Integrand size = 25, antiderivative size = 295 \[ \int \frac {1}{(d \csc (a+b x))^{3/2} (c \sec (a+b x))^{5/2}} \, dx=-\frac {c}{4 b d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac {1}{16 b c d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}-\frac {3 \arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{32 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}+\frac {3 \arctan \left (1+\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{32 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}+\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (a+b x)}}{1+\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{32 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}} \] Output:
-1/4*c/b/d/(d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(7/2)+1/16/b/c/d/(d*csc(b*x +a))^(1/2)/(c*sec(b*x+a))^(3/2)+3/64*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^2/(c*sec(b*x+a))^(1/2 )+3/64*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^2/(c*sec(b*x+a))^(1/2)+3/64*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/d^2/(c*sec(b*x+a))^(1/2)
Time = 1.77 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.56 \[ \int \frac {1}{(d \csc (a+b x))^{3/2} (c \sec (a+b x))^{5/2}} \, dx=-\frac {\left (4+6 \cos (2 (a+b x))+2 \cos (4 (a+b x))+3 \sqrt {2} \arctan \left (\frac {-1+\sqrt {\cot ^2(a+b x)}}{\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}\right ) \sqrt [4]{\cot ^2(a+b x)}-3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}{1+\sqrt {\cot ^2(a+b x)}}\right ) \sqrt [4]{\cot ^2(a+b x)}\right ) \sqrt {c \sec (a+b x)}}{64 b c^3 d \sqrt {d \csc (a+b x)}} \] Input:
Integrate[1/((d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x])^(5/2)),x]
Output:
-1/64*((4 + 6*Cos[2*(a + b*x)] + 2*Cos[4*(a + b*x)] + 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[(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*d*Sqrt [d*Csc[a + b*x]])
Time = 0.76 (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, 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.
| \(\displaystyle \int \frac {1}{(c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3107 |
| \(\displaystyle \frac {\int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}}dx}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}}dx}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 3108 |
| \(\displaystyle \frac {\frac {3 \int \frac {\sqrt {d \csc (a+b x)}}{\sqrt {c \sec (a+b x)}}dx}{4 c^2}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \int \frac {\sqrt {d \csc (a+b x)}}{\sqrt {c \sec (a+b x)}}dx}{4 c^2}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 3109 |
| \(\displaystyle \frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\tan (a+b x)}}dx}{4 c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\tan (a+b x)}}dx}{4 c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\tan (a+b x)} \left (\tan ^2(a+b x)+1\right )}d\tan (a+b x)}{4 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \int \frac {\tan (a+b x)+1}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \int \frac {1}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (a+b x)}+1}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2}}\right )\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {d}{2 b c (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}\) |
Input:
Int[1/((d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x])^(5/2)),x]
Output:
-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]]*((-(ArcT an[1 - Sqrt[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]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]/(2*Sqr t[2]))/2)*Sqrt[Tan[a + b*x]])/(2*b*c^2*Sqrt[c*Sec[a + b*x]]))/(8*d^2)
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[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]
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]
Time = 9.57 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.54
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (3 \ln \left (-\frac {\cos \left (b x +a \right ) \cot \left (b x +a \right )-2 \cot \left (b x +a \right )+2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{\cos \left (b x +a \right )-1}\right )-3 \ln \left (\frac {2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-\cos \left (b x +a \right ) \cot \left (b x +a \right )+\sin \left (b x +a \right )+2 \cos \left (b x +a \right )-\csc \left (b x +a \right )+2 \cot \left (b x +a \right )-2}{\cos \left (b x +a \right )-1}\right )+6 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right )+6 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-\cos \left (b x +a \right )+1}{\cos \left (b x +a \right )-1}\right )+\cos \left (b x +a \right ) \left (16 \cos \left (b x +a \right )^{3}+16 \cos \left (b x +a \right )^{2}-4 \cos \left (b x +a \right )-4\right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\right ) \sin \left (b x +a \right )^{6} \sec \left (\frac {b x}{2}+\frac {a}{2}\right )^{8} \csc \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}}{16384 b d \,c^{2} \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {d \csc \left (b x +a \right )}\, \sqrt {c \sec \left (b x +a \right )}}\) | \(455\) |
Input:
int(1/(d*csc(b*x+a))^(3/2)/(c*sec(b*x+a))^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/16384/b*2^(1/2)/d/c^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))-3*ln((2*(-2*sin(b*x+a)*cos(b*x+a)/(co s(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))+6*arctan(((-2*sin(b*x+a)*co s(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)-cos (b*x+a)+1)/(cos(b*x+a)-1))+cos(b*x+a)*(16*cos(b*x+a)^3+16*cos(b*x+a)^2-4*c os(b*x+a)-4)*(-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
Leaf count of result is larger than twice the leaf count of optimal. 537 vs. \(2 (240) = 480\).
