Integrand size = 25, antiderivative size = 251 \[ \int \frac {(d \csc (a+b x))^{5/2}}{(c \sec (a+b x))^{5/2}} \, dx=-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}+\frac {d^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{\sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {d^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{\sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {d^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (a+b x)}}{1+\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{\sqrt {2} b c^2 \sqrt {c \sec (a+b x)}} \] Output:
-2/3*d*(d*csc(b*x+a))^(3/2)/b/c/(c*sec(b*x+a))^(3/2)-1/2*d^2*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)-1/2*d^2*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)-1/2*d^2*ar ctanh(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 = 1.63 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.61 \[ \int \frac {(d \csc (a+b x))^{5/2}}{(c \sec (a+b x))^{5/2}} \, dx=-\frac {d^3 \left (4 \cot ^2(a+b x)-3 \sqrt {2} \arctan \left (\frac {-1+\sqrt {\cot ^2(a+b x)}}{\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}\right ) \sqrt [4]{\cot ^2(a+b x)}+3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}{1+\sqrt {\cot ^2(a+b x)}}\right ) \sqrt [4]{\cot ^2(a+b x)}\right ) \sqrt {c \sec (a+b x)}}{6 b c^3 \sqrt {d \csc (a+b x)}} \] Input:
Integrate[(d*Csc[a + b*x])^(5/2)/(c*Sec[a + b*x])^(5/2),x]
Output:
-1/6*(d^3*(4*Cot[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]*ArcT anh[(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*Sec[a + b*x]])/(b*c^3*Sqrt[d*Csc[a + b*x]])
Time = 0.61 (sec) , antiderivative size = 213, 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, 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.
| \(\displaystyle \int \frac {(d \csc (a+b x))^{5/2}}{(c \sec (a+b x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(d \csc (a+b x))^{5/2}}{(c \sec (a+b x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3103 |
| \(\displaystyle -\frac {d^2 \int \frac {\sqrt {d \csc (a+b x)}}{\sqrt {c \sec (a+b x)}}dx}{c^2}-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {d^2 \int \frac {\sqrt {d \csc (a+b x)}}{\sqrt {c \sec (a+b x)}}dx}{c^2}-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3109 |
| \(\displaystyle -\frac {d^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\tan (a+b x)}}dx}{c^2 \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {d^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\tan (a+b x)}}dx}{c^2 \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle -\frac {d^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\tan (a+b x)} \left (\tan ^2(a+b x)+1\right )}d\tan (a+b x)}{b c^2 \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {2 d^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}}{b c^2 \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle -\frac {2 d^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \int \frac {\tan (a+b x)+1}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}\right )}{b c^2 \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {2 d^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \int \frac {1}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}\right )\right )}{b c^2 \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {2 d^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}\right )\right )}{b c^2 \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {2 d^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{b c^2 \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {2 d^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{b c^2 \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 d^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{b c^2 \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 d^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (a+b x)}+1}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{b c^2 \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {2 d^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2}}\right )\right )}{b c^2 \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}\) |
Input:
Int[(d*Csc[a + b*x])^(5/2)/(c*Sec[a + b*x])^(5/2),x]
Output:
(-2*d*(d*Csc[a + b*x])^(3/2))/(3*b*c*(c*Sec[a + b*x])^(3/2)) - (2*d^2*Sqrt [d*Csc[a + b*x]]*((-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*x]]]/Sqrt[2]) + Arc Tan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]]]/Sqrt[2])/2 + (-1/2*Log[1 - Sqrt[2]*Sqr t[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]])/(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)/(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]
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. \(436\) vs. \(2(209)=418\).
Time = 6.99 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.74
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (\left (3 \cos \left (b x +a \right )-3\right ) \ln \left (-\frac {\cos \left (b x +a \right ) \cot \left (b x +a \right )-2 \cot \left (b x +a \right )+2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{\cos \left (b x +a \right )-1}\right )+\left (-3 \cos \left (b x +a \right )+3\right ) \ln \left (-\frac {\cos \left (b x +a \right ) \cot \left (b x +a \right )-2 \cot \left (b x +a \right )-2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{\cos \left (b x +a \right )-1}\right )+\left (-6 \cos \left (b x +a \right )+6\right ) \arctan \left (\frac {-\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right )+\left (6 \cos \left (b x +a \right )-6\right ) \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right )+4 \cos \left (b x +a \right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\right ) d^{2} \sqrt {d \csc \left (b x +a \right )}\, \csc \left (b x +a \right )}{12 b \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, c^{2} \sqrt {c \sec \left (b x +a \right )}}\) | \(437\) |
Input:
int((d*csc(b*x+a))^(5/2)/(c*sec(b*x+a))^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/12/b*2^(1/2)*((3*cos(b*x+a)-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*cos(b*x+a)+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* cos(b*x+a)+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))+(6*cos(b*x+a)-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))+4*cos(b*x+a)*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2))*d^ 2*(d*csc(b*x+a))^(1/2)/(-sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)/c^2 /(c*sec(b*x+a))^(1/2)*csc(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (209) = 418\).
