\(\int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx\) [222]

Optimal result

Integrand size = 19, antiderivative size = 100 \[ \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx=\frac {d^{9/2} \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{b}-\frac {d^{9/2} \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{b}+\frac {2 d^3 (d \cos (a+b x))^{3/2}}{3 b}+\frac {2 d (d \cos (a+b x))^{7/2}}{7 b} \] Output:

d^(9/2)*arctan((d*cos(b*x+a))^(1/2)/d^(1/2))/b-d^(9/2)*arctanh((d*cos(b*x+ 
a))^(1/2)/d^(1/2))/b+2/3*d^3*(d*cos(b*x+a))^(3/2)/b+2/7*d*(d*cos(b*x+a))^( 
7/2)/b
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.83 \[ \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx=\frac {d^4 \sqrt {d \cos (a+b x)} \left (21 \arctan \left (\sqrt {\cos (a+b x)}\right )-21 \text {arctanh}\left (\sqrt {\cos (a+b x)}\right )+2 \cos ^{\frac {3}{2}}(a+b x) \left (7+3 \cos ^2(a+b x)\right )\right )}{21 b \sqrt {\cos (a+b x)}} \] Input:

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

Output:

(d^4*Sqrt[d*Cos[a + b*x]]*(21*ArcTan[Sqrt[Cos[a + b*x]]] - 21*ArcTanh[Sqrt 
[Cos[a + b*x]]] + 2*Cos[a + b*x]^(3/2)*(7 + 3*Cos[a + b*x]^2)))/(21*b*Sqrt 
[Cos[a + b*x]])
 

Rubi [A] (warning: unable to verify)

Time = 0.46 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3042, 3045, 27, 262, 262, 266, 827, 216, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(d \cos (a+b x))^{9/2}}{\sin (a+b x)}dx\)

\(\Big \downarrow \) 3045

\(\displaystyle -\frac {\int \frac {d^2 (d \cos (a+b x))^{9/2}}{d^2-d^2 \cos ^2(a+b x)}d(d \cos (a+b x))}{b d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {d \int \frac {(d \cos (a+b x))^{9/2}}{d^2-d^2 \cos ^2(a+b x)}d(d \cos (a+b x))}{b}\)

\(\Big \downarrow \) 262

\(\displaystyle -\frac {d \left (d^2 \int \frac {(d \cos (a+b x))^{5/2}}{d^2-d^2 \cos ^2(a+b x)}d(d \cos (a+b x))-\frac {2}{7} (d \cos (a+b x))^{7/2}\right )}{b}\)

\(\Big \downarrow \) 262

\(\displaystyle -\frac {d \left (d^2 \left (d^2 \int \frac {\sqrt {d \cos (a+b x)}}{d^2-d^2 \cos ^2(a+b x)}d(d \cos (a+b x))-\frac {2}{3} (d \cos (a+b x))^{3/2}\right )-\frac {2}{7} (d \cos (a+b x))^{7/2}\right )}{b}\)

\(\Big \downarrow \) 266

\(\displaystyle -\frac {d \left (d^2 \left (2 d^2 \int \frac {d^2 \cos ^2(a+b x)}{d^2-d^4 \cos ^4(a+b x)}d\sqrt {d \cos (a+b x)}-\frac {2}{3} (d \cos (a+b x))^{3/2}\right )-\frac {2}{7} (d \cos (a+b x))^{7/2}\right )}{b}\)

