Integrand size = 25, antiderivative size = 249 \[ \int \frac {1}{\sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}} \, dx=\frac {d}{2 b c (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {c \sec (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {c \sec (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (a+b x)}}{1+\tan (a+b x)}\right ) \sqrt {c \sec (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}} \] Output:
1/2*d/b/c/(d*csc(b*x+a))^(3/2)/(c*sec(b*x+a))^(1/2)+1/8*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)+1/8*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)-1/8*arctanh(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*cs c(b*x+a))^(1/2)/tan(b*x+a)^(1/2)
Time = 1.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}} \, dx=\frac {d \left (4 \cos ^2(a+b x)-\sqrt {2} \arctan \left (\frac {-1+\sqrt {\cot ^2(a+b x)}}{\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}\right ) \cot ^2(a+b x)^{3/4}-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}}{1+\sqrt {\cot ^2(a+b x)}}\right ) \cot ^2(a+b x)^{3/4}\right ) \sec ^3(a+b x)}{8 b (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{3/2}} \] Input:
Integrate[1/(Sqrt[d*Csc[a + b*x]]*(c*Sec[a + b*x])^(3/2)),x]
Output:
(d*(4*Cos[a + b*x]^2 - Sqrt[2]*ArcTan[(-1 + Sqrt[Cot[a + b*x]^2])/(Sqrt[2] *(Cot[a + b*x]^2)^(1/4))]*(Cot[a + b*x]^2)^(3/4) - 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)/(8*b*(d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x])^(3/2))
Time = 0.61 (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, 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.
| \(\displaystyle \int \frac {1}{(c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}dx\) |
| \(\Big \downarrow \) 3108 |
| \(\displaystyle \frac {\int \frac {\sqrt {c \sec (a+b x)}}{\sqrt {d \csc (a+b x)}}dx}{4 c^2}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {c \sec (a+b x)}}{\sqrt {d \csc (a+b x)}}dx}{4 c^2}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3109 |
| \(\displaystyle \frac {\sqrt {c \sec (a+b x)} \int \sqrt {\tan (a+b x)}dx}{4 c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {c \sec (a+b x)} \int \sqrt {\tan (a+b x)}dx}{4 c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {\sqrt {c \sec (a+b x)} \int \frac {\sqrt {\tan (a+b x)}}{\tan ^2(a+b x)+1}d\tan (a+b x)}{4 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\sqrt {c \sec (a+b x)} \int \frac {\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \int \frac {\tan (a+b x)+1}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}+\frac {1}{2} \int \frac {1}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}\right )-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (a+b x)}}{\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (a+b x)}+1}{\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1}d\sqrt {\tan (a+b x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\sqrt {c \sec (a+b x)} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2}}\right )\right )}{2 b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}\) |
Input:
Int[1/(Sqrt[d*Csc[a + b*x]]*(c*Sec[a + b*x])^(3/2)),x]
Output:
d/(2*b*c*(d*Csc[a + b*x])^(3/2)*Sqrt[c*Sec[a + b*x]]) + (((-(ArcTan[1 - Sq rt[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]])/(2*b*c^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[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[(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[(-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. \(429\) vs. \(2(200)=400\).
Time = 10.82 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.73
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (\ln \left (-\frac {\cos \left (b x +a \right ) \cot \left (b x +a \right )-2 \cot \left (b x +a \right )-2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{\cos \left (b x +a \right )-1}\right )-\ln \left (-\frac {\cos \left (b x +a \right ) \cot \left (b x +a \right )-2 \cot \left (b x +a \right )+2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{\cos \left (b x +a \right )-1}\right )+2 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-\cos \left (b x +a \right )+1}{\cos \left (b x +a \right )-1}\right )+2 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right )+\left (4 \cos \left (b x +a \right )+4\right ) \sin \left (b x +a \right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\right ) \sin \left (b x +a \right )^{3} \sec \left (\frac {b x}{2}+\frac {a}{2}\right )^{5} \csc \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{256 b c \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {d \csc \left (b x +a \right )}\, \sqrt {c \sec \left (b x +a \right )}}\) | \(430\) |
Input:
int(1/(d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/256/b*2^(1/2)/c*(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)+cs c(b*x+a)+2)/(cos(b*x+a)-1))-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))+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))+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))+(4*cos(b*x+a)+4)*sin(b*x+a)*(-2*sin(b*x+a)*cos(b*x+a)/(co s(b*x+a)+1)^2)^(1/2))*sin(b*x+a)^3/(-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)^5*cs c(1/2*b*x+1/2*a)^3
Leaf count of result is larger than twice the leaf count of optimal. 510 vs. \(2 (200) = 400\).
Time = 0.12 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.05 \[ \int \frac {1}{\sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}} \, dx=\frac {2 \, \sqrt {2} c d \sqrt {\frac {1}{c d}} \arctan \left (-\frac {1}{2} \, \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}} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )}\right ) + \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 ) + \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 ) + \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 ) - \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 (\cos \left (b x + a\right )^{3} - \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 )}}}{32 \, b c^{2} d} \] Input:
integrate(1/(d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(3/2),x, algorithm="fricas ")
Output:
1/32*(2*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))*(cos(b*x + a) - sin(b*x + a))) + 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))) + sqrt(2)*c*d*sqrt(1/(c*d))*arctan(-1/2*(sqrt(2)*sqrt(c/co s(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))) + sqrt(2)*c*d*sqrt(1/(c*d))*log(2*s qrt(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) - sqrt(2)*c*d*sqrt(1/(c*d))*log(-2*sqrt(2)*(cos(b*x + a)^3 - c os(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*(cos(b*x + a)^3 - cos(b*x + a))*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a)))/(b*c^2*d )
\[ \int \frac {1}{\sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}} \, dx=\int \frac {1}{\left (c \sec {\left (a + b x \right )}\right )^{\frac {3}{2}} \sqrt {d \csc {\left (a + b x \right )}}}\, dx \] Input:
integrate(1/(d*csc(b*x+a))**(1/2)/(c*sec(b*x+a))**(3/2),x)
Output:
Integral(1/((c*sec(a + b*x))**(3/2)*sqrt(d*csc(a + b*x))), x)
\[ \int \frac {1}{\sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}} \, dx=\int { \frac {1}{\sqrt {d \csc \left (b x + a\right )} \left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(3/2),x, algorithm="maxima ")
Output:
integrate(1/(sqrt(d*csc(b*x + a))*(c*sec(b*x + a))^(3/2)), x)
\[ \int \frac {1}{\sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}} \, dx=\int { \frac {1}{\sqrt {d \csc \left (b x + a\right )} \left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(3/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(d*csc(b*x + a))*(c*sec(b*x + a))^(3/2)), x)
Timed out. \[ \int \frac {1}{\sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}} \, dx=\int \frac {1}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{3/2}\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}} \,d x \] Input:
int(1/((c/cos(a + b*x))^(3/2)*(d/sin(a + b*x))^(1/2)),x)
Output:
int(1/((c/cos(a + b*x))^(3/2)*(d/sin(a + b*x))^(1/2)), x)
\[ \int \frac {1}{\sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}} \, dx=\frac {\sqrt {d}\, \sqrt {c}\, \left (\int \frac {\sqrt {\sec \left (b x +a \right )}\, \sqrt {\csc \left (b x +a \right )}}{\csc \left (b x +a \right ) \sec \left (b x +a \right )^{2}}d x \right )}{c^{2} d} \] Input:
int(1/(d*csc(b*x+a))^(1/2)/(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))/(c**2*d)