Integrand size = 25, antiderivative size = 249 \[ \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}} \, dx=\frac {d}{2 b c \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}-\frac {3 \arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}+\frac {3 \arctan \left (1+\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}+\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (a+b x)}}{1+\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}} \] Output:
1/2*d/b/c/(d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(3/2)+3/8*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/(c*s ec(b*x+a))^(1/2)+3/8*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/(c*sec(b*x+a))^(1/2)+3/8*arctanh(2^(1/2) *tan(b*x+a)^(1/2)/(1+tan(b*x+a)))*(d*csc(b*x+a))^(1/2)*tan(b*x+a)^(1/2)*2^ (1/2)/b/c^2/(c*sec(b*x+a))^(1/2)
Time = 1.68 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}} \, dx=\frac {d \left (4 \cos ^2(a+b x)-3 \sqrt {2} \arctan \left (\frac {-1+\sqrt {\cot ^2(a+b x)}}{\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}\right ) \sqrt [4]{\cot ^2(a+b x)}+3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}{1+\sqrt {\cot ^2(a+b x)}}\right ) \sqrt [4]{\cot ^2(a+b x)}\right ) \sqrt {c \sec (a+b x)}}{8 b c^3 \sqrt {d \csc (a+b x)}} \] Input:
Integrate[Sqrt[d*Csc[a + b*x]]/(c*Sec[a + b*x])^(5/2),x]
Output:
(d*(4*Cos[a + b*x]^2 - 3*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) + 3*Sqrt[2]*ArcTanh[(Sq rt[2]*(Cot[a + b*x]^2)^(1/4))/(1 + Sqrt[Cot[a + b*x]^2])]*(Cot[a + b*x]^2) ^(1/4))*Sqrt[c*Sec[a + b*x]])/(8*b*c^3*Sqrt[d*Csc[a + b*x]])
Time = 0.60 (sec) , antiderivative size = 212, 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, 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.
| \(\displaystyle \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3108 |
| \(\displaystyle \frac {3 \int \frac {\sqrt {d \csc (a+b x)}}{\sqrt {c \sec (a+b x)}}dx}{4 c^2}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {\sqrt {d \csc (a+b x)}}{\sqrt {c \sec (a+b x)}}dx}{4 c^2}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 3109 |
| \(\displaystyle \frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\tan (a+b x)}}dx}{4 c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\tan (a+b x)}}dx}{4 c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\tan (a+b x)} \left (\tan ^2(a+b x)+1\right )}d\tan (a+b x)}{4 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \int \frac {\tan (a+b x)+1}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \int \frac {1}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (a+b x)}+1}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}\) |
Input:
Int[Sqrt[d*Csc[a + b*x]]/(c*Sec[a + b*x])^(5/2),x]
Output:
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 - Sqrt[2]*Sqrt[Tan[a + b*x]]]/Sqrt[2]) + ArcTan[1 + Sqr t[2]*Sqrt[Tan[a + b*x]]]/Sqrt[2])/2 + (-1/2*Log[1 - Sqrt[2]*Sqrt[Tan[a + b *x]] + Tan[a + b*x]]/Sqrt[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]])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]]))
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]
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]
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]
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]
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]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(431\) vs. \(2(200)=400\).