Time = 0.15 (sec) , antiderivative size = 561, normalized size of antiderivative = 2.24 \[ \int \frac {(d \csc (a+b x))^{5/2}}{(c \sec (a+b x))^{5/2}} \, dx=\frac {6 \, \sqrt {2} c d^{2} \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 ) \sin \left (b x + a\right ) - 3 \, \sqrt {2} c d^{2} \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 ) \sin \left (b x + a\right ) - 3 \, \sqrt {2} c d^{2} \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 ) \sin \left (b x + a\right ) + 3 \, \sqrt {2} c d^{2} \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 ) \sin \left (b x + a\right ) - 3 \, \sqrt {2} c d^{2} \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 ) \sin \left (b x + a\right ) - 16 \, d^{2} \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}}{24 \, b c^{3} \sin \left (b x + a\right )} \] Input:
integrate((d*csc(b*x+a))^(5/2)/(c*sec(b*x+a))^(5/2),x, algorithm="fricas")
Output:
1/24*(6*sqrt(2)*c*d^2*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*si n(b*x + a)))*sin(b*x + a) - 3*sqrt(2)*c*d^2*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)))*sin(b*x + a) - 3*sqrt (2)*c*d^2*sqrt(d/c)*arctan(-1/2*(sqrt(2)*sqrt(d/c)*sqrt(c/cos(b*x + a))*sq rt(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)))*sin(b*x + a) + 3*sqrt(2)*c*d^2*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)*sq rt(c/cos(b*x + a))*sqrt(d/sin(b*x + a)) + 4*d*cos(b*x + a)*sin(b*x + a) + d)*sin(b*x + a) - 3*sqrt(2)*c*d^2*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)*sin(b*x + a) - 16*d^2*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a))*cos(b*x + a)^2)/(b*c^3 *sin(b*x + a))
Timed out. \[ \int \frac {(d \csc (a+b x))^{5/2}}{(c \sec (a+b x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((d*csc(b*x+a))**(5/2)/(c*sec(b*x+a))**(5/2),x)
Output:
Timed out
\[ \int \frac {(d \csc (a+b x))^{5/2}}{(c \sec (a+b x))^{5/2}} \, dx=\int { \frac {\left (d \csc \left (b x + a\right )\right )^{\frac {5}{2}}}{\left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*csc(b*x+a))^(5/2)/(c*sec(b*x+a))^(5/2),x, algorithm="maxima")
Output:
integrate((d*csc(b*x + a))^(5/2)/(c*sec(b*x + a))^(5/2), x)
\[ \int \frac {(d \csc (a+b x))^{5/2}}{(c \sec (a+b x))^{5/2}} \, dx=\int { \frac {\left (d \csc \left (b x + a\right )\right )^{\frac {5}{2}}}{\left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*csc(b*x+a))^(5/2)/(c*sec(b*x+a))^(5/2),x, algorithm="giac")
Output:
integrate((d*csc(b*x + a))^(5/2)/(c*sec(b*x + a))^(5/2), x)
Timed out. \[ \int \frac {(d \csc (a+b x))^{5/2}}{(c \sec (a+b x))^{5/2}} \, dx=\int \frac {{\left (\frac {d}{\sin \left (a+b\,x\right )}\right )}^{5/2}}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{5/2}} \,d x \] Input:
int((d/sin(a + b*x))^(5/2)/(c/cos(a + b*x))^(5/2),x)
Output:
int((d/sin(a + b*x))^(5/2)/(c/cos(a + b*x))^(5/2), x)
\[ \int \frac {(d \csc (a+b x))^{5/2}}{(c \sec (a+b x))^{5/2}} \, dx=\frac {\sqrt {d}\, \sqrt {c}\, \left (\int \frac {\sqrt {\sec \left (b x +a \right )}\, \sqrt {\csc \left (b x +a \right )}\, \csc \left (b x +a \right )^{2}}{\sec \left (b x +a \right )^{3}}d x \right ) d^{2}}{c^{3}} \] Input:
int((d*csc(b*x+a))^(5/2)/(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)** 2)/sec(a + b*x)**3,x)*d**2)/c**3