\(\int \frac {(d \csc (a+b x))^{3/2}}{(c \sec (a+b x))^{3/2}} \, dx\) [266]

Optimal result

Integrand size = 25, antiderivative size = 248 \[ \int \frac {(d \csc (a+b x))^{3/2}}{(c \sec (a+b x))^{3/2}} \, dx=-\frac {2 d \sqrt {d \csc (a+b x)}}{b c \sqrt {c \sec (a+b x)}}+\frac {d^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {c \sec (a+b x)}}{\sqrt {2} b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}-\frac {d^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {c \sec (a+b x)}}{\sqrt {2} b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}+\frac {d^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (a+b x)}}{1+\tan (a+b x)}\right ) \sqrt {c \sec (a+b x)}}{\sqrt {2} b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}} \] Output:

-2*d*(d*csc(b*x+a))^(1/2)/b/c/(c*sec(b*x+a))^(1/2)-1/2*d^2*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*csc(b*x+a))^(1/ 
2)/tan(b*x+a)^(1/2)-1/2*d^2*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*csc(b*x+a))^(1/2)/tan(b*x+a)^(1/2)+1/2*d^2*arct 
anh(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*csc(b*x+a))^(1/2)/tan(b*x+a)^(1/2)
 

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.65 \[ \int \frac {(d \csc (a+b x))^{3/2}}{(c \sec (a+b x))^{3/2}} \, dx=-\frac {d \left (4 \cot ^2(a+b x)-\sqrt {2} \arctan \left (\frac {-1+\sqrt {\cot ^2(a+b x)}}{\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}\right ) \cot ^2(a+b x)^{3/4}-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}{1+\sqrt {\cot ^2(a+b x)}}\right ) \cot ^2(a+b x)^{3/4}\right ) \sqrt {d \csc (a+b x)} \tan ^2(a+b x)}{2 b c \sqrt {c \sec (a+b x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.85, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3103, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3103

\(\displaystyle -\frac {d^2 \int \frac {\sqrt {c \sec (a+b x)}}{\sqrt {d \csc (a+b x)}}dx}{c^2}-\frac {2 d \sqrt {d \csc (a+b x)}}{b c \sqrt {c \sec (a+b x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {d^2 \int \frac {\sqrt {c \sec (a+b x)}}{\sqrt {d \csc (a+b x)}}dx}{c^2}-\frac {2 d \sqrt {d \csc (a+b x)}}{b c \sqrt {c \sec (a+b x)}}\)

\(\Big \downarrow \) 3109

\(\displaystyle -\frac {d^2 \sqrt {c \sec (a+b x)} \int \sqrt {\tan (a+b x)}dx}{c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}-\frac {2 d \sqrt {d \csc (a+b x)}}{b c \sqrt {c \sec (a+b x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {d^2 \sqrt {c \sec (a+b x)} \int \sqrt {\tan (a+b x)}dx}{c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}-\frac {2 d \sqrt {d \csc (a+b x)}}{b c \sqrt {c \sec (a+b x)}}\)

\(\Big \downarrow \) 3957

\(\displaystyle -\frac {d^2 \sqrt {c \sec (a+b x)} \int \frac {\sqrt {\tan (a+b x)}}{\tan ^2(a+b x)+1}d\tan (a+b x)}{b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}-\frac {2 d \sqrt {d \csc (a+b x)}}{b c \sqrt {c \sec (a+b x)}}\)

\(\Big \downarrow \) 266

\(\displaystyle -\frac {2 d^2 \sqrt {c \sec (a+b x)} \int \frac {\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}}{b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}-\frac {2 d \sqrt {d \csc (a+b x)}}{b c \sqrt {c \sec (a+b x)}}\)

\(\Big \downarrow \) 826

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(-2*d*Sqrt[d*Csc[a + b*x]])/(b*c*Sqrt[c*Sec[a + b*x]]) - (2*d^2*((-(ArcTan 
[1 - Sqrt[2]*Sqrt[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]) - Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]/(2*Sqrt[2 
]))/2)*Sqrt[c*Sec[a + b*x]])/(b*c^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x] 
])
 

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 3103

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)/(f*b*(m - 1))), x] + Simp[a^2*((n + 1)/(b^2*(m - 1)))   Int[(a*Csc[e + f 
*x])^(m - 2)*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && 
GtQ[m, 1] && LtQ[n, -1] && 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 [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(454\) vs. \(2(209)=418\).