Time = 7.90 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.73
| method | result | size |
| default | \(\frac {\sqrt {2}\, \sin \left (b x +a \right )^{4} \left (\cos \left (b x +a \right ) \left (4 \cos \left (b x +a \right )+4\right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+3 \ln \left (-\frac {\cos \left (b x +a \right ) \cot \left (b x +a \right )-2 \cot \left (b x +a \right )-2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{\cos \left (b x +a \right )-1}\right )-6 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-\cos \left (b x +a \right )+1}{\cos \left (b x +a \right )-1}\right )-3 \ln \left (-\frac {\cos \left (b x +a \right ) \cot \left (b x +a \right )-2 \cot \left (b x +a \right )+2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{\cos \left (b x +a \right )-1}\right )-6 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right )\right ) \sqrt {d \csc \left (b x +a \right )}\, \sec \left (\frac {b x}{2}+\frac {a}{2}\right )^{5} \csc \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{256 b \,c^{2} \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {c \sec \left (b x +a \right )}}\) | \(432\) |
Input:
int((d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(5/2),x,method=_RETURNVERBOSE)
Output:
1/256/b*2^(1/2)/c^2*sin(b*x+a)^4*(cos(b*x+a)*(4*cos(b*x+a)+4)*(-2*sin(b*x+ a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+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) )-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)/(c os(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)/(co s(b*x+a)-1))-6*arctan(((-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*s in(b*x+a)+cos(b*x+a)-1)/(cos(b*x+a)-1)))*(d*csc(b*x+a))^(1/2)/(-sin(b*x+a) *cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)/(c*sec(b*x+a))^(1/2)*sec(1/2*b*x+1/2*a )^5*csc(1/2*b*x+1/2*a)^3
Leaf count of result is larger than twice the leaf count of optimal. 511 vs. \(2 (200) = 400\).
Time = 0.13 (sec) , antiderivative size = 511, normalized size of antiderivative = 2.05 \[ \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}} \, dx=\frac {16 \, \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) - 6 \, \sqrt {2} c \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 )}} \cos \left (b x + a\right ) \sin \left (b x + a\right )}{d \cos \left (b x + a\right ) - d \sin \left (b x + a\right )}\right ) + 3 \, \sqrt {2} c \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 ) + 3 \, \sqrt {2} c \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 ) - 3 \, \sqrt {2} c \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 ) + 3 \, \sqrt {2} c \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 )}{32 \, b c^{3}} \] Input:
integrate((d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(5/2),x, algorithm="fricas")
Output:
1/32*(16*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a))*cos(b*x + a)^2*sin(b*x + a) - 6*sqrt(2)*c*sqrt(d/c)*arctan(-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*cos(b*x + a) - d*sin(b *x + a))) + 3*sqrt(2)*c*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))) + 3*sqrt(2)*c*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))) - 3*sqrt(2)*c*sqrt (d/c)*log(2*sqrt(2)*(cos(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) + 3*sqrt(2)*c*sqrt(d/c)*log(-2*sqrt(2)*(cos(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))/(b*c^3)
Timed out. \[ \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((d*csc(b*x+a))**(1/2)/(c*sec(b*x+a))**(5/2),x)
Output:
Timed out
\[ \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}} \, dx=\int { \frac {\sqrt {d \csc \left (b x + a\right )}}{\left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(5/2),x, algorithm="maxima")
Output:
integrate(sqrt(d*csc(b*x + a))/(c*sec(b*x + a))^(5/2), x)
\[ \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}} \, dx=\int { \frac {\sqrt {d \csc \left (b x + a\right )}}{\left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(5/2),x, algorithm="giac")
Output:
integrate(sqrt(d*csc(b*x + a))/(c*sec(b*x + a))^(5/2), x)
Timed out. \[ \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}} \, dx=\int \frac {\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{5/2}} \,d x \] Input:
int((d/sin(a + b*x))^(1/2)/(c/cos(a + b*x))^(5/2),x)
Output:
int((d/sin(a + b*x))^(1/2)/(c/cos(a + b*x))^(5/2), x)
\[ \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}} \, dx=\frac {\sqrt {d}\, \sqrt {c}\, \left (\int \frac {\sqrt {\sec \left (b x +a \right )}\, \sqrt {\csc \left (b x +a \right )}}{\sec \left (b x +a \right )^{3}}d x \right )}{c^{3}} \] Input:
int((d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(5/2),x)
Output:
(sqrt(d)*sqrt(c)*int((sqrt(sec(a + b*x))*sqrt(csc(a + b*x)))/sec(a + b*x)* *3,x))/c**3