\(\int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}} \, dx\) [270]

Optimal result

Integrand size = 25, antiderivative size = 295 \[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}} \, dx=-\frac {c}{4 b d (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{5/2}}+\frac {3}{16 b c d (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}-\frac {3 \arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {c \sec (a+b x)}}{32 \sqrt {2} b c^2 d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}+\frac {3 \arctan \left (1+\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {c \sec (a+b x)}}{32 \sqrt {2} b c^2 d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (a+b x)}}{1+\tan (a+b x)}\right ) \sqrt {c \sec (a+b x)}}{32 \sqrt {2} b c^2 d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}} \] Output:

-1/4*c/b/d/(d*csc(b*x+a))^(3/2)/(c*sec(b*x+a))^(5/2)+3/16/b/c/d/(d*csc(b*x 
+a))^(3/2)/(c*sec(b*x+a))^(1/2)+3/64*arctan(-1+2^(1/2)*tan(b*x+a)^(1/2))*( 
c*sec(b*x+a))^(1/2)*2^(1/2)/b/c^2/d^2/(d*csc(b*x+a))^(1/2)/tan(b*x+a)^(1/2 
)+3/64*arctan(1+2^(1/2)*tan(b*x+a)^(1/2))*(c*sec(b*x+a))^(1/2)*2^(1/2)/b/c 
^2/d^2/(d*csc(b*x+a))^(1/2)/tan(b*x+a)^(1/2)-3/64*arctanh(2^(1/2)*tan(b*x+ 
a)^(1/2)/(1+tan(b*x+a)))*(c*sec(b*x+a))^(1/2)*2^(1/2)/b/c^2/d^2/(d*csc(b*x 
+a))^(1/2)/tan(b*x+a)^(1/2)
 

Mathematica [A] (verified)

Time = 1.59 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}} \, dx=-\frac {\left (2 (\cos (2 (a+b x))+\cos (4 (a+b x)))+3 \sqrt {2} \arctan \left (\frac {-1+\sqrt {\cot ^2(a+b x)}}{\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}\right ) \cot ^2(a+b x)^{3/4}+3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}{1+\sqrt {\cot ^2(a+b x)}}\right ) \cot ^2(a+b x)^{3/4}\right ) \sec ^3(a+b x)}{64 b d (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{3/2}} \] Input:

Integrate[1/((d*Csc[a + b*x])^(5/2)*(c*Sec[a + b*x])^(3/2)),x]
 

Output:

-1/64*((2*(Cos[2*(a + b*x)] + Cos[4*(a + b*x)]) + 3*Sqrt[2]*ArcTan[(-1 + S 
qrt[Cot[a + b*x]^2])/(Sqrt[2]*(Cot[a + b*x]^2)^(1/4))]*(Cot[a + b*x]^2)^(3 
/4) + 3*Sqrt[2]*ArcTanh[(Sqrt[2]*(Cot[a + b*x]^2)^(1/4))/(1 + Sqrt[Cot[a + 
 b*x]^2])]*(Cot[a + b*x]^2)^(3/4))*Sec[a + b*x]^3)/(b*d*(d*Csc[a + b*x])^( 
3/2)*(c*Sec[a + b*x])^(3/2))
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.86, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3042, 3107, 3042, 3108, 3042, 3109, 3042, 3957, 266, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {1}{(c \sec (a+b x))^{3/2} (d \csc (a+b x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(c \sec (a+b x))^{3/2} (d \csc (a+b x))^{5/2}}dx\)

\(\Big \downarrow \) 3107

\(\displaystyle \frac {3 \int \frac {1}{\sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}dx}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \int \frac {1}{\sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}dx}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\)

\(\Big \downarrow \) 3108

\(\displaystyle \frac {3 \left (\frac {\int \frac {\sqrt {c \sec (a+b x)}}{\sqrt {d \csc (a+b x)}}dx}{4 c^2}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \left (\frac {\int \frac {\sqrt {c \sec (a+b x)}}{\sqrt {d \csc (a+b x)}}dx}{4 c^2}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\)

\(\Big \downarrow \) 3109

\(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \int \sqrt {\tan (a+b x)}dx}{4 c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \int \sqrt {\tan (a+b x)}dx}{4 c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\)

\(\Big \downarrow \) 3957

\(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \int \frac {\sqrt {\tan (a+b x)}}{\tan ^2(a+b x)+1}d\tan (a+b x)}{4 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \int \frac {\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \int \frac {\tan (a+b x)+1}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \int \frac {1}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}\right )-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (a+b x)}+1}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2}}\right )\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\)

Input:

Int[1/((d*Csc[a + b*x])^(5/2)*(c*Sec[a + b*x])^(3/2)),x]
 

Output:

