)}} \] Output:
-5/4*(a-b)*(a+b)^(1/2)*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)/d+1/4*(a+b)^(1/2)*(5*a+2*b)*cot(d*x+c)*Ell ipticF((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)/d-1/4 *(a+b)^(1/2)*(3*a^2+4*b^2)*cot(d*x+c)*EllipticPi((a+b*cos(d*x+c))^(1/2)/(a +b)^(1/2)/cos(d*x+c)^(1/2),(a+b)/b,(-(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)/b/d+3/4*a*(a+b*cos(d*x+c))^(1 /2)*sin(d*x+c)/d/cos(d*x+c)^(1/2)+1/2*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/d/ cos(d*x+c)^(1/2)
Time = 12.66 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.01 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \, dx=\frac {\sqrt {\cos (c+d x)} \left (4 b (a+b \cos (c+d x)) \sin (c+d x)+\frac {10 a (a+b) \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right )-4 \left (4 a^2-a b+2 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )+12 a^2 \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )+16 b^2 \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )+5 a b \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sec \left (\frac {1}{2} (c+d x)\right ) \sin \left (\frac {3}{2} (c+d x)\right )+10 a^2 \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \tan \left (\frac {1}{2} (c+d x)\right )-5 a b \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}}}\right )}{8 d \sqrt {a+b \cos (c+d x)}} \] Input:
Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2),x]
Output:
(Sqrt[Cos[c + d*x]]*(4*b*(a + b*Cos[c + d*x])*Sin[c + d*x] + (10*a*(a + b) *Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[ Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*(4*a^2 - a*b + 2*b^2)*Sqrt[(a + b *Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x )/2]], (-a + b)/(a + b)] + 12*a^2*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 16*b^2*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Elliptic Pi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 5*a*b*Sqrt[Cos[c + d* x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sin[(3*(c + d*x))/2] + 10*a^2*Sqrt [Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2] - 5*a*b*Sqrt[Cos[c + d* x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2])/Sqrt[Cos[c + d*x]/(1 + Cos[c + d* x])]))/(8*d*Sqrt[a + b*Cos[c + d*x]])
Time = 2.20 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.03, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3042, 3300, 27, 3042, 3526, 27, 3042, 3540, 25, 3042, 3532, 3042, 3288, 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.
\(\displaystyle \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx\) |
\(\Big \downarrow \) 3300 |
\(\displaystyle \frac {\int -\frac {\sqrt {a+b \cos (c+d x)} \left (-2 \cos (c+d x) b^2-3 a \cos ^2(c+d x) b+a b\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx}{2 b}+\frac {\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt {\cos (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt {\cos (c+d x)}}-\frac {\int \frac {\sqrt {a+b \cos (c+d x)} \left (-2 \cos (c+d x) b^2-3 a \cos ^2(c+d x) b+a b\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx}{4 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt {\cos (c+d x)}}-\frac {\int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (-2 \sin \left (c+d x+\frac {\pi }{2}\right ) b^2-3 a \sin \left (c+d x+\frac {\pi }{2}\right )^2 b+a b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx}{4 b}\) |
\(\Big \downarrow \) 3526 |
\(\displaystyle \frac {\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt {\cos (c+d x)}}-\frac {2 \int -\frac {5 a \cos ^2(c+d x) b^2+a b^2+2 \left (2 a^2+b^2\right ) \cos (c+d x) b}{2 \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {2 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{4 b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt {\cos (c+d x)}}-\frac {\frac {2 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\int \frac {5 a \cos ^2(c+d x) b^2+a b^2+2 \left (2 a^2+b^2\right ) \cos (c+d x) b}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx}{4 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt {\cos (c+d x)}}-\frac {\frac {2 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\int \frac {5 a \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^2+a b^2+2 \left (2 a^2+b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 b}\) |
\(\Big \downarrow \) 3540 |
\(\displaystyle \frac {\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt {\cos (c+d x)}}-\frac {-\frac {\int -\frac {-2 a \cos (c+d x) b^3+5 a^2 b^2-\left (3 a^2+4 b^2\right ) \cos ^2(c+d x) b^2}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}-\frac {3 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{4 b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt {\cos (c+d x)}}-\frac {\frac {\int \frac {-2 a \cos (c+d x) b^3+5 a^2 b^2-\left (3 a^2+4 b^2\right ) \cos ^2(c+d x) b^2}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}-\frac {3 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{4 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt {\cos (c+d x)}}-\frac {\frac {\int \frac {-2 a \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+5 a^2 b^2-\left (3 a^2+4 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}-\frac {3 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{4 b}\) |
\(\Big \downarrow \) 3532 |
\(\displaystyle \frac {\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt {\cos (c+d x)}}-\frac {\frac {\int \frac {5 a^2 b^2-2 a b^3 \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx-b^2 \left (3 a^2+4 b^2\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}}dx}{2 b}-\frac {3 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{4 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt {\cos (c+d x)}}-\frac {\frac {\int \frac {5 a^2 b^2-2 a b^3 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-b^2 \left (3 a^2+4 b^2\right ) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}-\frac {3 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{4 b}\) |
\(\Big \downarrow \) 3288 |
\(\displaystyle \frac {\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt {\cos (c+d x)}}-\frac {\frac {\int \frac {5 a^2 b^2-2 a b^3 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \sqrt {a+b} \left (3 a^2+4 b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}-\frac {3 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{4 b}\) |
\(\Big \downarrow \) 3477 |
