Integrand size = 25, antiderivative size = 371 \[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}} \, dx=-\frac {c}{4 b d (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{5/2}}+\frac {3}{16 b c d (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}-\frac {3 \arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {c \sec (a+b x)}}{32 \sqrt {2} b c^2 d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}+\frac {3 \arctan \left (1+\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {c \sec (a+b x)}}{32 \sqrt {2} b c^2 d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}+\frac {3 \log \left (1-\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {c \sec (a+b x)}}{64 \sqrt {2} b c^2 d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}-\frac {3 \log \left (1+\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {c \sec (a+b x)}}{64 \sqrt {2} b c^2 d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}} \]
-1/4*c/b/d/(d*csc(b*x+a))^(3/2)/(c*sec(b*x+a))^(5/2)+3/16/b/c/d/(d*csc(b*x +a))^(3/2)/(c*sec(b*x+a))^(1/2)+3/64*arctan(-1+2^(1/2)*tan(b*x+a)^(1/2))*( c*sec(b*x+a))^(1/2)/b/c^2/d^2*2^(1/2)/(d*csc(b*x+a))^(1/2)/tan(b*x+a)^(1/2 )+3/64*arctan(1+2^(1/2)*tan(b*x+a)^(1/2))*(c*sec(b*x+a))^(1/2)/b/c^2/d^2*2 ^(1/2)/(d*csc(b*x+a))^(1/2)/tan(b*x+a)^(1/2)+3/128*ln(1-2^(1/2)*tan(b*x+a) ^(1/2)+tan(b*x+a))*(c*sec(b*x+a))^(1/2)/b/c^2/d^2*2^(1/2)/(d*csc(b*x+a))^( 1/2)/tan(b*x+a)^(1/2)-3/128*ln(1+2^(1/2)*tan(b*x+a)^(1/2)+tan(b*x+a))*(c*s ec(b*x+a))^(1/2)/b/c^2/d^2*2^(1/2)/(d*csc(b*x+a))^(1/2)/tan(b*x+a)^(1/2)
Time = 2.66 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.45 \[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}} \, dx=-\frac {\left (2 (\cos (2 (a+b x))+\cos (4 (a+b x)))+3 \sqrt {2} \arctan \left (\frac {-1+\sqrt {\cot ^2(a+b x)}}{\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}\right ) \cot ^2(a+b x)^{3/4}+3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}{1+\sqrt {\cot ^2(a+b x)}}\right ) \cot ^2(a+b x)^{3/4}\right ) \sec ^3(a+b x)}{64 b d (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{3/2}} \]
-1/64*((2*(Cos[2*(a + b*x)] + Cos[4*(a + b*x)]) + 3*Sqrt[2]*ArcTan[(-1 + S qrt[Cot[a + b*x]^2])/(Sqrt[2]*(Cot[a + b*x]^2)^(1/4))]*(Cot[a + b*x]^2)^(3 /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)^(3/4))*Sec[a + b*x]^3)/(b*d*(d*Csc[a + b*x])^( 3/2)*(c*Sec[a + b*x])^(3/2))
Time = 0.74 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.69, 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, 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.
| \(\displaystyle \int \frac {1}{(c \sec (a+b x))^{3/2} (d \csc (a+b x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(c \sec (a+b x))^{3/2} (d \csc (a+b x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3107 |
| \(\displaystyle \frac {3 \int \frac {1}{\sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}dx}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {1}{\sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}dx}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3108 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {\sqrt {c \sec (a+b x)}}{\sqrt {d \csc (a+b x)}}dx}{4 c^2}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {\sqrt {c \sec (a+b x)}}{\sqrt {d \csc (a+b x)}}dx}{4 c^2}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3109 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \int \sqrt {\tan (a+b x)}dx}{4 c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \int \sqrt {\tan (a+b x)}dx}{4 c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \int \frac {\sqrt {\tan (a+b x)}}{\tan ^2(a+b x)+1}d\tan (a+b x)}{4 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \int \frac {\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \int \frac {\tan (a+b x)+1}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \int \frac {1}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}\right )-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (a+b x)}+1}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2}}\right )\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\right )}{8 d^2}-\frac {c}{4 b d (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{3/2}}\) |
-1/4*c/(b*d*(d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x])^(5/2)) + (3*(d/(2*b*c* (d*Csc[a + b*x])^(3/2)*Sqrt[c*Sec[a + b*x]]) + (((-(ArcTan[1 - Sqrt[2]*Sqr t[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]) - Lo g[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]/(2*Sqrt[2]))/2)*Sqrt[c*Se c[a + b*x]])/(2*b*c^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])))/(8*d^2)
3.3.70.3.1 Defintions of rubi rules used
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[(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]]))
