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]

-4/5*csc(b*x+a)*sin(2*b*x+2*a)^(1/2)/b-1/5*csc(b*x+a)^3*sin(2*b*x+2*a)^(1/2)/b

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Rubi [A]  time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.09, 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]

-1/5*(Csc[a + b*x]*(4 + Csc[a + b*x]^2)*Sqrt[Sin[2*(a + b*x)]])/b

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fricas [A]  time = 0.48, size = 76, normalized size = 1.38 \[ -\frac {\sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{2} - 5\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 4 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )}{5 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (b x + a\right )^{3}}{\sqrt {\sin \left (2 \, b x + 2 \, a\right )}}\,{d x} \]

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)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{3}\left (b x +a \right )}{\sqrt {\sin \left (2 b x +2 a \right )}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (b x + a\right )^{3}}{\sqrt {\sin \left (2 \, b x + 2 \, a\right )}}\,{d x} \]

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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mupad [B]  time = 3.16, size = 93, normalized size = 1.69 \[ -\frac {8\,{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}}\,\left (-{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,3{}\mathrm {i}+{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}{5\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(8*exp(a*1i + b*x*1i)*((exp(- a*2i - b*x*2i)*1i)/2 - (exp(a*2i + b*x*2i)*1i)/2)^(1/2)*(exp(a*4i + b*x*4i)*1i
- exp(a*2i + b*x*2i)*3i + 1i))/(5*b*(exp(a*2i + b*x*2i) - 1)^3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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