3.239 \(\int \frac{\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{3 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{d \cos (a+b x)}}{b d^2 \sqrt{\cos (a+b x)}}+\frac{3 \sin (a+b x)}{b d \sqrt{d \cos (a+b x)}}-\frac{\csc (a+b x)}{b d \sqrt{d \cos (a+b x)}} \]

[Out]

-(Csc[a + b*x]/(b*d*Sqrt[d*Cos[a + b*x]])) - (3*Sqrt[d*Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2])/(b*d^2*Sqrt[Co
s[a + b*x]]) + (3*Sin[a + b*x])/(b*d*Sqrt[d*Cos[a + b*x]])

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Rubi [A]  time = 0.0804821, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2570, 2636, 2640, 2639} \[ -\frac{3 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{d \cos (a+b x)}}{b d^2 \sqrt{\cos (a+b x)}}+\frac{3 \sin (a+b x)}{b d \sqrt{d \cos (a+b x)}}-\frac{\csc (a+b x)}{b d \sqrt{d \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^2/(d*Cos[a + b*x])^(3/2),x]

[Out]

-(Csc[a + b*x]/(b*d*Sqrt[d*Cos[a + b*x]])) - (3*Sqrt[d*Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2])/(b*d^2*Sqrt[Co
s[a + b*x]]) + (3*Sin[a + b*x])/(b*d*Sqrt[d*Cos[a + b*x]])

Rule 2570

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[((b*Cos[e + f
*x])^(n + 1)*(a*Sin[e + f*x])^(m + 1))/(a*b*f*(m + 1)), x] + Dist[(m + n + 2)/(a^2*(m + 1)), Int[(b*Cos[e + f*
x])^n*(a*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]

Rule 2636

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

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]

Rubi steps

\begin{align*} \int \frac{\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx &=-\frac{\csc (a+b x)}{b d \sqrt{d \cos (a+b x)}}+\frac{3}{2} \int \frac{1}{(d \cos (a+b x))^{3/2}} \, dx\\ &=-\frac{\csc (a+b x)}{b d \sqrt{d \cos (a+b x)}}+\frac{3 \sin (a+b x)}{b d \sqrt{d \cos (a+b x)}}-\frac{3 \int \sqrt{d \cos (a+b x)} \, dx}{2 d^2}\\ &=-\frac{\csc (a+b x)}{b d \sqrt{d \cos (a+b x)}}+\frac{3 \sin (a+b x)}{b d \sqrt{d \cos (a+b x)}}-\frac{\left (3 \sqrt{d \cos (a+b x)}\right ) \int \sqrt{\cos (a+b x)} \, dx}{2 d^2 \sqrt{\cos (a+b x)}}\\ &=-\frac{\csc (a+b x)}{b d \sqrt{d \cos (a+b x)}}-\frac{3 \sqrt{d \cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b d^2 \sqrt{\cos (a+b x)}}+\frac{3 \sin (a+b x)}{b d \sqrt{d \cos (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.171588, size = 65, normalized size = 0.69 \[ \frac{2 \sin (a+b x)-\cos (a+b x) \cot (a+b x)-3 \sqrt{\cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b d \sqrt{d \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^2/(d*Cos[a + b*x])^(3/2),x]

[Out]

(-(Cos[a + b*x]*Cot[a + b*x]) - 3*Sqrt[Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2] + 2*Sin[a + b*x])/(b*d*Sqrt[d*C
os[a + b*x]])

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Maple [A]  time = 0.319, size = 209, normalized size = 2.2 \begin{align*} -{\frac{1}{2\,{d}^{3}b}\sqrt{d \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}} \left ( -2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}d+ \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}d \right ) ^{{\frac{3}{2}}} \left ( 6\,\sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \cos \left ( 1/2\,bx+a/2 \right ) +12\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}-12\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1 \right ) \left ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-5} \left ( 2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) ^{-2}{\frac{1}{\sqrt{d \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^2/(d*cos(b*x+a))^(3/2),x)

[Out]

-1/2*(d*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)/d^3/cos(1/2*b*x+1/2*a)/sin(1/2*b*x+1/2*a)^5/(2*
sin(1/2*b*x+1/2*a)^2-1)^2*(-2*sin(1/2*b*x+1/2*a)^4*d+sin(1/2*b*x+1/2*a)^2*d)^(3/2)*(6*(2*sin(1/2*b*x+1/2*a)^2-
1)^(1/2)*(sin(1/2*b*x+1/2*a)^2)^(1/2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*cos(1/2*b*x+1/2*a)+12*sin(1/2*b*x+
1/2*a)^4-12*sin(1/2*b*x+1/2*a)^2+1)/(d*(2*cos(1/2*b*x+1/2*a)^2-1))^(1/2)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csc(b*x + a)^2/(d*cos(b*x + a))^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d \cos \left (b x + a\right )} \csc \left (b x + a\right )^{2}}{d^{2} \cos \left (b x + a\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(d*cos(b*x + a))*csc(b*x + a)^2/(d^2*cos(b*x + a)^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**2/(d*cos(b*x+a))**(3/2),x)

[Out]

Integral(csc(a + b*x)**2/(d*cos(a + b*x))**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csc(b*x + a)^2/(d*cos(b*x + a))^(3/2), x)