\(\int \frac {1}{(d \csc (a+b x))^{7/2} (c \sec (a+b x))^{5/2}} \, dx\) [279]

Optimal result

Integrand size = 25, antiderivative size = 330 \[ \int \frac {1}{(d \csc (a+b x))^{7/2} (c \sec (a+b x))^{5/2}} \, dx=-\frac {c}{6 b d (d \csc (a+b x))^{5/2} (c \sec (a+b x))^{7/2}}-\frac {5 c}{48 b d^3 \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac {5}{192 b c d^3 \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}-\frac {5 \arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{128 \sqrt {2} b c^2 d^4 \sqrt {c \sec (a+b x)}}+\frac {5 \arctan \left (1+\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{128 \sqrt {2} b c^2 d^4 \sqrt {c \sec (a+b x)}}+\frac {5 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (a+b x)}}{1+\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{128 \sqrt {2} b c^2 d^4 \sqrt {c \sec (a+b x)}} \] Output:

-1/6*c/b/d/(d*csc(b*x+a))^(5/2)/(c*sec(b*x+a))^(7/2)-5/48*c/b/d^3/(d*csc(b 
*x+a))^(1/2)/(c*sec(b*x+a))^(7/2)+5/192/b/c/d^3/(d*csc(b*x+a))^(1/2)/(c*se 
c(b*x+a))^(3/2)+5/256*arctan(-1+2^(1/2)*tan(b*x+a)^(1/2))*(d*csc(b*x+a))^( 
1/2)*tan(b*x+a)^(1/2)*2^(1/2)/b/c^2/d^4/(c*sec(b*x+a))^(1/2)+5/256*arctan( 
1+2^(1/2)*tan(b*x+a)^(1/2))*(d*csc(b*x+a))^(1/2)*tan(b*x+a)^(1/2)*2^(1/2)/ 
b/c^2/d^4/(c*sec(b*x+a))^(1/2)+5/256*arctanh(2^(1/2)*tan(b*x+a)^(1/2)/(1+t 
an(b*x+a)))*(d*csc(b*x+a))^(1/2)*tan(b*x+a)^(1/2)*2^(1/2)/b/c^2/d^4/(c*sec 
(b*x+a))^(1/2)
 

Mathematica [A] (verified)

Time = 1.95 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.53 \[ \int \frac {1}{(d \csc (a+b x))^{7/2} (c \sec (a+b x))^{5/2}} \, dx=-\frac {\left (28+34 \cos (2 (a+b x))+2 \cos (4 (a+b x))-4 \cos (6 (a+b x))+15 \sqrt {2} \arctan \left (\frac {-1+\sqrt {\cot ^2(a+b x)}}{\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}\right ) \sqrt [4]{\cot ^2(a+b x)}-15 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}{1+\sqrt {\cot ^2(a+b x)}}\right ) \sqrt [4]{\cot ^2(a+b x)}\right ) \sqrt {c \sec (a+b x)}}{768 b c^3 d^3 \sqrt {d \csc (a+b x)}} \] Input:

Integrate[1/((d*Csc[a + b*x])^(7/2)*(c*Sec[a + b*x])^(5/2)),x]
 

Output:

-1/768*((28 + 34*Cos[2*(a + b*x)] + 2*Cos[4*(a + b*x)] - 4*Cos[6*(a + b*x) 
] + 15*Sqrt[2]*ArcTan[(-1 + Sqrt[Cot[a + b*x]^2])/(Sqrt[2]*(Cot[a + b*x]^2 
)^(1/4))]*(Cot[a + b*x]^2)^(1/4) - 15*Sqrt[2]*ArcTanh[(Sqrt[2]*(Cot[a + b* 
x]^2)^(1/4))/(1 + Sqrt[Cot[a + b*x]^2])]*(Cot[a + b*x]^2)^(1/4))*Sqrt[c*Se 
c[a + b*x]])/(b*c^3*d^3*Sqrt[d*Csc[a + b*x]])
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.90, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.760, Rules used = {3042, 3107, 3042, 3107, 3042, 3108, 3042, 3109, 3042, 3957, 266, 755, 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.

