3.231 \(\int \frac{\csc (a+b x)}{(d \cos (a+b x))^{9/2}} \, dx\)

Optimal. Leaf size=103 \[ \frac{2}{3 b d^3 (d \cos (a+b x))^{3/2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{b d^{9/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{b d^{9/2}}+\frac{2}{7 b d (d \cos (a+b x))^{7/2}} \]

[Out]

-(ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]]/(b*d^(9/2))) - ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]]/(b*d^(9/2)) + 2/(7
*b*d*(d*Cos[a + b*x])^(7/2)) + 2/(3*b*d^3*(d*Cos[a + b*x])^(3/2))

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Rubi [A]  time = 0.0752345, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2565, 325, 329, 212, 206, 203} \[ \frac{2}{3 b d^3 (d \cos (a+b x))^{3/2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{b d^{9/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{b d^{9/2}}+\frac{2}{7 b d (d \cos (a+b x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]/(d*Cos[a + b*x])^(9/2),x]

[Out]

-(ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]]/(b*d^(9/2))) - ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]]/(b*d^(9/2)) + 2/(7
*b*d*(d*Cos[a + b*x])^(7/2)) + 2/(3*b*d^3*(d*Cos[a + b*x])^(3/2))

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [C]  time = 0.0826734, size = 38, normalized size = 0.37 \[ \frac{2 \, _2F_1\left (-\frac{7}{4},1;-\frac{3}{4};\cos ^2(a+b x)\right )}{7 b d (d \cos (a+b x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]/(d*Cos[a + b*x])^(9/2),x]

[Out]

(2*Hypergeometric2F1[-7/4, 1, -3/4, Cos[a + b*x]^2])/(7*b*d*(d*Cos[a + b*x])^(7/2))

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Maple [B]  time = 0.223, size = 1082, normalized size = 10.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)/(d*cos(b*x+a))^(9/2),x)

[Out]

1/42/d^(19/2)/(-d)^(1/2)/(16*sin(1/2*b*x+1/2*a)^8-32*sin(1/2*b*x+1/2*a)^6+24*sin(1/2*b*x+1/2*a)^4-8*sin(1/2*b*
x+1/2*a)^2+1)*(42*ln(2/cos(1/2*b*x+1/2*a)*((-d)^(1/2)*(-2*sin(1/2*b*x+1/2*a)^2*d+d)^(1/2)-d))*d^(11/2)+40*(-2*
sin(1/2*b*x+1/2*a)^2*d+d)^(1/2)*d^(9/2)*(-d)^(1/2)-21*ln(2/(cos(1/2*b*x+1/2*a)-1)*(d^(1/2)*(-2*sin(1/2*b*x+1/2
*a)^2*d+d)^(1/2)+2*d*cos(1/2*b*x+1/2*a)-d))*(-d)^(1/2)*d^5-21*ln(2/(cos(1/2*b*x+1/2*a)+1)*(d^(1/2)*(-2*sin(1/2
*b*x+1/2*a)^2*d+d)^(1/2)-2*d*cos(1/2*b*x+1/2*a)-d))*(-d)^(1/2)*d^5-336*(-2*ln(2/cos(1/2*b*x+1/2*a)*((-d)^(1/2)
*(-2*sin(1/2*b*x+1/2*a)^2*d+d)^(1/2)-d))*d^(11/2)+ln(2/(cos(1/2*b*x+1/2*a)+1)*(d^(1/2)*(-2*sin(1/2*b*x+1/2*a)^
2*d+d)^(1/2)-2*d*cos(1/2*b*x+1/2*a)-d))*(-d)^(1/2)*d^5+ln(2/(cos(1/2*b*x+1/2*a)-1)*(d^(1/2)*(-2*sin(1/2*b*x+1/
2*a)^2*d+d)^(1/2)+2*d*cos(1/2*b*x+1/2*a)-d))*(-d)^(1/2)*d^5)*sin(1/2*b*x+1/2*a)^8+672*(-2*ln(2/cos(1/2*b*x+1/2
*a)*((-d)^(1/2)*(-2*sin(1/2*b*x+1/2*a)^2*d+d)^(1/2)-d))*d^(11/2)+ln(2/(cos(1/2*b*x+1/2*a)+1)*(d^(1/2)*(-2*sin(
1/2*b*x+1/2*a)^2*d+d)^(1/2)-2*d*cos(1/2*b*x+1/2*a)-d))*(-d)^(1/2)*d^5+ln(2/(cos(1/2*b*x+1/2*a)-1)*(d^(1/2)*(-2
*sin(1/2*b*x+1/2*a)^2*d+d)^(1/2)+2*d*cos(1/2*b*x+1/2*a)-d))*(-d)^(1/2)*d^5)*sin(1/2*b*x+1/2*a)^6+56*(-6*ln(2/c
os(1/2*b*x+1/2*a)*((-d)^(1/2)*(-2*sin(1/2*b*x+1/2*a)^2*d+d)^(1/2)-d))*d^(11/2)-2*(-2*sin(1/2*b*x+1/2*a)^2*d+d)
^(1/2)*d^(9/2)*(-d)^(1/2)+3*ln(2/(cos(1/2*b*x+1/2*a)+1)*(d^(1/2)*(-2*sin(1/2*b*x+1/2*a)^2*d+d)^(1/2)-2*d*cos(1
/2*b*x+1/2*a)-d))*(-d)^(1/2)*d^5+3*ln(2/(cos(1/2*b*x+1/2*a)-1)*(d^(1/2)*(-2*sin(1/2*b*x+1/2*a)^2*d+d)^(1/2)+2*
d*cos(1/2*b*x+1/2*a)-d))*(-d)^(1/2)*d^5)*sin(1/2*b*x+1/2*a)^2-56*(-18*ln(2/cos(1/2*b*x+1/2*a)*((-d)^(1/2)*(-2*
sin(1/2*b*x+1/2*a)^2*d+d)^(1/2)-d))*d^(11/2)-2*(-2*sin(1/2*b*x+1/2*a)^2*d+d)^(1/2)*d^(9/2)*(-d)^(1/2)+9*ln(2/(
cos(1/2*b*x+1/2*a)+1)*(d^(1/2)*(-2*sin(1/2*b*x+1/2*a)^2*d+d)^(1/2)-2*d*cos(1/2*b*x+1/2*a)-d))*(-d)^(1/2)*d^5+9
*ln(2/(cos(1/2*b*x+1/2*a)-1)*(d^(1/2)*(-2*sin(1/2*b*x+1/2*a)^2*d+d)^(1/2)+2*d*cos(1/2*b*x+1/2*a)-d))*(-d)^(1/2
)*d^5)*sin(1/2*b*x+1/2*a)^4)/b