-1/4*c/(b*d*(d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x])^(5/2)) + (3*(d/(2*b*c* 
(d*Csc[a + b*x])^(3/2)*Sqrt[c*Sec[a + b*x]]) + (((-(ArcTan[1 - Sqrt[2]*Sqr 
t[Tan[a + b*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]]]/Sqrt[2] 
)/2 + (Log[1 - Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]/(2*Sqrt[2]) - Lo 
g[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]/(2*Sqrt[2]))/2)*Sqrt[c*Se 
c[a + b*x]])/(2*b*c^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])))/(8*d^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 217

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

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 826

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

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 3042

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

rule 3107

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n 
_.), x_Symbol] :> Simp[b*(a*Csc[e + f*x])^(m + 1)*((b*Sec[e + f*x])^(n - 1) 
/(a*f*(m + n))), x] + Simp[(m + 1)/(a^2*(m + n))   Int[(a*Csc[e + f*x])^(m 
+ 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] 
&& NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
 

rule 3108

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*((b_.)*sec[(e_.) + (f_.)*(x_)])^( 
n_), x_Symbol] :> Simp[(-a)*(a*Csc[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n + 
 1)/(b*f*(m + n))), x] + Simp[(n + 1)/(b^2*(m + n))   Int[(a*Csc[e + f*x])^ 
m*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, - 
1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
 

rule 3109

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n 
_), x_Symbol] :> Simp[(a*Csc[e + f*x])^m*((b*Sec[e + f*x])^n/Tan[e + f*x]^n 
)   Int[Tan[e + f*x]^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !Integer 
Q[n] && EqQ[m + n, 0]
 

rule 3957

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Subst[Int 
[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && 
!IntegerQ[n]
 

Maple [A] (warning: unable to verify)

Time = 11.24 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.54

method	result	size
default	\(-\frac {\sqrt {2}\, \left (-3 \ln \left (-\frac {\cos \left (b x +a \right ) \cot \left (b x +a \right )-2 \cot \left (b x +a \right )-2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{\cos \left (b x +a \right )-1}\right )+3 \ln \left (-\frac {\cos \left (b x +a \right ) \cot \left (b x +a \right )-2 \cot \left (b x +a \right )+2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{\cos \left (b x +a \right )-1}\right )-6 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-\cos \left (b x +a \right )+1}{\cos \left (b x +a \right )-1}\right )-6 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right )+\left (16 \cos \left (b x +a \right )^{3}+16 \cos \left (b x +a \right )^{2}-12 \cos \left (b x +a \right )-12\right ) \sin \left (b x +a \right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\right ) \sin \left (b x +a \right )^{6} \sec \left (\frac {b x}{2}+\frac {a}{2}\right )^{8} \csc \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}}{16384 b \,d^{2} c \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {d \csc \left (b x +a \right )}\, \sqrt {c \sec \left (b x +a \right )}}\)	\(455\)

Input:

int(1/(d*csc(b*x+a))^(5/2)/(c*sec(b*x+a))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/16384/b*2^(1/2)/d^2/c*(-3*ln(-(cos(b*x+a)*cot(b*x+a)-2*cot(b*x+a)-2*(-2 
*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*sin(b*x+a)-2*cos(b*x+a)-sin 
(b*x+a)+csc(b*x+a)+2)/(cos(b*x+a)-1))+3*ln(-(cos(b*x+a)*cot(b*x+a)-2*cot(b 
*x+a)+2*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*sin(b*x+a)-2*cos 
(b*x+a)-sin(b*x+a)+csc(b*x+a)+2)/(cos(b*x+a)-1))-6*arctan(((-2*sin(b*x+a)* 
cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*sin(b*x+a)-cos(b*x+a)+1)/(cos(b*x+a)-1) 
)-6*arctan(((-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*sin(b*x+a)+c 
os(b*x+a)-1)/(cos(b*x+a)-1))+(16*cos(b*x+a)^3+16*cos(b*x+a)^2-12*cos(b*x+a 
)-12)*sin(b*x+a)*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2))*sin(b* 
x+a)^6/(-sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)/(d*csc(b*x+a))^(1/2 
)/(c*sec(b*x+a))^(1/2)*sec(1/2*b*x+1/2*a)^8*csc(1/2*b*x+1/2*a)^6
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 525 vs. \(2 (240) = 480\).

Time = 0.14 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.78 \[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}} \, dx=\frac {6 \, \sqrt {2} c d \sqrt {\frac {1}{c d}} \arctan \left (-\frac {1}{2} \, \sqrt {2} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} \sqrt {\frac {1}{c d}} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )}\right ) + 3 \, \sqrt {2} c d \sqrt {\frac {1}{c d}} \arctan \left (-\frac {\sqrt {2} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} \sqrt {\frac {1}{c d}} + 2 \, \cos \left (b x + a\right ) + 2 \, \sin \left (b x + a\right )}{2 \, {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )}}\right ) + 3 \, \sqrt {2} c d \sqrt {\frac {1}{c d}} \arctan \left (-\frac {\sqrt {2} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} \sqrt {\frac {1}{c d}} - 2 \, \cos \left (b x + a\right ) - 2 \, \sin \left (b x + a\right )}{2 \, {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )}}\right ) + 3 \, \sqrt {2} c d \sqrt {\frac {1}{c d}} \log \left (2 \, \sqrt {2} {\left (\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) - \cos \left (b x + a\right )\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} \sqrt {\frac {1}{c d}} + 4 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right ) - 3 \, \sqrt {2} c d \sqrt {\frac {1}{c d}} \log \left (-2 \, \sqrt {2} {\left (\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) - \cos \left (b x + a\right )\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} \sqrt {\frac {1}{c d}} + 4 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right ) + 16 \, {\left (4 \, \cos \left (b x + a\right )^{5} - 7 \, \cos \left (b x + a\right )^{3} + 3 \, \cos \left (b x + a\right )\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}}}{256 \, b c^{2} d^{3}} \] Input:

