Integrand size = 21, antiderivative size = 76 \[ \int \sqrt {d \cos (a+b x)} \csc ^p(a+b x) \, dx=\frac {d \sqrt [4]{\cos ^2(a+b x)} \csc ^{-1+p}(a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1-p}{2},\frac {3-p}{2},\sin ^2(a+b x)\right )}{b (1-p) \sqrt {d \cos (a+b x)}} \] Output:
d*(cos(b*x+a)^2)^(1/4)*csc(b*x+a)^(-1+p)*hypergeom([1/4, 1/2-1/2*p],[3/2-1 /2*p],sin(b*x+a)^2)/b/(1-p)/(d*cos(b*x+a))^(1/2)
Time = 10.50 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.92 \[ \int \sqrt {d \cos (a+b x)} \csc ^p(a+b x) \, dx=-\frac {2 (d \cos (a+b x))^{3/2} \csc ^{-1+p}(a+b x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {1+p}{2},\frac {7}{4},\cos ^2(a+b x)\right ) \sin ^2(a+b x)^{\frac {1}{2} (-1+p)}}{3 b d} \] Input:
Integrate[Sqrt[d*Cos[a + b*x]]*Csc[a + b*x]^p,x]
Output:
(-2*(d*Cos[a + b*x])^(3/2)*Csc[a + b*x]^(-1 + p)*Hypergeometric2F1[3/4, (1 + p)/2, 7/4, Cos[a + b*x]^2]*(Sin[a + b*x]^2)^((-1 + p)/2))/(3*b*d)
Time = 0.34 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3067, 3042, 3057}
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.
\(\displaystyle \int \sqrt {d \cos (a+b x)} \csc ^p(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {d \sin \left (a+b x+\frac {\pi }{2}\right )} \left (-\sec \left (a+b x+\frac {\pi }{2}\right )\right )^pdx\) |
\(\Big \downarrow \) 3067 |
\(\displaystyle \sin ^p(a+b x) \csc ^p(a+b x) \int \sqrt {d \cos (a+b x)} \sin ^{-p}(a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sin ^p(a+b x) \csc ^p(a+b x) \int \sqrt {d \cos (a+b x)} \sin (a+b x)^{-p}dx\) |
\(\Big \downarrow \) 3057 |
\(\displaystyle \frac {d \sqrt [4]{\cos ^2(a+b x)} \csc ^{p-1}(a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1-p}{2},\frac {3-p}{2},\sin ^2(a+b x)\right )}{b (1-p) \sqrt {d \cos (a+b x)}}\) |
Input:
Int[Sqrt[d*Cos[a + b*x]]*Csc[a + b*x]^p,x]
Output:
(d*(Cos[a + b*x]^2)^(1/4)*Csc[a + b*x]^(-1 + p)*Hypergeometric2F1[1/4, (1 - p)/2, (3 - p)/2, Sin[a + b*x]^2])/(b*(1 - p)*Sqrt[d*Cos[a + b*x]])
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[b^(2*IntPart[(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*Frac Part[(n - 1)/2])*((a*Sin[e + f*x])^(m + 1)/(a*f*(m + 1)*(Cos[e + f*x]^2)^Fr acPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[ e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[b^2*(b*Cos[e + f*x])^(n - 1)*(b*Sec[e + f*x])^(n - 1) Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n]
\[\int \sqrt {\cos \left (b x +a \right ) d}\, \csc \left (b x +a \right )^{p}d x\]
Input:
int((cos(b*x+a)*d)^(1/2)*csc(b*x+a)^p,x)
Output:
int((cos(b*x+a)*d)^(1/2)*csc(b*x+a)^p,x)
\[ \int \sqrt {d \cos (a+b x)} \csc ^p(a+b x) \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \csc \left (b x + a\right )^{p} \,d x } \] Input:
integrate((d*cos(b*x+a))^(1/2)*csc(b*x+a)^p,x, algorithm="fricas")
Output:
integral(sqrt(d*cos(b*x + a))*csc(b*x + a)^p, x)
\[ \int \sqrt {d \cos (a+b x)} \csc ^p(a+b x) \, dx=\int \sqrt {d \cos {\left (a + b x \right )}} \csc ^{p}{\left (a + b x \right )}\, dx \] Input:
integrate((d*cos(b*x+a))**(1/2)*csc(b*x+a)**p,x)
Output:
Integral(sqrt(d*cos(a + b*x))*csc(a + b*x)**p, x)
\[ \int \sqrt {d \cos (a+b x)} \csc ^p(a+b x) \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \csc \left (b x + a\right )^{p} \,d x } \] Input:
integrate((d*cos(b*x+a))^(1/2)*csc(b*x+a)^p,x, algorithm="maxima")
Output:
integrate(sqrt(d*cos(b*x + a))*csc(b*x + a)^p, x)
\[ \int \sqrt {d \cos (a+b x)} \csc ^p(a+b x) \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \csc \left (b x + a\right )^{p} \,d x } \] Input:
integrate((d*cos(b*x+a))^(1/2)*csc(b*x+a)^p,x, algorithm="giac")
Output:
integrate(sqrt(d*cos(b*x + a))*csc(b*x + a)^p, x)
Timed out. \[ \int \sqrt {d \cos (a+b x)} \csc ^p(a+b x) \, dx=\int \sqrt {d\,\cos \left (a+b\,x\right )}\,{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^p \,d x \] Input:
int((d*cos(a + b*x))^(1/2)*(1/sin(a + b*x))^p,x)
Output:
int((d*cos(a + b*x))^(1/2)*(1/sin(a + b*x))^p, x)
\[ \int \sqrt {d \cos (a+b x)} \csc ^p(a+b x) \, dx=\sqrt {d}\, \left (\int \csc \left (b x +a \right )^{p} \sqrt {\cos \left (b x +a \right )}d x \right ) \] Input:
int((d*cos(b*x+a))^(1/2)*csc(b*x+a)^p,x)
Output:
sqrt(d)*int(csc(a + b*x)**p*sqrt(cos(a + b*x)),x)