\(\Big \downarrow \) 827

\(\displaystyle -\frac {d \left (d^2 \left (2 d^2 \left (\frac {1}{2} \int \frac {1}{d-d^2 \cos ^2(a+b x)}d\sqrt {d \cos (a+b x)}-\frac {1}{2} \int \frac {1}{d^2 \cos ^2(a+b x)+d}d\sqrt {d \cos (a+b x)}\right )-\frac {2}{3} (d \cos (a+b x))^{3/2}\right )-\frac {2}{7} (d \cos (a+b x))^{7/2}\right )}{b}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {d \left (d^2 \left (2 d^2 \left (\frac {1}{2} \int \frac {1}{d-d^2 \cos ^2(a+b x)}d\sqrt {d \cos (a+b x)}-\frac {\arctan \left (\sqrt {d} \cos (a+b x)\right )}{2 \sqrt {d}}\right )-\frac {2}{3} (d \cos (a+b x))^{3/2}\right )-\frac {2}{7} (d \cos (a+b x))^{7/2}\right )}{b}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {d \left (d^2 \left (2 d^2 \left (\frac {\text {arctanh}\left (\sqrt {d} \cos (a+b x)\right )}{2 \sqrt {d}}-\frac {\arctan \left (\sqrt {d} \cos (a+b x)\right )}{2 \sqrt {d}}\right )-\frac {2}{3} (d \cos (a+b x))^{3/2}\right )-\frac {2}{7} (d \cos (a+b x))^{7/2}\right )}{b}\)

Input:

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

Output:

-((d*((-2*(d*Cos[a + b*x])^(7/2))/7 + d^2*(2*d^2*(-1/2*ArcTan[Sqrt[d]*Cos[ 
a + b*x]]/Sqrt[d] + ArcTanh[Sqrt[d]*Cos[a + b*x]]/(2*Sqrt[d])) - (2*(d*Cos 
[a + b*x])^(3/2))/3)))/b)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

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

rule 219

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

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 827

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3045

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ 
Symbol] :> Simp[-(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])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(330\) vs. \(2(80)=160\).

Time = 3.41 (sec) , antiderivative size = 331, normalized size of antiderivative = 3.31

method	result	size
default	\(-\frac {96 \sqrt {-d}\, \sqrt {-2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} d +d}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{6} d^{4}+21 d^{\frac {9}{2}} \ln \left (-\frac {2 \left (2 d \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-\sqrt {d}\, \sqrt {-2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} d +d}+d \right )}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )+1}\right ) \sqrt {-d}+21 d^{\frac {9}{2}} \ln \left (\frac {4 d \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+2 \sqrt {d}\, \sqrt {-2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} d +d}-2 d}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )-1}\right ) \sqrt {-d}-144 \sqrt {-d}\, \sqrt {-2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} d +d}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4} d^{4}+128 \sqrt {-d}\, \sqrt {-2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} d +d}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} d^{4}-40 d^{4} \sqrt {-2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} d +d}\, \sqrt {-d}+42 d^{5} \ln \left (\frac {2 \sqrt {-d}\, \sqrt {-2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} d +d}-2 d}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )}\right )}{42 \sqrt {-d}\, b}\)	\(331\)

Input:

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

Output:

-1/42/(-d)^(1/2)*(96*(-d)^(1/2)*(-2*sin(1/2*b*x+1/2*a)^2*d+d)^(1/2)*sin(1/ 
2*b*x+1/2*a)^6*d^4+21*d^(9/2)*ln(-2/(cos(1/2*b*x+1/2*a)+1)*(2*d*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)^(1/2)+21*d^( 
9/2)*ln(2/(cos(1/2*b*x+1/2*a)-1)*(2*d*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)^(1/2)-144*(-d)^(1/2)*(-2*sin(1/2*b*x+1 
/2*a)^2*d+d)^(1/2)*sin(1/2*b*x+1/2*a)^4*d^4+128*(-d)^(1/2)*(-2*sin(1/2*b*x 
+1/2*a)^2*d+d)^(1/2)*sin(1/2*b*x+1/2*a)^2*d^4-40*d^4*(-2*sin(1/2*b*x+1/2*a 
)^2*d+d)^(1/2)*(-d)^(1/2)+42*d^5*ln(2/cos(1/2*b*x+1/2*a)*((-d)^(1/2)*(-2*s 
in(1/2*b*x+1/2*a)^2*d+d)^(1/2)-d)))/b
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 322, normalized size of antiderivative = 3.22 \[ \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx=\left [\frac {42 \, \sqrt {-d} d^{4} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )} \sqrt {-d} {\left (\cos \left (b x + a\right ) + 1\right )}}{2 \, d \cos \left (b x + a\right )}\right ) + 21 \, \sqrt {-d} d^{4} \log \left (-\frac {d \cos \left (b x + a\right )^{2} + 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {-d} {\left (\cos \left (b x + a\right ) - 1\right )} - 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, {\left (3 \, d^{4} \cos \left (b x + a\right )^{3} + 7 \, d^{4} \cos \left (b x + a\right )\right )} \sqrt {d \cos \left (b x + a\right )}}{84 \, b}, \frac {42 \, d^{\frac {9}{2}} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - 1\right )}}{2 \, \sqrt {d} \cos \left (b x + a\right )}\right ) + 21 \, d^{\frac {9}{2}} \log \left (-\frac {d \cos \left (b x + a\right )^{2} - 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {d} {\left (\cos \left (b x + a\right ) + 1\right )} + 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, {\left (3 \, d^{4} \cos \left (b x + a\right )^{3} + 7 \, d^{4} \cos \left (b x + a\right )\right )} \sqrt {d \cos \left (b x + a\right )}}{84 \, b}\right ] \] Input:

