3.120 \(\int \frac{\csc ^3(a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx\)

Optimal. Leaf size=55 \[ -\frac{\sqrt{\sin (2 a+2 b x)} \csc ^3(a+b x)}{5 b}-\frac{4 \sqrt{\sin (2 a+2 b x)} \csc (a+b x)}{5 b} \]

[Out]

(-4*Csc[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(5*b) - (Csc[a + b*x]^3*Sqrt[Sin[2*a + 2*b*x]])/(5*b)

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Rubi [A]  time = 0.0533406, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4300, 4292} \[ -\frac{\sqrt{\sin (2 a+2 b x)} \csc ^3(a+b x)}{5 b}-\frac{4 \sqrt{\sin (2 a+2 b x)} \csc (a+b x)}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^3/Sqrt[Sin[2*a + 2*b*x]],x]

[Out]

(-4*Csc[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(5*b) - (Csc[a + b*x]^3*Sqrt[Sin[2*a + 2*b*x]])/(5*b)

Rule 4300

Int[((e_.)*sin[(a_.) + (b_.)*(x_)])^(m_)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Simp[((e*Sin[a + b
*x])^m*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(m + p + 1)), x] + Dist[(m + 2*p + 2)/(e^2*(m + p + 1)), Int[(e*Sin[a
+ b*x])^(m + 2)*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b,
2] &&  !IntegerQ[p] && LtQ[m, -1] && NeQ[m + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]

Rule 4292

Int[((e_.)*sin[(a_.) + (b_.)*(x_)])^(m_.)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Simp[((e*Sin[a +
b*x])^m*(g*Sin[c + d*x])^(p + 1))/(b*g*m), x] /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && Eq
Q[d/b, 2] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rubi steps

\begin{align*} \int \frac{\csc ^3(a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx &=-\frac{\csc ^3(a+b x) \sqrt{\sin (2 a+2 b x)}}{5 b}+\frac{4}{5} \int \frac{\csc (a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx\\ &=-\frac{4 \csc (a+b x) \sqrt{\sin (2 a+2 b x)}}{5 b}-\frac{\csc ^3(a+b x) \sqrt{\sin (2 a+2 b x)}}{5 b}\\ \end{align*}

Mathematica [A]  time = 0.0919217, size = 35, normalized size = 0.64 \[ -\frac{\sqrt{\sin (2 (a+b x))} \csc (a+b x) \left (\csc ^2(a+b x)+4\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^3/Sqrt[Sin[2*a + 2*b*x]],x]

[Out]

-(Csc[a + b*x]*(4 + Csc[a + b*x]^2)*Sqrt[Sin[2*(a + b*x)]])/(5*b)

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{ \left ( \csc \left ( bx+a \right ) \right ) ^{3}{\frac{1}{\sqrt{\sin \left ( 2\,bx+2\,a \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^3/sin(2*b*x+2*a)^(1/2),x)

[Out]

int(csc(b*x+a)^3/sin(2*b*x+2*a)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b x + a\right )^{3}}{\sqrt{\sin \left (2 \, b x + 2 \, a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3/sin(2*b*x+2*a)^(1/2),x, algorithm="maxima")

[Out]

integrate(csc(b*x + a)^3/sqrt(sin(2*b*x + 2*a)), x)

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Fricas [A]  time = 0.498063, size = 198, normalized size = 3.6 \begin{align*} -\frac{\sqrt{2}{\left (4 \, \cos \left (b x + a\right )^{2} - 5\right )} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 4 \,{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )}{5 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3/sin(2*b*x+2*a)^(1/2),x, algorithm="fricas")

[Out]

-1/5*(sqrt(2)*(4*cos(b*x + a)^2 - 5)*sqrt(cos(b*x + a)*sin(b*x + a)) + 4*(cos(b*x + a)^2 - 1)*sin(b*x + a))/((
b*cos(b*x + a)^2 - b)*sin(b*x + a))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**3/sin(2*b*x+2*a)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b x + a\right )^{3}}{\sqrt{\sin \left (2 \, b x + 2 \, a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3/sin(2*b*x+2*a)^(1/2),x, algorithm="giac")

[Out]

integrate(csc(b*x + a)^3/sqrt(sin(2*b*x + 2*a)), x)