integrate(1/(d*csc(b*x+a))^(5/2)/(c*sec(b*x+a))^(3/2),x, algorithm="fricas 
")
 

Output:

1/256*(6*sqrt(2)*c*d*sqrt(1/(c*d))*arctan(-1/2*sqrt(2)*sqrt(c/cos(b*x + a) 
)*sqrt(d/sin(b*x + a))*sqrt(1/(c*d))*(cos(b*x + a) - sin(b*x + a))) + 3*sq 
rt(2)*c*d*sqrt(1/(c*d))*arctan(-1/2*(sqrt(2)*sqrt(c/cos(b*x + a))*sqrt(d/s 
in(b*x + a))*sqrt(1/(c*d)) + 2*cos(b*x + a) + 2*sin(b*x + a))/(cos(b*x + a 
) - sin(b*x + a))) + 3*sqrt(2)*c*d*sqrt(1/(c*d))*arctan(-1/2*(sqrt(2)*sqrt 
(c/cos(b*x + a))*sqrt(d/sin(b*x + a))*sqrt(1/(c*d)) - 2*cos(b*x + a) - 2*s 
in(b*x + a))/(cos(b*x + a) - sin(b*x + a))) + 3*sqrt(2)*c*d*sqrt(1/(c*d))* 
log(2*sqrt(2)*(cos(b*x + a)^3 - cos(b*x + a)^2*sin(b*x + a) - cos(b*x + a) 
)*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a))*sqrt(1/(c*d)) + 4*cos(b*x + a) 
*sin(b*x + a) + 1) - 3*sqrt(2)*c*d*sqrt(1/(c*d))*log(-2*sqrt(2)*(cos(b*x + 
 a)^3 - cos(b*x + a)^2*sin(b*x + a) - cos(b*x + a))*sqrt(c/cos(b*x + a))*s 
qrt(d/sin(b*x + a))*sqrt(1/(c*d)) + 4*cos(b*x + a)*sin(b*x + a) + 1) + 16* 
(4*cos(b*x + a)^5 - 7*cos(b*x + a)^3 + 3*cos(b*x + a))*sqrt(c/cos(b*x + a) 
)*sqrt(d/sin(b*x + a)))/(b*c^2*d^3)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(d*csc(b*x+a))**(5/2)/(c*sec(b*x+a))**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}} \, dx=\int { \frac {1}{\left (d \csc \left (b x + a\right )\right )^{\frac {5}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(d*csc(b*x+a))^(5/2)/(c*sec(b*x+a))^(3/2),x, algorithm="maxima 
")
 

Output:

integrate(1/((d*csc(b*x + a))^(5/2)*(c*sec(b*x + a))^(3/2)), x)
 

Giac [F]

\[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}} \, dx=\int { \frac {1}{\left (d \csc \left (b x + a\right )\right )^{\frac {5}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(d*csc(b*x+a))^(5/2)/(c*sec(b*x+a))^(3/2),x, algorithm="giac")
 

Output:

integrate(1/((d*csc(b*x + a))^(5/2)*(c*sec(b*x + a))^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}} \, dx=\int \frac {1}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{3/2}\,{\left (\frac {d}{\sin \left (a+b\,x\right )}\right )}^{5/2}} \,d x \] Input:

int(1/((c/cos(a + b*x))^(3/2)*(d/sin(a + b*x))^(5/2)),x)
 

Output:

int(1/((c/cos(a + b*x))^(3/2)*(d/sin(a + b*x))^(5/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}} \, dx=\frac {\sqrt {d}\, \sqrt {c}\, \left (\int \frac {\sqrt {\sec \left (b x +a \right )}\, \sqrt {\csc \left (b x +a \right )}}{\csc \left (b x +a \right )^{3} \sec \left (b x +a \right )^{2}}d x \right )}{c^{2} d^{3}} \] Input:

int(1/(d*csc(b*x+a))^(5/2)/(c*sec(b*x+a))^(3/2),x)
 

Output:

(sqrt(d)*sqrt(c)*int((sqrt(sec(a + b*x))*sqrt(csc(a + b*x)))/(csc(a + b*x) 
**3*sec(a + b*x)**2),x))/(c**2*d**3)