\(\displaystyle \int \frac {1}{(c \sec (a+b x))^{5/2} (d \csc (a+b x))^{7/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(c \sec (a+b x))^{5/2} (d \csc (a+b x))^{7/2}}dx\)

\(\Big \downarrow \) 3107

\(\displaystyle \frac {5 \int \frac {1}{(d \csc (a+b x))^{3/2} (c \sec (a+b x))^{5/2}}dx}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5 \int \frac {1}{(d \csc (a+b x))^{3/2} (c \sec (a+b x))^{5/2}}dx}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 3107

\(\displaystyle \frac {5 \left (\frac {\int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}}dx}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\right )}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5 \left (\frac {\int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}}dx}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\right )}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 3108

\(\displaystyle \frac {5 \left (\frac {\frac {3 \int \frac {\sqrt {d \csc (a+b x)}}{\sqrt {c \sec (a+b x)}}dx}{4 c^2}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\right )}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5 \left (\frac {\frac {3 \int \frac {\sqrt {d \csc (a+b x)}}{\sqrt {c \sec (a+b x)}}dx}{4 c^2}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\right )}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 3109

\(\displaystyle \frac {5 \left (\frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\tan (a+b x)}}dx}{4 c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\right )}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5 \left (\frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\tan (a+b x)}}dx}{4 c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\right )}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 3957

\(\displaystyle \frac {5 \left (\frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\tan (a+b x)} \left (\tan ^2(a+b x)+1\right )}d\tan (a+b x)}{4 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\right )}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {5 \left (\frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\right )}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {5 \left (\frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \int \frac {\tan (a+b x)+1}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\right )}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {5 \left (\frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \int \frac {1}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\right )}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {5 \left (\frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\right )}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {5 \left (\frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\right )}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {5 \left (\frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\right )}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {5 \left (\frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\right )}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 \left (\frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (a+b x)}+1}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\right )}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {5 \left (\frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\right )}{12 d^2}-\frac {c}{6 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{5/2}}\)

Input:

Int[1/((d*Csc[a + b*x])^(7/2)*(c*Sec[a + b*x])^(5/2)),x]
 

Output:

-1/6*c/(b*d*(d*Csc[a + b*x])^(5/2)*(c*Sec[a + b*x])^(7/2)) + (5*(-1/4*c/(b 
*d*Sqrt[d*Csc[a + b*x]]*(c*Sec[a + b*x])^(7/2)) + (d/(2*b*c*Sqrt[d*Csc[a + 
 b*x]]*(c*Sec[a + b*x])^(3/2)) + (3*Sqrt[d*Csc[a + b*x]]*((-(ArcTan[1 - Sq 
rt[2]*Sqrt[Tan[a + b*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] 
]/Sqrt[2])/2 + (-1/2*Log[1 - Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]/Sq 
rt[2] + Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]/(2*Sqrt[2]))/2) 
*Sqrt[Tan[a + b*x]])/(2*b*c^2*Sqrt[c*Sec[a + b*x]]))/(8*d^2)))/(12*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 755

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

Time = 9.95 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.36

method	result	size
default	\(\frac {\sqrt {2}\, \left (15 \ln \left (-2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \cot \left (b x +a \right )+2-2 \cot \left (b x +a \right )-2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \csc \left (b x +a \right )\right )-15 \ln \left (2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \cot \left (b x +a \right )+2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \csc \left (b x +a \right )-2 \cot \left (b x +a \right )+2\right )-30 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right )+30 \arctan \left (\frac {-\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right )+4 \cos \left (b x +a \right ) \left (5+32 \cos \left (b x +a \right )^{5}+32 \cos \left (b x +a \right )^{4}-52 \cos \left (b x +a \right )^{3}-52 \cos \left (b x +a \right )^{2}+5 \cos \left (b x +a \right )\right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\right ) \sin \left (b x +a \right )^{9} \sec \left (\frac {b x}{2}+\frac {a}{2}\right )^{11} \csc \left (\frac {b x}{2}+\frac {a}{2}\right )^{9}}{1572864 b \,d^{3} c^{2} \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {d \csc \left (b x +a \right )}\, \sqrt {c \sec \left (b x +a \right )}}\)	\(449\)

Input:

int(1/(d*csc(b*x+a))^(7/2)/(c*sec(b*x+a))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/1572864/b*2^(1/2)/d^3/c^2*(15*ln(-2*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a 
)+1)^2)^(1/2)*cot(b*x+a)+2-2*cot(b*x+a)-2*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b 
*x+a)+1)^2)^(1/2)*csc(b*x+a))-15*ln(2*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a 
)+1)^2)^(1/2)*cot(b*x+a)+2*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/ 
2)*csc(b*x+a)-2*cot(b*x+a)+2)-30*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))+30*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)/(co 
s(b*x+a)-1))+4*cos(b*x+a)*(5+32*cos(b*x+a)^5+32*cos(b*x+a)^4-52*cos(b*x+a) 
^3-52*cos(b*x+a)^2+5*cos(b*x+a))*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^ 
2)^(1/2))*sin(b*x+a)^9/(-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)^11*csc(1/2*b*x+1 
/2*a)^9
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (269) = 538\).

