3.58 \(\int \csc ^6(a+b x) \sqrt {d \tan (a+b x)} \, dx\)

Optimal. Leaf size=63 \[ -\frac {2 d^5}{9 b (d \tan (a+b x))^{9/2}}-\frac {4 d^3}{5 b (d \tan (a+b x))^{5/2}}-\frac {2 d}{b \sqrt {d \tan (a+b x)}} \]

[Out]

-2*d/b/(d*tan(b*x+a))^(1/2)-2/9*d^5/b/(d*tan(b*x+a))^(9/2)-4/5*d^3/b/(d*tan(b*x+a))^(5/2)

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Rubi [A]  time = 0.05, antiderivative size = 63, 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} \[ -\frac {2 d^5}{9 b (d \tan (a+b x))^{9/2}}-\frac {4 d^3}{5 b (d \tan (a+b x))^{5/2}}-\frac {2 d}{b \sqrt {d \tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^6*Sqrt[d*Tan[a + b*x]],x]

[Out]

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

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

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Mathematica [A]  time = 0.17, size = 50, normalized size = 0.79 \[ \frac {2 d (20 \cos (2 (a+b x))-4 \cos (4 (a+b x))-21) \csc ^4(a+b x)}{45 b \sqrt {d \tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^6*Sqrt[d*Tan[a + b*x]],x]

[Out]

(2*d*(-21 + 20*Cos[2*(a + b*x)] - 4*Cos[4*(a + b*x)])*Csc[a + b*x]^4)/(45*b*Sqrt[d*Tan[a + b*x]])

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fricas [A]  time = 0.47, size = 82, normalized size = 1.30 \[ -\frac {2 \, {\left (32 \, \cos \left (b x + a\right )^{5} - 72 \, \cos \left (b x + a\right )^{3} + 45 \, \cos \left (b x + a\right )\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{45 \, {\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )} \sin \left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^6*(d*tan(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

-2/45*(32*cos(b*x + a)^5 - 72*cos(b*x + a)^3 + 45*cos(b*x + a))*sqrt(d*sin(b*x + a)/cos(b*x + a))/((b*cos(b*x
+ a)^4 - 2*b*cos(b*x + a)^2 + b)*sin(b*x + a))

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giac [A]  time = 0.52, size = 58, normalized size = 0.92 \[ -\frac {2 \, {\left (45 \, d^{6} \tan \left (b x + a\right )^{4} + 18 \, d^{6} \tan \left (b x + a\right )^{2} + 5 \, d^{6}\right )}}{45 \, \sqrt {d \tan \left (b x + a\right )} b d^{5} \tan \left (b x + a\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^6*(d*tan(b*x+a))^(1/2),x, algorithm="giac")

[Out]

-2/45*(45*d^6*tan(b*x + a)^4 + 18*d^6*tan(b*x + a)^2 + 5*d^6)/(sqrt(d*tan(b*x + a))*b*d^5*tan(b*x + a)^4)

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maple [A]  time = 0.58, size = 60, normalized size = 0.95 \[ -\frac {2 \left (32 \left (\cos ^{4}\left (b x +a \right )\right )-72 \left (\cos ^{2}\left (b x +a \right )\right )+45\right ) \cos \left (b x +a \right ) \sqrt {\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}}}{45 b \sin \left (b x +a \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45/b*(32*cos(b*x+a)^4-72*cos(b*x+a)^2+45)*cos(b*x+a)*(d*sin(b*x+a)/cos(b*x+a))^(1/2)/sin(b*x+a)^5

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maxima [A]  time = 0.32, size = 48, normalized size = 0.76 \[ -\frac {2 \, {\left (45 \, d^{4} \tan \left (b x + a\right )^{4} + 18 \, d^{4} \tan \left (b x + a\right )^{2} + 5 \, d^{4}\right )} d}{45 \, \left (d \tan \left (b x + a\right )\right )^{\frac {9}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^6*(d*tan(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

-2/45*(45*d^4*tan(b*x + a)^4 + 18*d^4*tan(b*x + a)^2 + 5*d^4)*d/((d*tan(b*x + a))^(9/2)*b)

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mupad [B]  time = 7.02, size = 356, normalized size = 5.65 \[ -\frac {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{45\,b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}+\frac {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{45\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^2}-\frac {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}\,32{}\mathrm {i}}{15\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^3}-\frac {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{9\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^4}-\frac {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}\,32{}\mathrm {i}}{9\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*tan(a + b*x))^(1/2)/sin(a + b*x)^6,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)*64i)/(45*b*(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)*64i)/(45*b*(exp(a*2i + b*x*2i) - 1)) - ((exp(a*2i + b*x*2i) + 1)*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(ex
p(a*2i + b*x*2i) + 1))^(1/2)*32i)/(15*b*(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)*64i)/(9*b*(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)*32i)/(9*b*(exp(a*2i + b*x*2i) - 1)
^5)

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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)**6*(d*tan(b*x+a))**(1/2),x)

[Out]

Timed out

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