Time = 0.15 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.82 \[ \int \frac {1}{(d \csc (a+b x))^{3/2} (c \sec (a+b x))^{5/2}} \, dx=-\frac {6 \, \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 ) - 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 )^{4} - \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 )}{256 \, b c^{3} d^{2}} \] Input:
integrate(1/(d*csc(b*x+a))^(3/2)/(c*sec(b*x+a))^(5/2),x, algorithm="fricas ")
Output:
-1/256*(6*sqrt(2)*c*d*sqrt(1/(c*d))*arctan(-sqrt(2)*sqrt(c/cos(b*x + a))*s qrt(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))) - 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*sin (b*x + a))/(cos(b*x + a) - sin(b*x + a))) - 3*sqrt(2)*c*d*sqrt(1/(c*d))*ar ctan(-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))) + 3*sqr t(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))*lo g(-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*(4*cos(b*x + a)^4 - cos(b*x + a)^2)*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a))*sin(b*x + a))/(b*c^3*d^2)
Timed out. \[ \int \frac {1}{(d \csc (a+b x))^{3/2} (c \sec (a+b x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(d*csc(b*x+a))**(3/2)/(c*sec(b*x+a))**(5/2),x)
Output:
Timed out
\[ \int \frac {1}{(d \csc (a+b x))^{3/2} (c \sec (a+b x))^{5/2}} \, dx=\int { \frac {1}{\left (d \csc \left (b x + a\right )\right )^{\frac {3}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(d*csc(b*x+a))^(3/2)/(c*sec(b*x+a))^(5/2),x, algorithm="maxima ")
Output:
integrate(1/((d*csc(b*x + a))^(3/2)*(c*sec(b*x + a))^(5/2)), x)
\[ \int \frac {1}{(d \csc (a+b x))^{3/2} (c \sec (a+b x))^{5/2}} \, dx=\int { \frac {1}{\left (d \csc \left (b x + a\right )\right )^{\frac {3}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(d*csc(b*x+a))^(3/2)/(c*sec(b*x+a))^(5/2),x, algorithm="giac")
Output:
integrate(1/((d*csc(b*x + a))^(3/2)*(c*sec(b*x + a))^(5/2)), x)
Timed out. \[ \int \frac {1}{(d \csc (a+b x))^{3/2} (c \sec (a+b x))^{5/2}} \, dx=\int \frac {1}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{5/2}\,{\left (\frac {d}{\sin \left (a+b\,x\right )}\right )}^{3/2}} \,d x \] Input:
int(1/((c/cos(a + b*x))^(5/2)*(d/sin(a + b*x))^(3/2)),x)
Output:
int(1/((c/cos(a + b*x))^(5/2)*(d/sin(a + b*x))^(3/2)), x)
\[ \int \frac {1}{(d \csc (a+b x))^{3/2} (c \sec (a+b x))^{5/2}} \, dx=\frac {\sqrt {d}\, \sqrt {c}\, \left (\int \frac {\sqrt {\sec \left (b x +a \right )}\, \sqrt {\csc \left (b x +a \right )}}{\csc \left (b x +a \right )^{2} \sec \left (b x +a \right )^{3}}d x \right )}{c^{3} d^{2}} \] Input:
int(1/(d*csc(b*x+a))^(3/2)/(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))/(c**3*d**2)