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)/(d*cos(b*x+a))^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.69262, size = 946, normalized size = 9.18 \begin{align*} \left [\frac{42 \, \sqrt{-d} \arctan \left (\frac{\sqrt{d \cos \left (b x + a\right )} \sqrt{-d}{\left (\cos \left (b x + a\right ) + 1\right )}}{2 \, d \cos \left (b x + a\right )}\right ) \cos \left (b x + a\right )^{4} - 21 \, \sqrt{-d} \cos \left (b x + a\right )^{4} \log \left (\frac{d \cos \left (b x + a\right )^{2} + 4 \, \sqrt{d \cos \left (b x + a\right )} \sqrt{-d}{\left (\cos \left (b x + a\right ) - 1\right )} - 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, \sqrt{d \cos \left (b x + a\right )}{\left (7 \, \cos \left (b x + a\right )^{2} + 3\right )}}{84 \, b d^{5} \cos \left (b x + a\right )^{4}}, -\frac{42 \, \sqrt{d} \arctan \left (\frac{\sqrt{d \cos \left (b x + a\right )}{\left (\cos \left (b x + a\right ) - 1\right )}}{2 \, \sqrt{d} \cos \left (b x + a\right )}\right ) \cos \left (b x + a\right )^{4} - 21 \, \sqrt{d} \cos \left (b x + a\right )^{4} \log \left (\frac{d \cos \left (b x + a\right )^{2} - 4 \, \sqrt{d \cos \left (b x + a\right )} \sqrt{d}{\left (\cos \left (b x + a\right ) + 1\right )} + 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1}\right ) - 8 \, \sqrt{d \cos \left (b x + a\right )}{\left (7 \, \cos \left (b x + a\right )^{2} + 3\right )}}{84 \, b d^{5} \cos \left (b x + a\right )^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)/(d*cos(b*x+a))^(9/2),x, algorithm="fricas")

[Out]

[1/84*(42*sqrt(-d)*arctan(1/2*sqrt(d*cos(b*x + a))*sqrt(-d)*(cos(b*x + a) + 1)/(d*cos(b*x + a)))*cos(b*x + a)^
4 - 21*sqrt(-d)*cos(b*x + a)^4*log((d*cos(b*x + a)^2 + 4*sqrt(d*cos(b*x + a))*sqrt(-d)*(cos(b*x + a) - 1) - 6*
d*cos(b*x + a) + d)/(cos(b*x + a)^2 + 2*cos(b*x + a) + 1)) + 8*sqrt(d*cos(b*x + a))*(7*cos(b*x + a)^2 + 3))/(b
*d^5*cos(b*x + a)^4), -1/84*(42*sqrt(d)*arctan(1/2*sqrt(d*cos(b*x + a))*(cos(b*x + a) - 1)/(sqrt(d)*cos(b*x +
a)))*cos(b*x + a)^4 - 21*sqrt(d)*cos(b*x + a)^4*log((d*cos(b*x + a)^2 - 4*sqrt(d*cos(b*x + a))*sqrt(d)*(cos(b*
x + a) + 1) + 6*d*cos(b*x + a) + d)/(cos(b*x + a)^2 - 2*cos(b*x + a) + 1)) - 8*sqrt(d*cos(b*x + a))*(7*cos(b*x
+ a)^2 + 3))/(b*d^5*cos(b*x + a)^4)]

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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)/(d*cos(b*x+a))**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.12811, size = 130, normalized size = 1.26 \begin{align*} \frac{d{\left (\frac{21 \, \arctan \left (\frac{\sqrt{d \cos \left (b x + a\right )}}{\sqrt{-d}}\right )}{\sqrt{-d} d^{5}} - \frac{21 \, \arctan \left (\frac{\sqrt{d \cos \left (b x + a\right )}}{\sqrt{d}}\right )}{d^{\frac{11}{2}}} + \frac{2 \,{\left (7 \, d^{2} \cos \left (b x + a\right )^{2} + 3 \, d^{2}\right )}}{\sqrt{d \cos \left (b x + a\right )} d^{7} \cos \left (b x + a\right )^{3}}\right )}}{21 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)/(d*cos(b*x+a))^(9/2),x, algorithm="giac")

[Out]

1/21*d*(21*arctan(sqrt(d*cos(b*x + a))/sqrt(-d))/(sqrt(-d)*d^5) - 21*arctan(sqrt(d*cos(b*x + a))/sqrt(d))/d^(1
1/2) + 2*(7*d^2*cos(b*x + a)^2 + 3*d^2)/(sqrt(d*cos(b*x + a))*d^7*cos(b*x + a)^3))/b