Time = 0.16 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.66 \[ \int \frac {1}{(d \csc (a+b x))^{7/2} (c \sec (a+b x))^{5/2}} \, dx=-\frac {30 \, \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}} \cos \left (b x + a\right ) \sin \left (b x + a\right )}{\cos \left (b x + a\right ) - \sin \left (b x + a\right )}\right ) - 15 \, \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 ) - 15 \, \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 ) + 15 \, \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 ) - 15 \, \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 (32 \, \cos \left (b x + a\right )^{6} - 52 \, \cos \left (b x + a\right )^{4} + 5 \, \cos \left (b x + a\right )^{2}\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} \sin \left (b x + a\right )}{3072 \, b c^{3} d^{4}} \] Input:

integrate(1/(d*csc(b*x+a))^(7/2)/(c*sec(b*x+a))^(5/2),x, algorithm="fricas 
")
 

Output:

-1/3072*(30*sqrt(2)*c*d*sqrt(1/(c*d))*arctan(-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)/(cos(b*x + a 
) - sin(b*x + a))) - 15*sqrt(2)*c*d*sqrt(1/(c*d))*arctan(-1/2*(sqrt(2)*sqr 
t(c/cos(b*x + a))*sqrt(d/sin(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))) - 15*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*sin(b*x + a))/(cos(b*x + a) - sin(b*x + a))) + 1 
5*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))*sq 
rt(1/(c*d)) + 4*cos(b*x + a)*sin(b*x + a) + 1) - 15*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) - 16*(32*cos(b*x + a)^6 - 52*cos(b*x + a)^4 + 5*co 
s(b*x + a)^2)*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a))*sin(b*x + a))/(b*c 
^3*d^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d \csc (a+b x))^{7/2} (c \sec (a+b x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(d*csc(b*x+a))**(7/2)/(c*sec(b*x+a))**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(d \csc (a+b x))^{7/2} (c \sec (a+b x))^{5/2}} \, dx=\int { \frac {1}{\left (d \csc \left (b x + a\right )\right )^{\frac {7}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(d*csc(b*x+a))^(7/2)/(c*sec(b*x+a))^(5/2),x, algorithm="maxima 
")
 

Output:

integrate(1/((d*csc(b*x + a))^(7/2)*(c*sec(b*x + a))^(5/2)), x)
 

Giac [F]

\[ \int \frac {1}{(d \csc (a+b x))^{7/2} (c \sec (a+b x))^{5/2}} \, dx=\int { \frac {1}{\left (d \csc \left (b x + a\right )\right )^{\frac {7}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(d*csc(b*x+a))^(7/2)/(c*sec(b*x+a))^(5/2),x, algorithm="giac")
 

Output:

integrate(1/((d*csc(b*x + a))^(7/2)*(c*sec(b*x + a))^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d \csc (a+b x))^{7/2} (c \sec (a+b x))^{5/2}} \, dx=\int \frac {1}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{5/2}\,{\left (\frac {d}{\sin \left (a+b\,x\right )}\right )}^{7/2}} \,d x \] Input:

int(1/((c/cos(a + b*x))^(5/2)*(d/sin(a + b*x))^(7/2)),x)
 

Output:

int(1/((c/cos(a + b*x))^(5/2)*(d/sin(a + b*x))^(7/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(d \csc (a+b x))^{7/2} (c \sec (a+b x))^{5/2}} \, dx=\frac {\sqrt {d}\, \sqrt {c}\, \left (\int \frac {\sqrt {\sec \left (b x +a \right )}\, \sqrt {\csc \left (b x +a \right )}}{\csc \left (b x +a \right )^{4} \sec \left (b x +a \right )^{3}}d x \right )}{c^{3} d^{4}} \] Input:

int(1/(d*csc(b*x+a))^(7/2)/(c*sec(b*x+a))^(5/2),x)
 

Output:

(sqrt(d)*sqrt(c)*int((sqrt(sec(a + b*x))*sqrt(csc(a + b*x)))/(csc(a + b*x) 
**4*sec(a + b*x)**3),x))/(c**3*d**4)