3.98 \(\int \frac {\csc ^6(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx\)

Optimal. Leaf size=65 \[ -\frac {2 d^5}{13 b (d \tan (a+b x))^{13/2}}-\frac {4 d^3}{9 b (d \tan (a+b x))^{9/2}}-\frac {2 d}{5 b (d \tan (a+b x))^{5/2}} \]

[Out]

-2/13*d^5/b/(d*tan(b*x+a))^(13/2)-4/9*d^3/b/(d*tan(b*x+a))^(9/2)-2/5*d/b/(d*tan(b*x+a))^(5/2)

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Rubi [A]  time = 0.05, antiderivative size = 65, 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, 270} \[ -\frac {2 d^5}{13 b (d \tan (a+b x))^{13/2}}-\frac {4 d^3}{9 b (d \tan (a+b x))^{9/2}}-\frac {2 d}{5 b (d \tan (a+b x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^6/(d*Tan[a + b*x])^(3/2),x]

[Out]

(-2*d^5)/(13*b*(d*Tan[a + b*x])^(13/2)) - (4*d^3)/(9*b*(d*Tan[a + b*x])^(9/2)) - (2*d)/(5*b*(d*Tan[a + b*x])^(
5/2))

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

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 ^6(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx &=\frac {d \operatorname {Subst}\left (\int \frac {\left (d^2+x^2\right )^2}{x^{15/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=\frac {d \operatorname {Subst}\left (\int \left (\frac {d^4}{x^{15/2}}+\frac {2 d^2}{x^{11/2}}+\frac {1}{x^{7/2}}\right ) \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac {2 d^5}{13 b (d \tan (a+b x))^{13/2}}-\frac {4 d^3}{9 b (d \tan (a+b x))^{9/2}}-\frac {2 d}{5 b (d \tan (a+b x))^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 54, normalized size = 0.83 \[ \frac {-90 \csc ^6(a+b x)+10 \csc ^4(a+b x)+16 \csc ^2(a+b x)+64}{585 b d \sqrt {d \tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^6/(d*Tan[a + b*x])^(3/2),x]

[Out]

(64 + 16*Csc[a + b*x]^2 + 10*Csc[a + b*x]^4 - 90*Csc[a + b*x]^6)/(585*b*d*Sqrt[d*Tan[a + b*x]])

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fricas [B]  time = 0.51, size = 109, normalized size = 1.68 \[ \frac {2 \, {\left (32 \, \cos \left (b x + a\right )^{7} - 104 \, \cos \left (b x + a\right )^{5} + 117 \, \cos \left (b x + a\right )^{3}\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{585 \, {\left (b d^{2} \cos \left (b x + a\right )^{6} - 3 \, b d^{2} \cos \left (b x + a\right )^{4} + 3 \, b d^{2} \cos \left (b x + a\right )^{2} - b d^{2}\right )} \sin \left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^6/(d*tan(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

2/585*(32*cos(b*x + a)^7 - 104*cos(b*x + a)^5 + 117*cos(b*x + a)^3)*sqrt(d*sin(b*x + a)/cos(b*x + a))/((b*d^2*
cos(b*x + a)^6 - 3*b*d^2*cos(b*x + a)^4 + 3*b*d^2*cos(b*x + a)^2 - b*d^2)*sin(b*x + a))

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giac [A]  time = 2.56, size = 58, normalized size = 0.89 \[ -\frac {2 \, {\left (117 \, d^{6} \tan \left (b x + a\right )^{4} + 130 \, d^{6} \tan \left (b x + a\right )^{2} + 45 \, d^{6}\right )}}{585 \, \sqrt {d \tan \left (b x + a\right )} b d^{7} \tan \left (b x + a\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^6/(d*tan(b*x+a))^(3/2),x, algorithm="giac")

[Out]

-2/585*(117*d^6*tan(b*x + a)^4 + 130*d^6*tan(b*x + a)^2 + 45*d^6)/(sqrt(d*tan(b*x + a))*b*d^7*tan(b*x + a)^6)

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maple [A]  time = 0.66, size = 60, normalized size = 0.92 \[ -\frac {2 \left (32 \left (\cos ^{4}\left (b x +a \right )\right )-104 \left (\cos ^{2}\left (b x +a \right )\right )+117\right ) \cos \left (b x +a \right )}{585 b \sin \left (b x +a \right )^{5} \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^6/(d*tan(b*x+a))^(3/2),x)

[Out]

-2/585/b*(32*cos(b*x+a)^4-104*cos(b*x+a)^2+117)*cos(b*x+a)/sin(b*x+a)^5/(d*sin(b*x+a)/cos(b*x+a))^(3/2)

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maxima [A]  time = 0.45, size = 48, normalized size = 0.74 \[ -\frac {2 \, {\left (117 \, d^{4} \tan \left (b x + a\right )^{4} + 130 \, d^{4} \tan \left (b x + a\right )^{2} + 45 \, d^{4}\right )} d}{585 \, \left (d \tan \left (b x + a\right )\right )^{\frac {13}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^6/(d*tan(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

-2/585*(117*d^4*tan(b*x + a)^4 + 130*d^4*tan(b*x + a)^2 + 45*d^4)*d/((d*tan(b*x + a))^(13/2)*b)

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mupad [B]  time = 16.41, size = 987, normalized size = 15.18 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(a + b*x)^6*(d*tan(a + b*x))^(3/2)),x)

[Out]

(128*(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^2*(e
xp(a*2i + b*x*2i)*1i - 1i)^3) - ((exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i + b*x*2
i) + 1))^(1/2)*294464i)/(45045*b*d^2*(exp(a*2i + b*x*2i) - 1)^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)*24608i)/(2145*b*d^2*(exp(a*2i + b*x*2i) - 1)^3) - ((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)*135104i)/(9009*b*d^2*(exp(a
*2i + b*x*2i) - 1)^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)*13088i)/(1287*b*d^2*(exp(a*2i + b*x*2i) - 1)^5) - ((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)*384i)/(143*b*d^2*(exp(a*2i + b*x*2i) - 1)^6) - (55808*(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))/(6435*b*d^2*(exp(a*2i + b*x*2i)*1
i - 1i)) - ((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)*7424i)
/(1155*b*d^2*(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))/(e
xp(a*2i + b*x*2i) + 1))^(1/2)*18368i)/(2145*b*d^2*(exp(a*2i + b*x*2i) - 1)) + ((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)*228736i)/(9009*b*d^2*(exp(a*2i + b*x*2i)*1i - 1i)
^4) - (17152*(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))/(429
*b*d^2*(exp(a*2i + b*x*2i)*1i - 1i)^5) - ((exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2
i + b*x*2i) + 1))^(1/2)*4608i)/(143*b*d^2*(exp(a*2i + b*x*2i)*1i - 1i)^6) + (128*(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))/(13*b*d^2*(exp(a*2i + b*x*2i)*1i - 1i)^7)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{6}{\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**6/(d*tan(b*x+a))**(3/2),x)

[Out]

Integral(csc(a + b*x)**6/(d*tan(a + b*x))**(3/2), x)

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