Integrand size = 35, antiderivative size = 116 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {b (4 A+3 C) x \sqrt {b \cos (c+d x)}}{8 \sqrt {\cos (c+d x)}}+\frac {b (4 A+3 C) \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{8 d}+\frac {b C \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d} \] Output:
1/8*b*(4*A+3*C)*x*(b*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2)+1/8*b*(4*A+3*C)*co s(d*x+c)^(1/2)*(b*cos(d*x+c))^(1/2)*sin(d*x+c)/d+1/4*b*C*cos(d*x+c)^(5/2)* (b*cos(d*x+c))^(1/2)*sin(d*x+c)/d
Time = 0.61 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.58 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {(b \cos (c+d x))^{3/2} (4 (4 A+3 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+C \sin (4 (c+d x)))}{32 d \cos ^{\frac {3}{2}}(c+d x)} \] Input:
Integrate[Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2) ,x]
Output:
((b*Cos[c + d*x])^(3/2)*(4*(4*A + 3*C)*(c + d*x) + 8*(A + C)*Sin[2*(c + d* x)] + C*Sin[4*(c + d*x)]))/(32*d*Cos[c + d*x]^(3/2))
Time = 0.33 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.72, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2031, 3042, 3493, 3042, 3115, 24}
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.
\(\displaystyle \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 2031 |
\(\displaystyle \frac {b \sqrt {b \cos (c+d x)} \int \cos ^2(c+d x) \left (C \cos ^2(c+d x)+A\right )dx}{\sqrt {\cos (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b \sqrt {b \cos (c+d x)} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )dx}{\sqrt {\cos (c+d x)}}\) |
\(\Big \downarrow \) 3493 |
\(\displaystyle \frac {b \sqrt {b \cos (c+d x)} \left (\frac {1}{4} (4 A+3 C) \int \cos ^2(c+d x)dx+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{\sqrt {\cos (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b \sqrt {b \cos (c+d x)} \left (\frac {1}{4} (4 A+3 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{\sqrt {\cos (c+d x)}}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {b \sqrt {b \cos (c+d x)} \left (\frac {1}{4} (4 A+3 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{\sqrt {\cos (c+d x)}}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {b \sqrt {b \cos (c+d x)} \left (\frac {1}{4} (4 A+3 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{\sqrt {\cos (c+d x)}}\) |
Input:
Int[Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]
Output:
(b*Sqrt[b*Cos[c + d*x]]*((C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + ((4*A + 3 *C)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4))/Sqrt[Cos[c + d*x]]
Int[(Fx_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Simp[a^(m + 1/ 2)*b^(n - 1/2)*(Sqrt[b*v]/Sqrt[a*v]) Int[v^(m + n)*Fx, x], x] /; FreeQ[{a , b, m}, x] && !IntegerQ[m] && IGtQ[n + 1/2, 0] && IntegerQ[m + n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*( x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f *(m + 2))), x] + Simp[(A*(m + 2) + C*(m + 1))/(m + 2) Int[(b*Sin[e + f*x] )^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && !LtQ[m, -1]
Time = 0.44 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.72
method | result | size |
default | \(\frac {b \left (4 A \left (d x +c \right )+3 C \left (d x +c \right )+4 A \cos \left (d x +c \right ) \sin \left (d x +c \right )+\sin \left (d x +c \right ) \cos \left (d x +c \right ) \left (2 \cos \left (d x +c \right )^{2}+3\right ) C \right ) \sqrt {b \cos \left (d x +c \right )}}{8 d \sqrt {\cos \left (d x +c \right )}}\) | \(83\) |
parts | \(\frac {A \left (\cos \left (d x +c \right ) \sin \left (d x +c \right )+d x +c \right ) b \sqrt {b \cos \left (d x +c \right )}}{2 d \sqrt {\cos \left (d x +c \right )}}+\frac {C \left (2 \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )+3 \cos \left (d x +c \right ) \sin \left (d x +c \right )+3 d x +3 c \right ) b \sqrt {b \cos \left (d x +c \right )}}{8 d \sqrt {\cos \left (d x +c \right )}}\) | \(108\) |
risch | \(\frac {b \sqrt {b \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}\, {\mathrm e}^{i \left (d x +c \right )} \left (8 A +6 C \right ) x}{8 \,{\mathrm e}^{2 i \left (d x +c \right )}+8}-\frac {i b \sqrt {b \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}\, {\mathrm e}^{5 i \left (d x +c \right )} C}{32 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}+\frac {i b \sqrt {b \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}\, {\mathrm e}^{-i \left (d x +c \right )} \left (A +C \right )}{4 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}-\frac {i b \sqrt {b \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}\, \left (8 A +7 C \right ) \cos \left (3 d x +3 c \right )}{32 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}+\frac {b \sqrt {b \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}\, \left (9 C +8 A \right ) \sin \left (3 d x +3 c \right )}{32 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}\) | \(258\) |
