Integrand size = 19, antiderivative size = 258 \[ \int \cos ^2(a+b x) \sqrt {\cos (c+d x)} \, dx=\frac {E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {i e^{-\frac {1}{2} i (4 a+c)-\frac {1}{2} i (4 b+d) x+\frac {1}{2} i (c+d x)} \sqrt {\cos (c+d x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4} \left (-1-\frac {4 b}{d}\right ),\frac {1}{4} \left (3-\frac {4 b}{d}\right ),-e^{2 i (c+d x)}\right )}{2 (4 b+d) \sqrt {1+e^{2 i (c+d x)}}}-\frac {i e^{\frac {1}{2} i (4 a-c)+\frac {1}{2} i (4 b-d) x+\frac {1}{2} i (c+d x)} \sqrt {\cos (c+d x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4} \left (-1+\frac {4 b}{d}\right ),\frac {1}{4} \left (3+\frac {4 b}{d}\right ),-e^{2 i (c+d x)}\right )}{2 (4 b-d) \sqrt {1+e^{2 i (c+d x)}}} \] Output:
EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+1/2*I*exp(-1/2*I*(4*a+c)-1/2*I*(4* b+d)*x+1/2*I*(d*x+c))*cos(d*x+c)^(1/2)*hypergeom([-1/2, -1/4-b/d],[3/4-b/d ],-exp(2*I*(d*x+c)))/(4*b+d)/(1+exp(2*I*(d*x+c)))^(1/2)-1/2*I*exp(1/2*I*(4 *a-c)+1/2*I*(4*b-d)*x+1/2*I*(d*x+c))*cos(d*x+c)^(1/2)*hypergeom([-1/2, -1/ 4+b/d],[3/4+b/d],-exp(2*I*(d*x+c)))/(4*b-d)/(1+exp(2*I*(d*x+c)))^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3372\) vs. \(2(258)=516\).
Time = 13.85 (sec) , antiderivative size = 3372, normalized size of antiderivative = 13.07 \[ \int \cos ^2(a+b x) \sqrt {\cos (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[Cos[a + b*x]^2*Sqrt[Cos[c + d*x]],x]
Output:
(2*Sqrt[2]*b^2*Csc[c]*(3*(1 + E^((2*I)*(c + d*x))) + E^((2*I)*(c + d*x))*( -1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4 , 7/4, -E^((2*I)*(c + d*x))]))/(3*d*E^(I*(2*c + d*x))*Sqrt[(1 + E^((2*I)*( c + d*x)))/E^(I*(c + d*x))]*(4*b*Cos[c] - I*d*Sin[c])*(4*b*Cos[c] + I*d*Si n[c])) - (d*Csc[c]*(3*(1 + E^((2*I)*(c + d*x))) + E^((2*I)*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7 /4, -E^((2*I)*(c + d*x))]))/(12*Sqrt[2]*E^(I*(2*c + d*x))*Sqrt[(1 + E^((2* I)*(c + d*x)))/E^(I*(c + d*x))]*(4*b*Cos[c] - I*d*Sin[c])*(4*b*Cos[c] + I* d*Sin[c])) - (2*Sqrt[2]*b^2*Csc[c]*(-3*E^((4*I)*c)*(1 + E^((2*I)*(c + d*x) )) + E^((2*I)*d*x)*(-1 + E^((6*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hyperg eometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/(3*d*E^(I*(2*c + d*x))* Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(4*b*Cos[c] - I*d*Sin[c])* (4*b*Cos[c] + I*d*Sin[c])) + (d*Csc[c]*(-3*E^((4*I)*c)*(1 + E^((2*I)*(c + d*x))) + E^((2*I)*d*x)*(-1 + E^((6*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hy pergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/(12*Sqrt[2]*E^(I*(2 *c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(4*b*Cos[c] - I *d*Sin[c])*(4*b*Cos[c] + I*d*Sin[c])) + (d^2*Sqrt[(1 + E^((2*I)*(c + d*x)) )/E^(I*(c + d*x))]*Csc[c]*(-((4*b - d)*E^((2*I)*d*x)*Hypergeometric2F1[1, 5/4 - b/d, 7/4 - b/d, -E^((2*I)*(c + d*x))]) - (4*b - 3*d)*E^((4*I)*(a + b *x))*Hypergeometric2F1[1, 1/4 + b/d, 3/4 + b/d, -E^((2*I)*(c + d*x))]))...
