3.356 \(\int x \cos (a+b x) \sqrt {\csc (a+b x)} \, dx\)

Optimal. Leaf size=58 \[ \frac {2 x}{b \sqrt {\csc (a+b x)}}-\frac {4 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b^2} \]

[Out]

2*x/b/csc(b*x+a)^(1/2)+4*(sin(1/2*a+1/4*Pi+1/2*b*x)^2)^(1/2)/sin(1/2*a+1/4*Pi+1/2*b*x)*EllipticE(cos(1/2*a+1/4
*Pi+1/2*b*x),2^(1/2))*csc(b*x+a)^(1/2)*sin(b*x+a)^(1/2)/b^2

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Rubi [A]  time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4213, 3771, 2639} \[ \frac {2 x}{b \sqrt {\csc (a+b x)}}-\frac {4 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*Cos[a + b*x]*Sqrt[Csc[a + b*x]],x]

[Out]

(2*x)/(b*Sqrt[Csc[a + b*x]]) - (4*Sqrt[Csc[a + b*x]]*EllipticE[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/b^2

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 4213

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*Csc[(a_.) + (b_.)*(x_)^(n_.)]^(p_)*(x_)^(m_.), x_Symbol] :> -Simp[(x^(m - n
+ 1)*Csc[a + b*x^n]^(p - 1))/(b*n*(p - 1)), x] + Dist[(m - n + 1)/(b*n*(p - 1)), Int[x^(m - n)*Csc[a + b*x^n]^
(p - 1), x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]

Rubi steps

\begin {align*} \int x \cos (a+b x) \sqrt {\csc (a+b x)} \, dx &=\frac {2 x}{b \sqrt {\csc (a+b x)}}-\frac {2 \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx}{b}\\ &=\frac {2 x}{b \sqrt {\csc (a+b x)}}-\frac {\left (2 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \sqrt {\sin (a+b x)} \, dx}{b}\\ &=\frac {2 x}{b \sqrt {\csc (a+b x)}}-\frac {4 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{b^2}\\ \end {align*}

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Mathematica [C]  time = 0.72, size = 106, normalized size = 1.83 \[ \frac {4 \sin \left (\frac {1}{2} (a+b x)\right ) \cos \left (\frac {1}{2} (a+b x)\right ) \sqrt {\csc (a+b x)} \left (2 \tan \left (\frac {1}{2} (a+b x)\right ) \sqrt {\sec ^2\left (\frac {1}{2} (a+b x)\right )} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-6 \tan \left (\frac {1}{2} (a+b x)\right )+3 b x\right )}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cos[a + b*x]*Sqrt[Csc[a + b*x]],x]

[Out]

(4*Cos[(a + b*x)/2]*Sqrt[Csc[a + b*x]]*Sin[(a + b*x)/2]*(3*b*x - 6*Tan[(a + b*x)/2] + 2*Hypergeometric2F1[1/2,
3/4, 7/4, -Tan[(a + b*x)/2]^2]*Sqrt[Sec[(a + b*x)/2]^2]*Tan[(a + b*x)/2]))/(3*b^2)

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(b*x+a)*csc(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cos \left (b x + a\right ) \sqrt {\csc \left (b x + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(b*x+a)*csc(b*x+a)^(1/2),x, algorithm="giac")

[Out]

integrate(x*cos(b*x + a)*sqrt(csc(b*x + a)), x)

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maple [C]  time = 0.12, size = 308, normalized size = 5.31 \[ -\frac {i \left (b x +2 i\right ) \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) \sqrt {2}\, \sqrt {\frac {i {\mathrm e}^{i \left (b x +a \right )}}{{\mathrm e}^{2 i \left (b x +a \right )}-1}}\, {\mathrm e}^{-i \left (b x +a \right )}}{b^{2}}-\frac {2 \left (-\frac {2 i \left (-i+i {\mathrm e}^{2 i \left (b x +a \right )}\right )}{\sqrt {{\mathrm e}^{i \left (b x +a \right )} \left (-i+i {\mathrm e}^{2 i \left (b x +a \right )}\right )}}-\frac {\sqrt {{\mathrm e}^{i \left (b x +a \right )}+1}\, \sqrt {-2 \,{\mathrm e}^{i \left (b x +a \right )}+2}\, \sqrt {-{\mathrm e}^{i \left (b x +a \right )}}\, \left (-2 \EllipticE \left (\sqrt {{\mathrm e}^{i \left (b x +a \right )}+1}, \frac {\sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {{\mathrm e}^{i \left (b x +a \right )}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {i {\mathrm e}^{3 i \left (b x +a \right )}-i {\mathrm e}^{i \left (b x +a \right )}}}\right ) \sqrt {2}\, \sqrt {\frac {i {\mathrm e}^{i \left (b x +a \right )}}{{\mathrm e}^{2 i \left (b x +a \right )}-1}}\, \sqrt {i \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{i \left (b x +a \right )}}\, {\mathrm e}^{-i \left (b x +a \right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(b*x+a)*csc(b*x+a)^(1/2),x)

[Out]

-I*(b*x+2*I)*(exp(I*(b*x+a))^2-1)/b^2*2^(1/2)*(I*exp(I*(b*x+a))/(exp(I*(b*x+a))^2-1))^(1/2)/exp(I*(b*x+a))-2/b
^2*(-2*I*(-I+I*exp(I*(b*x+a))^2)/(exp(I*(b*x+a))*(-I+I*exp(I*(b*x+a))^2))^(1/2)-(exp(I*(b*x+a))+1)^(1/2)*(-2*e
xp(I*(b*x+a))+2)^(1/2)*(-exp(I*(b*x+a)))^(1/2)/(I*exp(I*(b*x+a))^3-I*exp(I*(b*x+a)))^(1/2)*(-2*EllipticE((exp(
I*(b*x+a))+1)^(1/2),1/2*2^(1/2))+EllipticF((exp(I*(b*x+a))+1)^(1/2),1/2*2^(1/2))))*2^(1/2)*(I*exp(I*(b*x+a))/(
exp(I*(b*x+a))^2-1))^(1/2)*(I*(exp(I*(b*x+a))^2-1)*exp(I*(b*x+a)))^(1/2)/exp(I*(b*x+a))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cos \left (b x + a\right ) \sqrt {\csc \left (b x + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(b*x+a)*csc(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(x*cos(b*x + a)*sqrt(csc(b*x + a)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\cos \left (a+b\,x\right )\,\sqrt {\frac {1}{\sin \left (a+b\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(a + b*x)*(1/sin(a + b*x))^(1/2),x)

[Out]

int(x*cos(a + b*x)*(1/sin(a + b*x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cos {\left (a + b x \right )} \sqrt {\csc {\left (a + b x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(b*x+a)*csc(b*x+a)**(1/2),x)

[Out]

Integral(x*cos(a + b*x)*sqrt(csc(a + b*x)), x)

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