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

Optimal. Leaf size=63 \[ -\frac{2 d^5}{7 b (d \tan (a+b x))^{7/2}}-\frac{4 d^3}{3 b (d \tan (a+b x))^{3/2}}+\frac{2 d \sqrt{d \tan (a+b x)}}{b} \]

[Out]

(-2*d^5)/(7*b*(d*Tan[a + b*x])^(7/2)) - (4*d^3)/(3*b*(d*Tan[a + b*x])^(3/2)) + (2*d*Sqrt[d*Tan[a + b*x]])/b

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Rubi [A]  time = 0.0547959, antiderivative size = 63, normalized size of antiderivative = 1., 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} \[ -\frac{2 d^5}{7 b (d \tan (a+b x))^{7/2}}-\frac{4 d^3}{3 b (d \tan (a+b x))^{3/2}}+\frac{2 d \sqrt{d \tan (a+b x)}}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^5)/(7*b*(d*Tan[a + b*x])^(7/2)) - (4*d^3)/(3*b*(d*Tan[a + b*x])^(3/2)) + (2*d*Sqrt[d*Tan[a + b*x]])/b

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.141006, size = 42, normalized size = 0.67 \[ -\frac{2 d \left (3 \csc ^4(a+b x)+8 \csc ^2(a+b x)-32\right ) \sqrt{d \tan (a+b x)}}{21 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d*(-32 + 8*Csc[a + b*x]^2 + 3*Csc[a + b*x]^4)*Sqrt[d*Tan[a + b*x]])/(21*b)

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Maple [A]  time = 0.192, size = 60, normalized size = 1. \begin{align*}{\frac{ \left ( 64\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}-112\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}+42 \right ) \cos \left ( bx+a \right ) }{21\,b \left ( \sin \left ( bx+a \right ) \right ) ^{5}} \left ({\frac{d\sin \left ( bx+a \right ) }{\cos \left ( bx+a \right ) }} \right ) ^{{\frac{3}{2}}}} \end{align*}

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]

2/21/b*(32*cos(b*x+a)^4-56*cos(b*x+a)^2+21)*cos(b*x+a)*(d*sin(b*x+a)/cos(b*x+a))^(3/2)/sin(b*x+a)^5

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Maxima [A]  time = 1.59114, size = 78, normalized size = 1.24 \begin{align*} \frac{2 \, d^{5}{\left (\frac{21 \, \sqrt{d \tan \left (b x + a\right )}}{d^{4}} - \frac{14 \, d^{2} \tan \left (b x + a\right )^{2} + 3 \, d^{2}}{\left (d \tan \left (b x + a\right )\right )^{\frac{7}{2}} d^{2}}\right )}}{21 \, b} \end{align*}

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]

2/21*d^5*(21*sqrt(d*tan(b*x + a))/d^4 - (14*d^2*tan(b*x + a)^2 + 3*d^2)/((d*tan(b*x + a))^(7/2)*d^2))/b

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Fricas [A]  time = 1.49198, size = 182, normalized size = 2.89 \begin{align*} \frac{2 \,{\left (32 \, d \cos \left (b x + a\right )^{4} - 56 \, d \cos \left (b x + a\right )^{2} + 21 \, d\right )} \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{21 \,{\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )}} \end{align*}

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]

2/21*(32*d*cos(b*x + a)^4 - 56*d*cos(b*x + a)^2 + 21*d)*sqrt(d*sin(b*x + a)/cos(b*x + a))/(b*cos(b*x + a)^4 -
2*b*cos(b*x + a)^2 + b)

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

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]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \tan \left (b x + a\right )\right )^{\frac{3}{2}} \csc \left (b x + a\right )^{6}\,{d x} \end{align*}

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]

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