∫ csc6(a + bx)(d tan(a + bx))5∕2 dx [77]

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

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

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Rubi [A]
time = 0.04, 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 = {2671, 276} \begin {gather*} -\frac {2 d^5}{5 b (d \tan (a+b x))^{5/2}}-\frac {4 d^3}{b \sqrt {d \tan (a+b x)}}+\frac {2 d (d \tan (a+b x))^{3/2}}{3 b} \end {gather*}

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

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,

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]/ff

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

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Mathematica [A]
time = 0.24, size = 42, normalized size = 0.67 \begin {gather*} -\frac {2 d \left (-5+3 \cot ^2(a+b x) \left (9+\csc ^2(a+b x)\right )\right ) (d \tan (a+b x))^{3/2}}{15 b} \end {gather*}

(-2*d*(-5 + 3*Cot[a + b*x]^2*(9 + Csc[a + b*x]^2))*(d*Tan[a + b*x])^(3/2))/(15*b)

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method	result	size

default	\(\frac {2 \left (32 \left (\cos ^{4}\left (b x +a \right )\right )-40 \left (\cos ^{2}\left (b x +a \right )\right )+5\right ) \cos \left (b x +a \right ) \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {5}{2}}}{15 b \sin \left (b x +a \right )^{5}}\)	\(60\)

2/15/b*(32*cos(b*x+a)^4-40*cos(b*x+a)^2+5)*cos(b*x+a)*(d*sin(b*x+a)/cos(b*x+a))^(5/2)/sin(b*x+a)^5

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Maxima [A]
time = 0.27, size = 56, normalized size = 0.89 \begin {gather*} \frac {2 \, d^{5} {\left (\frac {5 \, \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}}{d^{4}} - \frac {3 \, {\left (10 \, d^{2} \tan \left (b x + a\right )^{2} + d^{2}\right )}}{\left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}} d^{2}}\right )}}{15 \, b} \end {gather*}

integrate(csc(b*x+a)^6*(d*tan(b*x+a))^(5/2),x, algorithm="maxima")

2/15*d^5*(5*(d*tan(b*x + a))^(3/2)/d^4 - 3*(10*d^2*tan(b*x + a)^2 + d^2)/((d*tan(b*x + a))^(5/2)*d^2))/b

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Fricas [A]
time = 0.39, size = 82, normalized size = 1.30 \begin {gather*} -\frac {2 \, {\left (32 \, d^{2} \cos \left (b x + a\right )^{4} - 40 \, d^{2} \cos \left (b x + a\right )^{2} + 5 \, d^{2}\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{15 \, {\left (b \cos \left (b x + a\right )^{3} - b \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )} \end {gather*}

integrate(csc(b*x+a)^6*(d*tan(b*x+a))^(5/2),x, algorithm="fricas")

-2/15*(32*d^2*cos(b*x + a)^4 - 40*d^2*cos(b*x + a)^2 + 5*d^2)*sqrt(d*sin(b*x + a)/cos(b*x + a))/((b*cos(b*x +

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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Giac [A]
time = 0.59, size = 70, normalized size = 1.11 \begin {gather*} \frac {2}{15} \, d^{2} {\left (\frac {5 \, \sqrt {d \tan \left (b x + a\right )} \tan \left (b x + a\right )}{b} - \frac {3 \, {\left (10 \, d^{3} \tan \left (b x + a\right )^{2} + d^{3}\right )}}{\sqrt {d \tan \left (b x + a\right )} b d^{2} \tan \left (b x + a\right )^{2}}\right )} \end {gather*}

2/15*d^2*(5*sqrt(d*tan(b*x + a))*tan(b*x + a)/b - 3*(10*d^3*tan(b*x + a)^2 + d^3)/(sqrt(d*tan(b*x + a))*b*d^2*

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Mupad [B]
time = 5.46, size = 134, normalized size = 2.13 \begin {gather*} \frac {32\,d^2\,\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}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,2{}\mathrm {i}+{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,3{}\mathrm {i}+{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}\,2{}\mathrm {i}-{\mathrm {e}}^{a\,8{}\mathrm {i}+b\,x\,8{}\mathrm {i}}\,2{}\mathrm {i}-2{}\mathrm {i}\right )}{15\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^3\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )} \end {gather*}

(32*d^2*(-(d*(exp(a*2i + b*x*2i)*1i - 1i))/(exp(a*2i + b*x*2i) + 1))^(1/2)*(exp(a*2i + b*x*2i)*2i + exp(a*4i +

b*x*4i)*3i + exp(a*6i + b*x*6i)*2i - exp(a*8i + b*x*8i)*2i - 2i))/(15*b*(exp(a*2i + b*x*2i) - 1)^3*(exp(a*2i

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3.1.77 \(\int \csc ^6(a+b x) (d \tan (a+b x))^{5/2} \, dx\) [77]