Input:
int(cos(d*x+c)^(1/2)*(b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2),x,method=_RET URNVERBOSE)
Output:
1/8*b/d*(4*A*(d*x+c)+3*C*(d*x+c)+4*A*cos(d*x+c)*sin(d*x+c)+sin(d*x+c)*cos( d*x+c)*(2*cos(d*x+c)^2+3)*C)*(b*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2)
Time = 0.12 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.80 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\left [\frac {{\left (4 \, A + 3 \, C\right )} \sqrt {-b} b \log \left (2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right ) + 2 \, {\left (2 \, C b \cos \left (d x + c\right )^{2} + {\left (4 \, A + 3 \, C\right )} b\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{16 \, d}, \frac {{\left (4 \, A + 3 \, C\right )} b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right ) + {\left (2 \, C b \cos \left (d x + c\right )^{2} + {\left (4 \, A + 3 \, C\right )} b\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{8 \, d}\right ] \] Input:
integrate(cos(d*x+c)^(1/2)*(b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2),x, algo rithm="fricas")
Output:
[1/16*((4*A + 3*C)*sqrt(-b)*b*log(2*b*cos(d*x + c)^2 - 2*sqrt(b*cos(d*x + c))*sqrt(-b)*sqrt(cos(d*x + c))*sin(d*x + c) - b) + 2*(2*C*b*cos(d*x + c)^ 2 + (4*A + 3*C)*b)*sqrt(b*cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c))/d , 1/8*((4*A + 3*C)*b^(3/2)*arctan(sqrt(b*cos(d*x + c))*sin(d*x + c)/(sqrt( b)*cos(d*x + c)^(3/2))) + (2*C*b*cos(d*x + c)^2 + (4*A + 3*C)*b)*sqrt(b*co s(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c))/d]
Timed out. \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(1/2)*(b*cos(d*x+c))**(3/2)*(A+C*cos(d*x+c)**2),x)
Output:
Timed out
Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.71 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {8 \, {\left (2 \, {\left (d x + c\right )} b + b \sin \left (2 \, d x + 2 \, c\right )\right )} A \sqrt {b} + {\left (12 \, {\left (d x + c\right )} b + b \sin \left (4 \, d x + 4 \, c\right ) + 8 \, b \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, d x + 4 \, c\right ), \cos \left (4 \, d x + 4 \, c\right )\right )\right )\right )} C \sqrt {b}}{32 \, d} \] Input:
integrate(cos(d*x+c)^(1/2)*(b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2),x, algo rithm="maxima")
Output:
1/32*(8*(2*(d*x + c)*b + b*sin(2*d*x + 2*c))*A*sqrt(b) + (12*(d*x + c)*b + b*sin(4*d*x + 4*c) + 8*b*sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4* c))))*C*sqrt(b))/d
Time = 0.16 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.41 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{32} \, {\left (4 \, {\left (4 \, A + 3 \, C\right )} x + \frac {C \sin \left (4 \, d x + 4 \, c\right )}{d} + \frac {8 \, {\left (A + C\right )} \sin \left (2 \, d x + 2 \, c\right )}{d}\right )} b^{\frac {3}{2}} \] Input:
integrate(cos(d*x+c)^(1/2)*(b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2),x, algo rithm="giac")
Output:
1/32*(4*(4*A + 3*C)*x + C*sin(4*d*x + 4*c)/d + 8*(A + C)*sin(2*d*x + 2*c)/ d)*b^(3/2)
Time = 36.89 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.97 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {b\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (8\,A\,\sin \left (c+d\,x\right )+8\,C\,\sin \left (c+d\,x\right )+8\,A\,\sin \left (3\,c+3\,d\,x\right )+9\,C\,\sin \left (3\,c+3\,d\,x\right )+C\,\sin \left (5\,c+5\,d\,x\right )+32\,A\,d\,x\,\cos \left (c+d\,x\right )+24\,C\,d\,x\,\cos \left (c+d\,x\right )\right )}{32\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \] Input:
int(cos(c + d*x)^(1/2)*(A + C*cos(c + d*x)^2)*(b*cos(c + d*x))^(3/2),x)
Output:
(b*cos(c + d*x)^(1/2)*(b*cos(c + d*x))^(1/2)*(8*A*sin(c + d*x) + 8*C*sin(c + d*x) + 8*A*sin(3*c + 3*d*x) + 9*C*sin(3*c + 3*d*x) + C*sin(5*c + 5*d*x) + 32*A*d*x*cos(c + d*x) + 24*C*d*x*cos(c + d*x)))/(32*d*(cos(2*c + 2*d*x) + 1))
Time = 0.16 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.57 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {b}\, b \left (-2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} c +4 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +5 \cos \left (d x +c \right ) \sin \left (d x +c \right ) c +4 a d x +3 c d x \right )}{8 d} \] Input:
int(cos(d*x+c)^(1/2)*(b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2),x)
Output:
(sqrt(b)*b*( - 2*cos(c + d*x)*sin(c + d*x)**3*c + 4*cos(c + d*x)*sin(c + d *x)*a + 5*cos(c + d*x)*sin(c + d*x)*c + 4*a*d*x + 3*c*d*x))/(8*d)