\(\displaystyle \frac {\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt {\cos (c+d x)}}-\frac {\frac {5 a^2 b^2 \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx-a b^2 (5 a+2 b) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {2 b \sqrt {a+b} \left (3 a^2+4 b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}-\frac {3 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{4 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt {\cos (c+d x)}}-\frac {\frac {5 a^2 b^2 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-a b^2 (5 a+2 b) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \sqrt {a+b} \left (3 a^2+4 b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}-\frac {3 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{4 b}\) |
\(\Big \downarrow \) 3295 |
\(\displaystyle \frac {\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt {\cos (c+d x)}}-\frac {\frac {5 a^2 b^2 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \sqrt {a+b} \left (3 a^2+4 b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}-\frac {2 b^2 \sqrt {a+b} (5 a+2 b) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}-\frac {3 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{4 b}\) |
\(\Big \downarrow \) 3473 |
\(\displaystyle \frac {\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt {\cos (c+d x)}}-\frac {\frac {\frac {2 b \sqrt {a+b} \left (3 a^2+4 b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}-\frac {2 b^2 \sqrt {a+b} (5 a+2 b) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}+\frac {10 b^2 (a-b) \sqrt {a+b} \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{d}}{2 b}-\frac {3 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{4 b}\) |
Input:
Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2),x]
Output:
((a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]) - (((10 *(a - b)*b^2*Sqrt[a + b]*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)])/d - (2*b^2*S qrt[a + b]*(5*a + 2*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d* x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Se c[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*b*Sqrt[ a + b]*(3*a^2 + 4*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqr t[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d) /(2*b) - (3*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x ]]))/(4*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[2*b*(Tan[e + f*x]/(d*f))*Rt[(c + d)/b, 2]*Sqrt[c *((1 + Csc[e + f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*Ellipti cPi[(c + d)/d, 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}, x] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/b]
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_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x] )^(m - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Simp[1/(d*(m + n)) I nt[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n - 1)*Simp[a^2*c*d*( m + n) + b*d*(b*c*(m - 1) + a*d*n) + (a*d*(2*b*c + a*d)*(m + n) - b*d*(a*c - b*d*(m + n - 1)))*Sin[e + f*x] + b*d*(b*c*n + a*d*(2*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[0, m, 2] && LtQ[-1, n, 2] && NeQ[m + n, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
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[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[C/b^2 Int[Sqrt[a + b*Sin[e + f*x]] /Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[1/b^2 Int[(A*b^2 - a^2*C + b*(b* B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x ]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] & & NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(Sqrt[c + d*Sin[e + f *x]]/(d*f*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[1/(2*d) Int[(1/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]))*Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - d*(A*b + a*B))*Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a *d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(833\) vs. \(2(382)=764\).
Time = 9.21 (sec) , antiderivative size = 834, normalized size of antiderivative = 1.93
Input:
int(cos(d*x+c)^(1/2)*(a+cos(d*x+c)*b)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/4/d*((-6*cos(d*x+c)^2-12*cos(d*x+c)-6)*((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*EllipticPi(cot(d*x+c)-c sc(d*x+c),-1,(-(a-b)/(a+b))^(1/2))+(-8*cos(d*x+c)^2-16*cos(d*x+c)-8)*(cos( d*x+c)/(cos(d*x+c)+1))^(1/2)*((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2) *b^2*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,(-(a-b)/(a+b))^(1/2))+(-5*cos(d*x +c)^2-10*cos(d*x+c)-5)*((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*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/( a+b))^(1/2))+(-5*cos(d*x+c)^2-10*cos(d*x+c)-5)*((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*EllipticE(cot(d*x +c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))+(8*cos(d*x+c)^2+16*cos(d*x+c)+8)*((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*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))+(-2*cos(d*x+c) ^2-4*cos(d*x+c)-2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*((a+cos(d*x+c)*b)/(co s(d*x+c)+1)/(a+b))^(1/2)*a*b*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b) )^(1/2))+(4*cos(d*x+c)^2+8*cos(d*x+c)+4)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2) *((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*b^2*EllipticF(cot(d*x+c)-cs c(d*x+c),(-(a-b)/(a+b))^(1/2))+5*a^2*cos(d*x+c)*sin(d*x+c)+sin(d*x+c)*cos( d*x+c)*(7*cos(d*x+c)+2)*a*b+sin(d*x+c)*cos(d*x+c)^2*(2+2*cos(d*x+c))*b^2)* (a+cos(d*x+c)*b)^(1/2)/cos(d*x+c)^(1/2)/(b*cos(d*x+c)^2+a*cos(d*x+c)+cos(d *x+c)*b+a)
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(3/2),x, algorithm="fricas")
Output:
integral((b*cos(d*x + c) + a)^(3/2)*sqrt(cos(d*x + c)), x)
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \, dx=\int \left (a + b \cos {\left (c + d x \right )}\right )^{\frac {3}{2}} \sqrt {\cos {\left (c + d x \right )}}\, dx \] Input:
integrate(cos(d*x+c)**(1/2)*(a+b*cos(d*x+c))**(3/2),x)
Output:
Integral((a + b*cos(c + d*x))**(3/2)*sqrt(cos(c + d*x)), x)
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(3/2),x, algorithm="maxima")
Output:
integrate((b*cos(d*x + c) + a)^(3/2)*sqrt(cos(d*x + c)), x)
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(3/2),x, algorithm="giac")
Output:
integrate((b*cos(d*x + c) + a)^(3/2)*sqrt(cos(d*x + c)), x)
Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \, dx=\int \sqrt {\cos \left (c+d\,x\right )}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \] Input:
int(cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(3/2),x)
Output:
int(cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(3/2), x)
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \, dx=\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}d x \right ) a \] Input:
int(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(3/2),x)
Output:
int(sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x),x)*b + int(sq rt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)),x)*a