3.241 \(\int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx\)

Optimal. Leaf size=126 \[ -\frac {21 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{5 b d^4 \sqrt {\cos (a+b x)}}+\frac {21 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {7 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}} \]

[Out]

-csc(b*x+a)/b/d/(d*cos(b*x+a))^(5/2)+7/5*sin(b*x+a)/b/d/(d*cos(b*x+a))^(5/2)+21/5*sin(b*x+a)/b/d^3/(d*cos(b*x+
a))^(1/2)-21/5*(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^4/cos(b*x+a)^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2570, 2636, 2640, 2639} \[ \frac {21 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}-\frac {21 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{5 b d^4 \sqrt {\cos (a+b x)}}+\frac {7 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^2/(d*Cos[a + b*x])^(7/2),x]

[Out]

-(Csc[a + b*x]/(b*d*(d*Cos[a + b*x])^(5/2))) - (21*Sqrt[d*Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2])/(5*b*d^4*Sq
rt[Cos[a + b*x]]) + (7*Sin[a + b*x])/(5*b*d*(d*Cos[a + b*x])^(5/2)) + (21*Sin[a + b*x])/(5*b*d^3*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 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 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]

Rubi steps

\begin {align*} \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx &=-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}+\frac {7}{2} \int \frac {1}{(d \cos (a+b x))^{7/2}} \, dx\\ &=-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}+\frac {7 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}+\frac {21 \int \frac {1}{(d \cos (a+b x))^{3/2}} \, dx}{10 d^2}\\ &=-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}+\frac {7 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}+\frac {21 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}-\frac {21 \int \sqrt {d \cos (a+b x)} \, dx}{10 d^4}\\ &=-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}+\frac {7 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}+\frac {21 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}-\frac {\left (21 \sqrt {d \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{10 d^4 \sqrt {\cos (a+b x)}}\\ &=-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}-\frac {21 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b d^4 \sqrt {\cos (a+b x)}}+\frac {7 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}+\frac {21 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 82, normalized size = 0.65 \[ \frac {16 \sin (a+b x)-5 \cos (a+b x) \cot (a+b x)-21 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+2 \tan (a+b x) \sec (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^2/(d*Cos[a + b*x])^(7/2),x]

[Out]

(-5*Cos[a + b*x]*Cot[a + b*x] - 21*Sqrt[Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2] + 16*Sin[a + b*x] + 2*Sec[a +
b*x]*Tan[a + b*x])/(5*b*d^3*Sqrt[d*Cos[a + b*x]])

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \cos \left (b x + a\right )} \csc \left (b x + a\right )^{2}}{d^{4} \cos \left (b x + a\right )^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^2/(d*cos(b*x+a))^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*cos(b*x + a))*csc(b*x + a)^2/(d^4*cos(b*x + a)^4), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^2/(d*cos(b*x+a))^(7/2),x, algorithm="giac")

[Out]

integrate(csc(b*x + a)^2/(d*cos(b*x + a))^(7/2), x)

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maple [B]  time = 0.39, size = 408, normalized size = 3.24 \[ -\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 (168 \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}}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+336 \left (\sin ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-168 \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}}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-672 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+42 \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}}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+448 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-112 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+5\right ) \left (-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d \right )^{\frac {3}{2}}}{10 d^{5} \left (2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{5} \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (8 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^2/(d*cos(b*x+a))^(7/2),x)

[Out]

-1/10*(d*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)/d^5/(2*sin(1/2*b*x+1/2*a)^2-1)/sin(1/2*b*x+1/2
*a)^5/cos(1/2*b*x+1/2*a)/(8*sin(1/2*b*x+1/2*a)^6-12*sin(1/2*b*x+1/2*a)^4+6*sin(1/2*b*x+1/2*a)^2-1)*(168*(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)*sin(1/2*b*x+1/2*a)^4+336*sin(1/2*b*x+1/2*a)^8-168*(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)*sin(1/2*b*x+1/2*a)^2-672*sin(1/2*b*x+1/2*a)^6+42
*(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)+448*sin(1/2*b*x+1/2*a)^4-112*sin(1/2*b*x+1/2*a)^2+5)*(-2*sin(1/2*b*x+1/2*a)^4*d+sin(1/2*b*x+1/2*a)^2
*d)^(3/2)/(d*(2*cos(1/2*b*x+1/2*a)^2-1))^(1/2)/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^2/(d*cos(b*x+a))^(7/2),x, algorithm="maxima")

[Out]

integrate(csc(b*x + a)^2/(d*cos(b*x + a))^(7/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\sin \left (a+b\,x\right )}^2\,{\left (d\,\cos \left (a+b\,x\right )\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(a + b*x)^2*(d*cos(a + b*x))^(7/2)),x)

[Out]

int(1/(sin(a + b*x)^2*(d*cos(a + b*x))^(7/2)), x)

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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)**2/(d*cos(b*x+a))**(7/2),x)

[Out]

Timed out

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