Time = 10.62 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.83

method	result	size
default	\(-\frac {\sqrt {2}\, \left (4 \cos \left (b x +a \right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+2 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right ) \sin \left (b x +a \right )-\ln \left (-\frac {\cos \left (b x +a \right ) \cot \left (b x +a \right )-2 \cot \left (b x +a \right )+2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{\cos \left (b x +a \right )-1}\right ) \sin \left (b x +a \right )+2 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-\cos \left (b x +a \right )+1}{\cos \left (b x +a \right )-1}\right ) \sin \left (b x +a \right )+\ln \left (\frac {2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-\cos \left (b x +a \right ) \cot \left (b x +a \right )+\sin \left (b x +a \right )+2 \cos \left (b x +a \right )-\csc \left (b x +a \right )+2 \cot \left (b x +a \right )-2}{\cos \left (b x +a \right )-1}\right ) \sin \left (b x +a \right )+4 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\right ) \sqrt {d \csc \left (b x +a \right )}\, d}{4 b \left (\cos \left (b x +a \right )+1\right ) \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, c \sqrt {c \sec \left (b x +a \right )}}\)	\(455\)

Input:

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

Output:

-1/4/b*2^(1/2)*(4*cos(b*x+a)*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^( 
1/2)+2*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))*sin(b*x+a)-ln(-(cos(b*x+a)*cot(b*x+a)-2*co 
t(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))*sin(b*x+a)+2*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))*sin(b*x+a)+ln((2*(-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)*cot(b*x+a)+sin(b*x+a)+2*cos(b*x+a)-csc(b*x+ 
a)+2*cot(b*x+a)-2)/(cos(b*x+a)-1))*sin(b*x+a)+4*(-2*sin(b*x+a)*cos(b*x+a)/ 
(cos(b*x+a)+1)^2)^(1/2))*(d*csc(b*x+a))^(1/2)*d/(cos(b*x+a)+1)/(-sin(b*x+a 
)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)/c/(c*sec(b*x+a))^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 492 vs. \(2 (209) = 418\).

Time = 0.13 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.98 \[ \int \frac {(d \csc (a+b x))^{3/2}}{(c \sec (a+b x))^{3/2}} \, dx=-\frac {2 \, \sqrt {2} c d \sqrt {\frac {d}{c}} \arctan \left (-\frac {\sqrt {2} \sqrt {\frac {d}{c}} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )}}{2 \, d}\right ) + \sqrt {2} c d \sqrt {\frac {d}{c}} \arctan \left (-\frac {\sqrt {2} \sqrt {\frac {d}{c}} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} + 2 \, d \cos \left (b x + a\right ) + 2 \, d \sin \left (b x + a\right )}{2 \, {\left (d \cos \left (b x + a\right ) - d \sin \left (b x + a\right )\right )}}\right ) + \sqrt {2} c d \sqrt {\frac {d}{c}} \arctan \left (-\frac {\sqrt {2} \sqrt {\frac {d}{c}} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} - 2 \, d \cos \left (b x + a\right ) - 2 \, d \sin \left (b x + a\right )}{2 \, {\left (d \cos \left (b x + a\right ) - d \sin \left (b x + a\right )\right )}}\right ) + \sqrt {2} c d \sqrt {\frac {d}{c}} \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 {d}{c}} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} + 4 \, d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + d\right ) - \sqrt {2} c d \sqrt {\frac {d}{c}} \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 {d}{c}} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} + 4 \, d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + d\right ) + 16 \, d \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} \cos \left (b x + a\right )}{8 \, b c^{2}} \] Input:

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

Output:

-1/8*(2*sqrt(2)*c*d*sqrt(d/c)*arctan(-1/2*sqrt(2)*sqrt(d/c)*sqrt(c/cos(b*x 
 + a))*sqrt(d/sin(b*x + a))*(cos(b*x + a) - sin(b*x + a))/d) + sqrt(2)*c*d 
*sqrt(d/c)*arctan(-1/2*(sqrt(2)*sqrt(d/c)*sqrt(c/cos(b*x + a))*sqrt(d/sin( 
b*x + a)) + 2*d*cos(b*x + a) + 2*d*sin(b*x + a))/(d*cos(b*x + a) - d*sin(b 
*x + a))) + sqrt(2)*c*d*sqrt(d/c)*arctan(-1/2*(sqrt(2)*sqrt(d/c)*sqrt(c/co 
s(b*x + a))*sqrt(d/sin(b*x + a)) - 2*d*cos(b*x + a) - 2*d*sin(b*x + a))/(d 
*cos(b*x + a) - d*sin(b*x + a))) + sqrt(2)*c*d*sqrt(d/c)*log(2*sqrt(2)*(co 
s(b*x + a)^3 - cos(b*x + a)^2*sin(b*x + a) - cos(b*x + a))*sqrt(d/c)*sqrt( 
c/cos(b*x + a))*sqrt(d/sin(b*x + a)) + 4*d*cos(b*x + a)*sin(b*x + a) + d) 
- sqrt(2)*c*d*sqrt(d/c)*log(-2*sqrt(2)*(cos(b*x + a)^3 - cos(b*x + a)^2*si 
n(b*x + a) - cos(b*x + a))*sqrt(d/c)*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + 
 a)) + 4*d*cos(b*x + a)*sin(b*x + a) + d) + 16*d*sqrt(c/cos(b*x + a))*sqrt 
(d/sin(b*x + a))*cos(b*x + a))/(b*c^2)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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