\(\int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx\) [239]

Optimal result

Integrand size = 21, antiderivative size = 94 \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=-\frac {\csc (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {3 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b d^2 \sqrt {\cos (a+b x)}}+\frac {3 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}} \]

[Out]

-csc(b*x+a)/b/d/(d*cos(b*x+a))^(1/2)+3*sin(b*x+a)/b/d/(d*cos(b*x+a))^(1/2)-3*(cos(1/2*a+1/2*b*x)^2)^(1/2)/cos(
1/2*a+1/2*b*x)*EllipticE(sin(1/2*a+1/2*b*x),2^(1/2))*(d*cos(b*x+a))^(1/2)/b/d^2/cos(b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 94, 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 = {2650, 2716, 2721, 2719} \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=-\frac {3 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{b d^2 \sqrt {\cos (a+b x)}}+\frac {3 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {\csc (a+b x)}{b d \sqrt {d \cos (a+b x)}} \]

[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 2650

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 2716

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 2719

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

Rule 2721

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

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Time = 0.30 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.69 \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\frac {-\cos (a+b x) \cot (a+b x)-3 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}} \]

[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]])

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.22

method	result	size
default	\(-\frac {\sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, {\left (-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}^{\frac {3}{2}} \left (6 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) E\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}+12 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1\right )}{2 d^{3} \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{5} {\left (2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}^{2} \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\)	\(209\)

[In]

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

[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+d*sin(1/2*b*x+1/2*a)^2)^(3/2)*(6*cos(1/2*b*x+1/2*a)*Ellip
ticE(cos(1/2*b*x+1/2*a),2^(1/2))*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*(sin(1/2*b*x+1/2*a)^2)^(1/2)+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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.39 \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\frac {-3 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right ) \sin \left (b x + a\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 3 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right ) \sin \left (b x + a\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) - 2 \, \sqrt {d \cos \left (b x + a\right )} {\left (3 \, \cos \left (b x + a\right )^{2} - 2\right )}}{2 \, b d^{2} \cos \left (b x + a\right ) \sin \left (b x + a\right )} \]

[In]

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

[Out]

1/2*(-3*I*sqrt(2)*sqrt(d)*cos(b*x + a)*sin(b*x + a)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*x
+ a) + I*sin(b*x + a))) + 3*I*sqrt(2)*sqrt(d)*cos(b*x + a)*sin(b*x + a)*weierstrassZeta(-4, 0, weierstrassPInv
erse(-4, 0, cos(b*x + a) - I*sin(b*x + a))) - 2*sqrt(d*cos(b*x + a))*(3*cos(b*x + a)^2 - 2))/(b*d^2*cos(b*x +
a)*sin(b*x + a))

Sympy [F]

\[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\int \frac {\csc ^{2}{\left (a + b x \right )}}{\left (d \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\int \frac {1}{{\sin \left (a+b\,x\right )}^2\,{\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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