Integrand size = 25, antiderivative size = 313 \[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx=-\frac {c^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} b d^{3/2}}+\frac {c^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} b d^{3/2}}+\frac {c^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (a+b x)-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{2 \sqrt {2} b d^{3/2}}-\frac {c^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (a+b x)+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{2 \sqrt {2} b d^{3/2}}+\frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}} \]
1/2*c^(3/2)*arctan(-1+2^(1/2)*c^(1/2)*(d*cos(b*x+a))^(1/2)/d^(1/2)/(c*sin( b*x+a))^(1/2))/b/d^(3/2)*2^(1/2)+1/2*c^(3/2)*arctan(1+2^(1/2)*c^(1/2)*(d*c os(b*x+a))^(1/2)/d^(1/2)/(c*sin(b*x+a))^(1/2))/b/d^(3/2)*2^(1/2)+1/4*c^(3/ 2)*ln(d^(1/2)+cot(b*x+a)*d^(1/2)-2^(1/2)*c^(1/2)*(d*cos(b*x+a))^(1/2)/(c*s in(b*x+a))^(1/2))/b/d^(3/2)*2^(1/2)-1/4*c^(3/2)*ln(d^(1/2)+cot(b*x+a)*d^(1 /2)+2^(1/2)*c^(1/2)*(d*cos(b*x+a))^(1/2)/(c*sin(b*x+a))^(1/2))/b/d^(3/2)*2 ^(1/2)+2*c*(c*sin(b*x+a))^(1/2)/b/d/(d*cos(b*x+a))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.21 \[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx=\frac {2 \sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {5}{4},\frac {9}{4},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{5/2}}{5 b c d \sqrt {d \cos (a+b x)}} \]
(2*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[5/4, 5/4, 9/4, Sin[a + b*x]^2] *(c*Sin[a + b*x])^(5/2))/(5*b*c*d*Sqrt[d*Cos[a + b*x]])
Time = 0.59 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3046, 3042, 3055, 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.
| \(\displaystyle \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3046 |
| \(\displaystyle \frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}}-\frac {c^2 \int \frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}dx}{d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}}-\frac {c^2 \int \frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}dx}{d^2}\) |
| \(\Big \downarrow \) 3055 |
| \(\displaystyle \frac {2 c^3 \int \frac {d \cot (a+b x)}{c \left (\cot ^2(a+b x) d^2+d^2\right )}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{b d}+\frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {2 c^3 \left (\frac {\int \frac {\cot (a+b x) d+d}{\cot ^2(a+b x) d^2+d^2}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 c}-\frac {\int \frac {d-d \cot (a+b x)}{\cot ^2(a+b x) d^2+d^2}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 c}\right )}{b d}+\frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 c^3 \left (\frac {\frac {\int \frac {1}{\frac {\cot (a+b x) d}{c}+\frac {d}{c}-\frac {\sqrt {2} \sqrt {d \cos (a+b x)} \sqrt {d}}{\sqrt {c} \sqrt {c \sin (a+b x)}}}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 c}+\frac {\int \frac {1}{\frac {\cot (a+b x) d}{c}+\frac {d}{c}+\frac {\sqrt {2} \sqrt {d \cos (a+b x)} \sqrt {d}}{\sqrt {c} \sqrt {c \sin (a+b x)}}}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 c}}{2 c}-\frac {\int \frac {d-d \cot (a+b x)}{\cot ^2(a+b x) d^2+d^2}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 c}\right )}{b d}+\frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 c^3 \left (\frac {\frac {\int \frac {1}{-\frac {d \cot (a+b x)}{c}-1}d\left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\int \frac {1}{-\frac {d \cot (a+b x)}{c}-1}d\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}-\frac {\int \frac {d-d \cot (a+b x)}{\cot ^2(a+b x) d^2+d^2}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 c}\right )}{b d}+\frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 c^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}-\frac {\int \frac {d-d \cot (a+b x)}{\cot ^2(a+b x) d^2+d^2}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 c}\right )}{b d}+\frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 c^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{\sqrt {c} \left (\frac {\cot (a+b x) d}{c}+\frac {d}{c}-\frac {\sqrt {2} \sqrt {d \cos (a+b x)} \sqrt {d}}{\sqrt {c} \sqrt {c \sin (a+b x)}}\right )}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{\sqrt {c} \left (\frac {\cot (a+b x) d}{c}+\frac {d}{c}+\frac {\sqrt {2} \sqrt {d \cos (a+b x)} \sqrt {d}}{\sqrt {c} \sqrt {c \sin (a+b x)}}\right )}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}\right )}{b d}+\frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{\sqrt {c} \left (\frac {\cot (a+b x) d}{c}+\frac {d}{c}-\frac {\sqrt {2} \sqrt {d \cos (a+b x)} \sqrt {d}}{\sqrt {c} \sqrt {c \sin (a+b x)}}\right )}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{\sqrt {c} \left (\frac {\cot (a+b x) d}{c}+\frac {d}{c}+\frac {\sqrt {2} \sqrt {d \cos (a+b x)} \sqrt {d}}{\sqrt {c} \sqrt {c \sin (a+b x)}}\right )}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}\right )}{b d}+\frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{\frac {\cot (a+b x) d}{c}+\frac {d}{c}-\frac {\sqrt {2} \sqrt {d \cos (a+b x)} \sqrt {d}}{\sqrt {c} \sqrt {c \sin (a+b x)}}}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 \sqrt {2} c \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{\frac {\cot (a+b x) d}{c}+\frac {d}{c}+\frac {\sqrt {2} \sqrt {d \cos (a+b x)} \sqrt {d}}{\sqrt {c} \sqrt {c \sin (a+b x)}}}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 c \sqrt {d}}}{2 c}\right )}{b d}+\frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 c^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}+d \cot (a+b x)+d\right )}{2 \sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}+d \cot (a+b x)+d\right )}{2 \sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}\right )}{b d}+\frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}}\) |
(2*c^3*((-(ArcTan[1 - (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a + b*x]])/(Sqrt[d]*Sqrt [c*Sin[a + b*x]])]/(Sqrt[2]*Sqrt[c]*Sqrt[d])) + ArcTan[1 + (Sqrt[2]*Sqrt[c ]*Sqrt[d*Cos[a + b*x]])/(Sqrt[d]*Sqrt[c*Sin[a + b*x]])]/(Sqrt[2]*Sqrt[c]*S qrt[d]))/(2*c) - (-1/2*Log[d + d*Cot[a + b*x] - (Sqrt[2]*Sqrt[c]*Sqrt[d]*S qrt[d*Cos[a + b*x]])/Sqrt[c*Sin[a + b*x]]]/(Sqrt[2]*Sqrt[c]*Sqrt[d]) + Log [d + d*Cot[a + b*x] + (Sqrt[2]*Sqrt[c]*Sqrt[d]*Sqrt[d*Cos[a + b*x]])/Sqrt[ c*Sin[a + b*x]]]/(2*Sqrt[2]*Sqrt[c]*Sqrt[d]))/(2*c)))/(b*d) + (2*c*Sqrt[c* Sin[a + b*x]])/(b*d*Sqrt[d*Cos[a + b*x]])
3.3.72.3.1 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[(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[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(a*Sin[e + f*x])^(m - 1)*((b*Cos[e + f*x])^(n + 1)/(b*f*(n + 1))), x] + Simp[a^2*((m - 1)/(b^2*(n + 1))) Int[(a*Sin[e + f *x])^(m - 2)*(b*Cos[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> With[{k = Denominator[m]}, Simp[(-k)*a*(b/f) Subst[Int[x ^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Cos[e + f*x])^(1/k)/(b*Sin[ e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
Leaf count of result is larger than twice the leaf count of optimal. \(686\) vs. \(2(237)=474\).
Time = 0.25 (sec) , antiderivative size = 687, normalized size of antiderivative = 2.19
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (\frac {c \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )+1}\right )^{\frac {3}{2}} \left (\sin ^{2}\left (b x +a \right )\right ) \left (\ln \left (-\frac {-\left (1-\cos \left (b x +a \right )\right )^{2} \csc \left (b x +a \right )+2 \sqrt {\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}\, \sin \left (b x +a \right )-2+2 \cos \left (b x +a \right )+\sin \left (b x +a \right )}{1-\cos \left (b x +a \right )}\right ) \sqrt {\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}+2 \arctan \left (\frac {\sqrt {\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}\, \sin \left (b x +a \right )+\cos \left (b x +a \right )-1}{1-\cos \left (b x +a \right )}\right ) \sqrt {\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}-\ln \left (\frac {\left (1-\cos \left (b x +a \right )\right )^{2} \csc \left (b x +a \right )+2 \sqrt {\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}\, \sin \left (b x +a \right )+2-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )}{1-\cos \left (b x +a \right )}\right ) \sqrt {\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}+2 \arctan \left (\frac {\sqrt {\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}\, \sin \left (b x +a \right )+1-\cos \left (b x +a \right )}{1-\cos \left (b x +a \right )}\right ) \sqrt {\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}+8 \csc \left (b x +a \right )-8 \cot \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right )}{4 b \left (1-\cos \left (b x +a \right )\right )^{2} {\left (-\frac {d \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right )}{\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )+1}\right )}^{\frac {3}{2}}}\) | \(687\) |
-1/4/b*2^(1/2)*(c/((1-cos(b*x+a))^2*csc(b*x+a)^2+1)*(csc(b*x+a)-cot(b*x+a) ))^(3/2)/(1-cos(b*x+a))^2*sin(b*x+a)^2*(ln(-1/(1-cos(b*x+a))*(-(1-cos(b*x+ a))^2*csc(b*x+a)+2*((1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*csc(b *x+a))^(1/2)*sin(b*x+a)-2+2*cos(b*x+a)+sin(b*x+a)))*((1-cos(b*x+a))*((1-co s(b*x+a))^2*csc(b*x+a)^2-1)*csc(b*x+a))^(1/2)+2*arctan(1/(1-cos(b*x+a))*(( (1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*csc(b*x+a))^(1/2)*sin(b*x +a)+cos(b*x+a)-1))*((1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*csc(b *x+a))^(1/2)-ln(1/(1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)+2*((1-cos(b* x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*csc(b*x+a))^(1/2)*sin(b*x+a)+2-2*c os(b*x+a)-sin(b*x+a)))*((1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*c sc(b*x+a))^(1/2)+2*arctan(1/(1-cos(b*x+a))*(((1-cos(b*x+a))*((1-cos(b*x+a) )^2*csc(b*x+a)^2-1)*csc(b*x+a))^(1/2)*sin(b*x+a)+1-cos(b*x+a)))*((1-cos(b* x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*csc(b*x+a))^(1/2)+8*csc(b*x+a)-8*c ot(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)/(-d*((1-cos(b*x+a))^2*csc(b*x +a)^2-1)/((1-cos(b*x+a))^2*csc(b*x+a)^2+1))^(3/2)
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 1092, normalized size of antiderivative = 3.49 \[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx=\text {Too large to display} \]
-1/8*(-I*b*d^2*(-c^6/(b^4*d^6))^(1/4)*cos(b*x + a)*log(2*b^2*c^2*d^3*sqrt( -c^6/(b^4*d^6))*cos(b*x + a)*sin(b*x + a) + 2*c^5*cos(b*x + a)^2 - c^5 - 2 *(I*b^3*d^4*(-c^6/(b^4*d^6))^(3/4)*cos(b*x + a) - I*b*c^3*d*(-c^6/(b^4*d^6 ))^(1/4)*sin(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))) + I*b*d^ 2*(-c^6/(b^4*d^6))^(1/4)*cos(b*x + a)*log(2*b^2*c^2*d^3*sqrt(-c^6/(b^4*d^6 ))*cos(b*x + a)*sin(b*x + a) + 2*c^5*cos(b*x + a)^2 - c^5 - 2*(-I*b^3*d^4* (-c^6/(b^4*d^6))^(3/4)*cos(b*x + a) + I*b*c^3*d*(-c^6/(b^4*d^6))^(1/4)*sin (b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))) - b*d^2*(-c^6/(b^4*d ^6))^(1/4)*cos(b*x + a)*log(-2*b^2*c^2*d^3*sqrt(-c^6/(b^4*d^6))*cos(b*x + a)*sin(b*x + a) + 2*c^5*cos(b*x + a)^2 - c^5 + 2*(b^3*d^4*(-c^6/(b^4*d^6)) ^(3/4)*cos(b*x + a) + b*c^3*d*(-c^6/(b^4*d^6))^(1/4)*sin(b*x + a))*sqrt(d* cos(b*x + a))*sqrt(c*sin(b*x + a))) + b*d^2*(-c^6/(b^4*d^6))^(1/4)*cos(b*x + a)*log(-2*b^2*c^2*d^3*sqrt(-c^6/(b^4*d^6))*cos(b*x + a)*sin(b*x + a) + 2*c^5*cos(b*x + a)^2 - c^5 - 2*(b^3*d^4*(-c^6/(b^4*d^6))^(3/4)*cos(b*x + a ) + b*c^3*d*(-c^6/(b^4*d^6))^(1/4)*sin(b*x + a))*sqrt(d*cos(b*x + a))*sqrt (c*sin(b*x + a))) + b*d^2*(-c^6/(b^4*d^6))^(1/4)*cos(b*x + a)*log(-c^5 + 2 *(b^3*d^4*(-c^6/(b^4*d^6))^(3/4)*cos(b*x + a) - b*c^3*d*(-c^6/(b^4*d^6))^( 1/4)*sin(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))) - b*d^2*(-c^ 6/(b^4*d^6))^(1/4)*cos(b*x + a)*log(-c^5 - 2*(b^3*d^4*(-c^6/(b^4*d^6))^(3/ 4)*cos(b*x + a) - b*c^3*d*(-c^6/(b^4*d^6))^(1/4)*sin(b*x + a))*sqrt(d*c...
\[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx=\int \frac {\left (c \sin {\left (a + b x \right )}\right )^{\frac {3}{2}}}{\left (d \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx=\int \frac {{\left (c\,\sin \left (a+b\,x\right )\right )}^{3/2}}{{\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}} \,d x \]