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 = 50.58 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.49
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (16 \sqrt {2}\, \cos \left (b x +a \right )^{3} \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )+16 \sqrt {2}\, \cos \left (b x +a \right )^{2} \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-12 \sqrt {2}\, \cos \left (b x +a \right ) \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-12 \sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+3 \ln \left (2 \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \cot \left (b x +a \right )+2 \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \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 )-6 \arctan \left (\frac {\sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right )+6 \arctan \left (\frac {-\sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right )-3 \ln \left (-2 \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \cot \left (b x +a \right )-2 \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \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 )\right ) \sin \left (b x +a \right )^{2}}{128 b \left (\cos \left (b x +a \right )-1\right ) \left (\cos \left (b x +a \right )+1\right )^{2} \sqrt {c \sec \left (b x +a \right )}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {d \csc \left (b x +a \right )}\, c \,d^{2}}\) | \(551\) |
1/128/b*2^(1/2)*(16*2^(1/2)*cos(b*x+a)^3*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+ a)+1)^2)^(1/2)*sin(b*x+a)+16*2^(1/2)*cos(b*x+a)^2*(-cos(b*x+a)*sin(b*x+a)/ (cos(b*x+a)+1)^2)^(1/2)*sin(b*x+a)-12*2^(1/2)*cos(b*x+a)*(-cos(b*x+a)*sin( b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*sin(b*x+a)-12*sin(b*x+a)*2^(1/2)*(-cos(b*x+ a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+3*ln(2*2^(1/2)*(-cos(b*x+a)*sin(b*x+ a)/(cos(b*x+a)+1)^2)^(1/2)*cot(b*x+a)+2*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(c os(b*x+a)+1)^2)^(1/2)*csc(b*x+a)-2*cot(b*x+a)+2)-6*arctan((sin(b*x+a)*2^(1 /2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+cos(b*x+a)-1)/(cos(b*x +a)-1))+6*arctan((-sin(b*x+a)*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+ 1)^2)^(1/2)+cos(b*x+a)-1)/(cos(b*x+a)-1))-3*ln(-2*2^(1/2)*(-cos(b*x+a)*sin (b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*cot(b*x+a)-2*2^(1/2)*(-cos(b*x+a)*sin(b*x+ a)/(cos(b*x+a)+1)^2)^(1/2)*csc(b*x+a)-2*cot(b*x+a)+2))/(cos(b*x+a)-1)/(cos (b*x+a)+1)^2/(c*sec(b*x+a))^(1/2)/(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2 )^(1/2)/(d*csc(b*x+a))^(1/2)/c/d^2*sin(b*x+a)^2
Result contains complex when optimal does not.
Time = 1.15 (sec) , antiderivative size = 1341, normalized size of antiderivative = 3.61 \[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}} \, dx=\text {Too large to display} \]
-1/256*(3*I*b*c^2*d^3*(-1/(b^4*c^6*d^10))^(1/4)*log(2*b^2*c^3*d^5*sqrt(-1/ (b^4*c^6*d^10))*cos(b*x + a)*sin(b*x + a) - 2*(I*b*c*d^2*(-1/(b^4*c^6*d^10 ))^(1/4)*cos(b*x + a)^2*sin(b*x + a) + (I*b^3*c^4*d^7*cos(b*x + a)^3 - I*b ^3*c^4*d^7*cos(b*x + a))*(-1/(b^4*c^6*d^10))^(3/4))*sqrt(c/cos(b*x + a))*s qrt(d/sin(b*x + a)) - 2*cos(b*x + a)^2 + 1) - 3*I*b*c^2*d^3*(-1/(b^4*c^6*d ^10))^(1/4)*log(2*b^2*c^3*d^5*sqrt(-1/(b^4*c^6*d^10))*cos(b*x + a)*sin(b*x + a) - 2*(-I*b*c*d^2*(-1/(b^4*c^6*d^10))^(1/4)*cos(b*x + a)^2*sin(b*x + a ) + (-I*b^3*c^4*d^7*cos(b*x + a)^3 + I*b^3*c^4*d^7*cos(b*x + a))*(-1/(b^4* c^6*d^10))^(3/4))*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a)) - 2*cos(b*x + a)^2 + 1) - 3*b*c^2*d^3*(-1/(b^4*c^6*d^10))^(1/4)*log(-2*b^2*c^3*d^5*sqrt( -1/(b^4*c^6*d^10))*cos(b*x + a)*sin(b*x + a) + 2*(b*c*d^2*(-1/(b^4*c^6*d^1 0))^(1/4)*cos(b*x + a)^2*sin(b*x + a) - (b^3*c^4*d^7*cos(b*x + a)^3 - b^3* c^4*d^7*cos(b*x + a))*(-1/(b^4*c^6*d^10))^(3/4))*sqrt(c/cos(b*x + a))*sqrt (d/sin(b*x + a)) - 2*cos(b*x + a)^2 + 1) + 3*b*c^2*d^3*(-1/(b^4*c^6*d^10)) ^(1/4)*log(-2*b^2*c^3*d^5*sqrt(-1/(b^4*c^6*d^10))*cos(b*x + a)*sin(b*x + a ) - 2*(b*c*d^2*(-1/(b^4*c^6*d^10))^(1/4)*cos(b*x + a)^2*sin(b*x + a) - (b^ 3*c^4*d^7*cos(b*x + a)^3 - b^3*c^4*d^7*cos(b*x + a))*(-1/(b^4*c^6*d^10))^( 3/4))*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a)) - 2*cos(b*x + a)^2 + 1) - 3*b*c^2*d^3*(-1/(b^4*c^6*d^10))^(1/4)*log(2*(b*c*d^2*(-1/(b^4*c^6*d^10))^( 1/4)*cos(b*x + a)^2*sin(b*x + a) + (b^3*c^4*d^7*cos(b*x + a)^3 - b^3*c^...
Timed out. \[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}} \, dx=\int { \frac {1}{\left (d \csc \left (b x + a\right )\right )^{\frac {5}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}} \, dx=\int { \frac {1}{\left (d \csc \left (b x + a\right )\right )^{\frac {5}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}} \, dx=\int \frac {1}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{3/2}\,{\left (\frac {d}{\sin \left (a+b\,x\right )}\right )}^{5/2}} \,d x \]