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

Output:

[1/84*(42*sqrt(-d)*d^4*arctan(1/2*sqrt(d*cos(b*x + a))*sqrt(-d)*(cos(b*x + 
 a) + 1)/(d*cos(b*x + a))) + 21*sqrt(-d)*d^4*log(-(d*cos(b*x + a)^2 + 4*sq 
rt(d*cos(b*x + a))*sqrt(-d)*(cos(b*x + a) - 1) - 6*d*cos(b*x + a) + d)/(co 
s(b*x + a)^2 + 2*cos(b*x + a) + 1)) + 8*(3*d^4*cos(b*x + a)^3 + 7*d^4*cos( 
b*x + a))*sqrt(d*cos(b*x + a)))/b, 1/84*(42*d^(9/2)*arctan(1/2*sqrt(d*cos( 
b*x + a))*(cos(b*x + a) - 1)/(sqrt(d)*cos(b*x + a))) + 21*d^(9/2)*log(-(d* 
cos(b*x + a)^2 - 4*sqrt(d*cos(b*x + a))*sqrt(d)*(cos(b*x + a) + 1) + 6*d*c 
os(b*x + a) + d)/(cos(b*x + a)^2 - 2*cos(b*x + a) + 1)) + 8*(3*d^4*cos(b*x 
 + a)^3 + 7*d^4*cos(b*x + a))*sqrt(d*cos(b*x + a)))/b]
 

Sympy [F(-1)]

Timed out. \[ \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.98 \[ \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx=\frac {42 \, d^{\frac {11}{2}} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )}}{\sqrt {d}}\right ) + 21 \, d^{\frac {11}{2}} \log \left (\frac {\sqrt {d \cos \left (b x + a\right )} - \sqrt {d}}{\sqrt {d \cos \left (b x + a\right )} + \sqrt {d}}\right ) + 12 \, \left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}} d^{2} + 28 \, \left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}} d^{4}}{42 \, b d} \] Input:

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

Output:

1/42*(42*d^(11/2)*arctan(sqrt(d*cos(b*x + a))/sqrt(d)) + 21*d^(11/2)*log(( 
sqrt(d*cos(b*x + a)) - sqrt(d))/(sqrt(d*cos(b*x + a)) + sqrt(d))) + 12*(d* 
cos(b*x + a))^(7/2)*d^2 + 28*(d*cos(b*x + a))^(3/2)*d^4)/(b*d)
 

Giac [F]

\[ \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \csc \left (b x + a\right ) \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx=\int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^{9/2}}{\sin \left (a+b\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx=\sqrt {d}\, \left (\int \sqrt {\cos \left (b x +a \right )}\, \cos \left (b x +a \right )^{4} \csc \left (b x +a \right )d x \right ) d^{4} \] Input:

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

Output:

sqrt(d)*int(sqrt(cos(a + b*x))*cos(a + b*x)**4*csc(a + b*x),x)*d**4