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

Optimal. Leaf size=20 \[ -\frac {2 d}{5 b (d \tan (a+b x))^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2591, 30} \[ -\frac {2 d}{5 b (d \tan (a+b x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d)/(5*b*(d*Tan[a + b*x])^(5/2))

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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

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Mathematica [A]  time = 0.12, size = 20, normalized size = 1.00 \[ -\frac {2 d}{5 b (d \tan (a+b x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d)/(5*b*(d*Tan[a + b*x])^(5/2))

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fricas [B]  time = 0.50, size = 58, normalized size = 2.90 \[ \frac {2 \, \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )^{3}}{5 \, {\left (b d^{2} \cos \left (b x + a\right )^{2} - b d^{2}\right )} \sin \left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.23, size = 26, normalized size = 1.30 \[ -\frac {2}{5 \, \sqrt {d \tan \left (b x + a\right )} b d \tan \left (b x + a\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.50, size = 38, normalized size = 1.90 \[ -\frac {2 \cos \left (b x +a \right )}{5 b \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}} \sin \left (b x +a \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.35, size = 23, normalized size = 1.15 \[ -\frac {2}{5 \, \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} b \tan \left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 6.45, size = 381, normalized size = 19.05 \[ -\frac {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}\,14{}\mathrm {i}}{5\,b\,d^2\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}-\frac {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}\,8{}\mathrm {i}}{15\,b\,d^2\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^2}-\frac {16\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}}{5\,b\,d^2\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}-\frac {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}\,32{}\mathrm {i}}{15\,b\,d^2\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}^2}+\frac {8\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1}}}{5\,b\,d^2\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*(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))/(5*b*d^2*(exp(
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*2i)
+ 1))^(1/2)*8i)/(15*b*d^2*(exp(a*2i + b*x*2i) - 1)^2) - (16*(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))/(5*b*d^2*(exp(a*2i + b*x*2i)*1i - 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)*32i)/(15*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))/(exp(a*2i + b*x*2i) + 1))^(1/2)*14i)/(5*b*d
^2*(exp(a*2i + b*x*2i) - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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