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(646\) vs. \(2(258)=516\).
Time = 1.16 (sec) , antiderivative size = 646, normalized size of antiderivative = 2.50, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5065, 2009}
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.
\(\displaystyle \int \cos ^2(a+b x) \sqrt {\cos (c+d x)} \, dx\) |
\(\Big \downarrow \) 5065 |
\(\displaystyle \frac {\int \left (e^{-2 i a-2 i b x} \sqrt {e^{-i (c+d x)}+e^{i (c+d x)}}+e^{2 i a+2 i b x} \sqrt {e^{-i (c+d x)}+e^{i (c+d x)}}+2 \sqrt {e^{-i (c+d x)}+e^{i (c+d x)}}\right )dx}{4 \sqrt {2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {2 i \sqrt {e^{-i (c+d x)}+e^{i (c+d x)}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4} \left (-\frac {4 b}{d}-1\right ),\frac {1}{4} \left (3-\frac {4 b}{d}\right ),-e^{2 i (c+d x)}\right ) \exp \left (-\frac {1}{2} i (4 a+c)-\frac {1}{2} i x (4 b+d)+\frac {1}{2} i (c+d x)\right )}{(4 b+d) \sqrt {1+e^{2 i c+2 i d x}}}-\frac {2 i \sqrt {e^{-i (c+d x)}+e^{i (c+d x)}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4} \left (\frac {4 b}{d}-1\right ),\frac {1}{4} \left (\frac {4 b}{d}+3\right ),-e^{2 i (c+d x)}\right ) \exp \left (\frac {1}{2} i (4 a-c)+\frac {1}{2} i x (4 b-d)+\frac {1}{2} i (c+d x)\right )}{(4 b-d) \sqrt {1+e^{2 i c+2 i d x}}}-\frac {4 i \sqrt {e^{i (c+d x)}} \left (1+e^{i (c+d x)}\right ) \sqrt {e^{-i (c+d x)}+e^{i (c+d x)}} \sqrt {\frac {1+e^{2 i (c+d x)}}{\left (1+e^{i (c+d x)}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {e^{i (c+d x)}}\right ),\frac {1}{2}\right )}{d \left (1+e^{2 i (c+d x)}\right )}+\frac {8 i \sqrt {e^{i (c+d x)}} \left (1+e^{i (c+d x)}\right ) \sqrt {e^{-i (c+d x)}+e^{i (c+d x)}} \sqrt {\frac {1+e^{2 i (c+d x)}}{\left (1+e^{i (c+d x)}\right )^2}} E\left (2 \arctan \left (\sqrt {e^{i (c+d x)}}\right )|\frac {1}{2}\right )}{d \left (1+e^{2 i (c+d x)}\right )}+\frac {4 i \sqrt {e^{-i (c+d x)}+e^{i (c+d x)}}}{d}-\frac {8 i e^{i (c+d x)} \sqrt {e^{-i (c+d x)}+e^{i (c+d x)}}}{d \left (1+e^{i (c+d x)}\right )}}{4 \sqrt {2}}\) |
Input:
Int[Cos[a + b*x]^2*Sqrt[Cos[c + d*x]],x]
Output:
(((4*I)*Sqrt[E^((-I)*(c + d*x)) + E^(I*(c + d*x))])/d - ((8*I)*E^(I*(c + d *x))*Sqrt[E^((-I)*(c + d*x)) + E^(I*(c + d*x))])/(d*(1 + E^(I*(c + d*x)))) + ((8*I)*Sqrt[E^(I*(c + d*x))]*(1 + E^(I*(c + d*x)))*Sqrt[E^((-I)*(c + d* x)) + E^(I*(c + d*x))]*Sqrt[(1 + E^((2*I)*(c + d*x)))/(1 + E^(I*(c + d*x)) )^2]*EllipticE[2*ArcTan[Sqrt[E^(I*(c + d*x))]], 1/2])/(d*(1 + E^((2*I)*(c + d*x)))) - ((4*I)*Sqrt[E^(I*(c + d*x))]*(1 + E^(I*(c + d*x)))*Sqrt[E^((-I )*(c + d*x)) + E^(I*(c + d*x))]*Sqrt[(1 + E^((2*I)*(c + d*x)))/(1 + E^(I*( c + d*x)))^2]*EllipticF[2*ArcTan[Sqrt[E^(I*(c + d*x))]], 1/2])/(d*(1 + E^( (2*I)*(c + d*x)))) + ((2*I)*E^((-1/2*I)*(4*a + c) - (I/2)*(4*b + d)*x + (I /2)*(c + d*x))*Sqrt[E^((-I)*(c + d*x)) + E^(I*(c + d*x))]*Hypergeometric2F 1[-1/2, (-1 - (4*b)/d)/4, (3 - (4*b)/d)/4, -E^((2*I)*(c + d*x))])/((4*b + d)*Sqrt[1 + E^((2*I)*c + (2*I)*d*x)]) - ((2*I)*E^((I/2)*(4*a - c) + (I/2)* (4*b - d)*x + (I/2)*(c + d*x))*Sqrt[E^((-I)*(c + d*x)) + E^(I*(c + d*x))]* Hypergeometric2F1[-1/2, (-1 + (4*b)/d)/4, (3 + (4*b)/d)/4, -E^((2*I)*(c + d*x))])/((4*b - d)*Sqrt[1 + E^((2*I)*c + (2*I)*d*x)]))/(4*Sqrt[2])
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*Cos[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Simp[1/2^(p + q) Int[ExpandIntegrand[(E^((-I)*(c + d*x)) + E^(I*(c + d *x)))^q, (E^((-I)*(a + b*x)) + E^(I*(a + b*x)))^p, x], x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && !IntegerQ[q]
\[\int \cos \left (b x +a \right )^{2} \sqrt {\cos \left (d x +c \right )}d x\]
Input:
int(cos(b*x+a)^2*cos(d*x+c)^(1/2),x)
Output:
int(cos(b*x+a)^2*cos(d*x+c)^(1/2),x)
Exception generated. \[ \int \cos ^2(a+b x) \sqrt {\cos (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cos(b*x+a)^2*cos(d*x+c)^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int \cos ^2(a+b x) \sqrt {\cos (c+d x)} \, dx=\int \cos ^{2}{\left (a + b x \right )} \sqrt {\cos {\left (c + d x \right )}}\, dx \] Input:
integrate(cos(b*x+a)**2*cos(d*x+c)**(1/2),x)
Output:
Integral(cos(a + b*x)**2*sqrt(cos(c + d*x)), x)
\[ \int \cos ^2(a+b x) \sqrt {\cos (c+d x)} \, dx=\int { \cos \left (b x + a\right )^{2} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(b*x+a)^2*cos(d*x+c)^(1/2),x, algorithm="maxima")
Output:
integrate(cos(b*x + a)^2*sqrt(cos(d*x + c)), x)
\[ \int \cos ^2(a+b x) \sqrt {\cos (c+d x)} \, dx=\int { \cos \left (b x + a\right )^{2} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(b*x+a)^2*cos(d*x+c)^(1/2),x, algorithm="giac")
Output:
integrate(cos(b*x + a)^2*sqrt(cos(d*x + c)), x)
Timed out. \[ \int \cos ^2(a+b x) \sqrt {\cos (c+d x)} \, dx=\int {\cos \left (a+b\,x\right )}^2\,\sqrt {\cos \left (c+d\,x\right )} \,d x \] Input:
int(cos(a + b*x)^2*cos(c + d*x)^(1/2),x)
Output:
int(cos(a + b*x)^2*cos(c + d*x)^(1/2), x)
\[ \int \cos ^2(a+b x) \sqrt {\cos (c+d x)} \, dx=\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (b x +a \right )^{2}d x \] Input:
int(cos(b*x+a)^2*cos(d*x+c)^(1/2),x)
Output:
int(sqrt(cos(c + d*x))*cos(a + b*x)**2,x)