3.106 \(\int \frac {\csc ^4(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx\)

Optimal. Leaf size=43 \[ -\frac {2 d^3}{11 b (d \tan (a+b x))^{11/2}}-\frac {2 d}{7 b (d \tan (a+b x))^{7/2}} \]

[Out]

-2/11*d^3/b/(d*tan(b*x+a))^(11/2)-2/7*d/b/(d*tan(b*x+a))^(7/2)

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Rubi [A]  time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2591, 14} \[ -\frac {2 d^3}{11 b (d \tan (a+b x))^{11/2}}-\frac {2 d}{7 b (d \tan (a+b x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^4/(d*Tan[a + b*x])^(5/2),x]

[Out]

(-2*d^3)/(11*b*(d*Tan[a + b*x])^(11/2)) - (2*d)/(7*b*(d*Tan[a + b*x])^(7/2))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2591

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> With[{ff = FreeFactors[Ta
n[e + f*x], x]}, Dist[(b*ff)/f, Subst[Int[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, (b*Tan[e + f*x])/f
f], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]

Rubi steps

\begin {align*} \int \frac {\csc ^4(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx &=\frac {d \operatorname {Subst}\left (\int \frac {d^2+x^2}{x^{13/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=\frac {d \operatorname {Subst}\left (\int \left (\frac {d^2}{x^{13/2}}+\frac {1}{x^{9/2}}\right ) \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac {2 d^3}{11 b (d \tan (a+b x))^{11/2}}-\frac {2 d}{7 b (d \tan (a+b x))^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 50, normalized size = 1.16 \[ \frac {2 (2 \cos (2 (a+b x))-9) \cot ^4(a+b x) \csc ^2(a+b x) \sqrt {d \tan (a+b x)}}{77 b d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^4/(d*Tan[a + b*x])^(5/2),x]

[Out]

(2*(-9 + 2*Cos[2*(a + b*x)])*Cot[a + b*x]^4*Csc[a + b*x]^2*Sqrt[d*Tan[a + b*x]])/(77*b*d^3)

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fricas [B]  time = 0.83, size = 91, normalized size = 2.12 \[ -\frac {2 \, {\left (4 \, \cos \left (b x + a\right )^{6} - 11 \, \cos \left (b x + a\right )^{4}\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{77 \, {\left (b d^{3} \cos \left (b x + a\right )^{6} - 3 \, b d^{3} \cos \left (b x + a\right )^{4} + 3 \, b d^{3} \cos \left (b x + a\right )^{2} - b d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^4/(d*tan(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

-2/77*(4*cos(b*x + a)^6 - 11*cos(b*x + a)^4)*sqrt(d*sin(b*x + a)/cos(b*x + a))/(b*d^3*cos(b*x + a)^6 - 3*b*d^3
*cos(b*x + a)^4 + 3*b*d^3*cos(b*x + a)^2 - b*d^3)

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giac [A]  time = 2.35, size = 45, normalized size = 1.05 \[ -\frac {2 \, {\left (11 \, d^{3} \tan \left (b x + a\right )^{2} + 7 \, d^{3}\right )}}{77 \, \sqrt {d \tan \left (b x + a\right )} b d^{5} \tan \left (b x + a\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^4/(d*tan(b*x+a))^(5/2),x, algorithm="giac")

[Out]

-2/77*(11*d^3*tan(b*x + a)^2 + 7*d^3)/(sqrt(d*tan(b*x + a))*b*d^5*tan(b*x + a)^5)

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maple [A]  time = 0.59, size = 50, normalized size = 1.16 \[ \frac {2 \left (4 \left (\cos ^{2}\left (b x +a \right )\right )-11\right ) \cos \left (b x +a \right )}{77 b \sin \left (b x +a \right )^{3} \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^4/(d*tan(b*x+a))^(5/2),x)

[Out]

2/77/b*(4*cos(b*x+a)^2-11)*cos(b*x+a)/sin(b*x+a)^3/(d*sin(b*x+a)/cos(b*x+a))^(5/2)

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maxima [A]  time = 0.53, size = 35, normalized size = 0.81 \[ -\frac {2 \, {\left (11 \, d^{2} \tan \left (b x + a\right )^{2} + 7 \, d^{2}\right )} d}{77 \, \left (d \tan \left (b x + a\right )\right )^{\frac {11}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^4/(d*tan(b*x+a))^(5/2),x, algorithm="maxima")

[Out]

-2/77*(11*d^2*tan(b*x + a)^2 + 7*d^2)*d/((d*tan(b*x + a))^(11/2)*b)

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mupad [B]  time = 12.14, size = 831, normalized size = 19.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(a + b*x)^4*(d*tan(a + b*x))^(5/2)),x)

[Out]

((exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i + b*x*2i) + 1))^(1/2)*2048i)/(165*b*d^3
*(exp(a*2i + b*x*2i)*1i - 1i)) - (7768*(exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i +
b*x*2i) + 1))^(1/2))/(945*b*d^3*(exp(a*2i + b*x*2i) - 1)^2) - (4232*(exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i +
b*x*2i)*1i - 1i))/(exp(a*2i + b*x*2i) + 1))^(1/2))/(495*b*d^3*(exp(a*2i + b*x*2i) - 1)^3) - (1328*(exp(a*2i +
b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i + b*x*2i) + 1))^(1/2))/(231*b*d^3*(exp(a*2i + b*x*2
i) - 1)^4) - (160*(exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i + b*x*2i) + 1))^(1/2))
/(99*b*d^3*(exp(a*2i + b*x*2i) - 1)^5) - (14456*(exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(e
xp(a*2i + b*x*2i) + 1))^(1/2))/(1155*b*d^3*(exp(a*2i + b*x*2i) - 1)) - (86528*(exp(a*2i + b*x*2i) + 1)*(-(d*(e
xp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i + b*x*2i) + 1))^(1/2))/(10395*b*d^3*(exp(a*2i + b*x*2i)*1i - 1i)^2) - ((
exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i + b*x*2i) + 1))^(1/2)*3904i)/(315*b*d^3*(
exp(a*2i + b*x*2i)*1i - 1i)^3) + (4160*(exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i +
b*x*2i) + 1))^(1/2))/(231*b*d^3*(exp(a*2i + b*x*2i)*1i - 1i)^4) + ((exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i +
b*x*2i)*1i - 1i))/(exp(a*2i + b*x*2i) + 1))^(1/2)*1600i)/(99*b*d^3*(exp(a*2i + b*x*2i)*1i - 1i)^5) - (64*(exp(
a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i + b*x*2i) + 1))^(1/2))/(11*b*d^3*(exp(a*2i +
b*x*2i)*1i - 1i)^6)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{4}{\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**4/(d*tan(b*x+a))**(5/2),x)

[Out]

Integral(csc(a + b*x)**4/(d*tan(a + b*x))**(5/2), x)

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