Integrand size = 25, antiderivative size = 320 \[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx=\frac {c^{3/2} \sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{4 \sqrt {2} b}-\frac {c^{3/2} \sqrt {d} \arctan \left (1+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{4 \sqrt {2} b}-\frac {c^{3/2} \sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (a+b x)-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{8 \sqrt {2} b}+\frac {c^{3/2} \sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (a+b x)+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{8 \sqrt {2} b}-\frac {c (d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)}}{2 b d} \]
-1/8*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))*d^(1/2)/b*2^(1/2)-1/8*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))*d^(1/2)/b*2^(1/2)-1/16*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))*d^(1/2)/b*2^(1/2)+1/16*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))*d^(1/2)/ b*2^(1/2)-1/2*c*(d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^(1/2)/b/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.21 \[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx=\frac {2 \sqrt {d \cos (a+b x)} \sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {5}{4},\frac {9}{4},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{3/2} \tan (a+b x)}{5 b} \]
(2*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, 5/4, 9/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(3/2)*Tan[a + b*x])/(5*b)
Time = 0.59 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3048, 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.
| \(\displaystyle \int (c \sin (a+b x))^{3/2} \sqrt {d \cos (a+b x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c \sin (a+b x))^{3/2} \sqrt {d \cos (a+b x)}dx\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {1}{4} c^2 \int \frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}dx-\frac {c \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2}}{2 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} c^2 \int \frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}dx-\frac {c \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2}}{2 b d}\) |
| \(\Big \downarrow \) 3055 |
| \(\displaystyle -\frac {c^3 d \int \frac {d \cot (a+b x)}{c \left (\cot ^2(a+b x) d^2+d^2\right )}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 b}-\frac {c \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2}}{2 b d}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {c^3 d \left (\frac {\int \frac {\cot (a+b x) d+d}{\cot ^2(a+b x) d^2+d^2}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 c}-\frac {\int \frac {d-d \cot (a+b x)}{\cot ^2(a+b x) d^2+d^2}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 c}\right )}{2 b}-\frac {c \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2}}{2 b d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {c^3 d \left (\frac {\frac {\int \frac {1}{\frac {\cot (a+b x) d}{c}+\frac {d}{c}-\frac {\sqrt {2} \sqrt {d \cos (a+b x)} \sqrt {d}}{\sqrt {c} \sqrt {c \sin (a+b x)}}}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 c}+\frac {\int \frac {1}{\frac {\cot (a+b x) d}{c}+\frac {d}{c}+\frac {\sqrt {2} \sqrt {d \cos (a+b x)} \sqrt {d}}{\sqrt {c} \sqrt {c \sin (a+b x)}}}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 c}}{2 c}-\frac {\int \frac {d-d \cot (a+b x)}{\cot ^2(a+b x) d^2+d^2}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 c}\right )}{2 b}-\frac {c \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2}}{2 b d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {c^3 d \left (\frac {\frac {\int \frac {1}{-\frac {d \cot (a+b x)}{c}-1}d\left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\int \frac {1}{-\frac {d \cot (a+b x)}{c}-1}d\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}-\frac {\int \frac {d-d \cot (a+b x)}{\cot ^2(a+b x) d^2+d^2}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 c}\right )}{2 b}-\frac {c \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2}}{2 b d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {c^3 d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}-\frac {\int \frac {d-d \cot (a+b x)}{\cot ^2(a+b x) d^2+d^2}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 c}\right )}{2 b}-\frac {c \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2}}{2 b d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {c^3 d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{\sqrt {c} \left (\frac {\cot (a+b x) d}{c}+\frac {d}{c}-\frac {\sqrt {2} \sqrt {d \cos (a+b x)} \sqrt {d}}{\sqrt {c} \sqrt {c \sin (a+b x)}}\right )}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{\sqrt {c} \left (\frac {\cot (a+b x) d}{c}+\frac {d}{c}+\frac {\sqrt {2} \sqrt {d \cos (a+b x)} \sqrt {d}}{\sqrt {c} \sqrt {c \sin (a+b x)}}\right )}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}\right )}{2 b}-\frac {c \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2}}{2 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {c^3 d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{\sqrt {c} \left (\frac {\cot (a+b x) d}{c}+\frac {d}{c}-\frac {\sqrt {2} \sqrt {d \cos (a+b x)} \sqrt {d}}{\sqrt {c} \sqrt {c \sin (a+b x)}}\right )}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{\sqrt {c} \left (\frac {\cot (a+b x) d}{c}+\frac {d}{c}+\frac {\sqrt {2} \sqrt {d \cos (a+b x)} \sqrt {d}}{\sqrt {c} \sqrt {c \sin (a+b x)}}\right )}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}\right )}{2 b}-\frac {c \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2}}{2 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^3 d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{\frac {\cot (a+b x) d}{c}+\frac {d}{c}-\frac {\sqrt {2} \sqrt {d \cos (a+b x)} \sqrt {d}}{\sqrt {c} \sqrt {c \sin (a+b x)}}}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 \sqrt {2} c \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{\frac {\cot (a+b x) d}{c}+\frac {d}{c}+\frac {\sqrt {2} \sqrt {d \cos (a+b x)} \sqrt {d}}{\sqrt {c} \sqrt {c \sin (a+b x)}}}d\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}}{2 c \sqrt {d}}}{2 c}\right )}{2 b}-\frac {c \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2}}{2 b d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {c^3 d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}+d \cot (a+b x)+d\right )}{2 \sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}+d \cot (a+b x)+d\right )}{2 \sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}\right )}{2 b}-\frac {c \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2}}{2 b d}\) |
-1/2*(c^3*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]*Sqrt[d])) + ArcTan[1 + (Sqrt[2]*S qrt[c]*Sqrt[d*Cos[a + b*x]])/(Sqrt[d]*Sqrt[c*Sin[a + b*x]])]/(Sqrt[2]*Sqrt [c]*Sqrt[d]))/(2*c) - (-1/2*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]]]/(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 - (c*(d*Cos[a + b*x])^(3/2)*Sqrt[c*Sin[a + b*x]])/(2*b*d)
3.3.71.3.1 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)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
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]
Time = 0.20 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.35
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (4 \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \sqrt {2}\, \left (\cos ^{2}\left (b x +a \right )\right )+4 \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \cos \left (b x +a \right )+2 \arctan \left (\frac {\sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}-\cos \left (b x +a \right )+1}{\cos \left (b x +a \right )-1}\right )-\ln \left (-2 \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \cot \left (b x +a \right )-2 \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \csc \left (b x +a \right )+2-2 \cot \left (b x +a \right )\right )+\ln \left (2 \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \cot \left (b x +a \right )+2 \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \csc \left (b x +a \right )+2-2 \cot \left (b x +a \right )\right )+2 \arctan \left (\frac {\sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right )\right ) \sqrt {c \sin \left (b x +a \right )}\, \sqrt {d \cos \left (b x +a \right )}\, c}{16 b \left (1+\cos \left (b x +a \right )\right ) \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}}\) | \(433\) |
-1/16/b*2^(1/2)*(4*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*2^(1/2) *cos(b*x+a)^2+4*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*co s(b*x+a)+2*arctan((sin(b*x+a)*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a ))^2)^(1/2)-cos(b*x+a)+1)/(cos(b*x+a)-1))-ln(-2*2^(1/2)*(-sin(b*x+a)*cos(b *x+a)/(1+cos(b*x+a))^2)^(1/2)*cot(b*x+a)-2*2^(1/2)*(-sin(b*x+a)*cos(b*x+a) /(1+cos(b*x+a))^2)^(1/2)*csc(b*x+a)+2-2*cot(b*x+a))+ln(2*2^(1/2)*(-sin(b*x +a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*cot(b*x+a)+2*2^(1/2)*(-sin(b*x+a)*c os(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*csc(b*x+a)+2-2*cot(b*x+a))+2*arctan((sin (b*x+a)*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)+cos(b*x+a) -1)/(cos(b*x+a)-1)))*(c*sin(b*x+a))^(1/2)*(d*cos(b*x+a))^(1/2)*c/(1+cos(b* x+a))/(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 1015, normalized size of antiderivative = 3.17 \[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx=\text {Too large to display} \]
-1/32*(16*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))*c*cos(b*x + a) - (-c^6 *d^2/b^4)^(1/4)*b*log(-2*c^5*d^2*cos(b*x + a)^2 + 2*sqrt(-c^6*d^2/b^4)*b^2 *c^2*d*cos(b*x + a)*sin(b*x + a) + c^5*d^2 + 2*((-c^6*d^2/b^4)^(1/4)*b*c^3 *d*sin(b*x + a) + (-c^6*d^2/b^4)^(3/4)*b^3*cos(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))) + (-c^6*d^2/b^4)^(1/4)*b*log(-2*c^5*d^2*cos(b*x + a)^2 + 2*sqrt(-c^6*d^2/b^4)*b^2*c^2*d*cos(b*x + a)*sin(b*x + a) + c^5*d^ 2 - 2*((-c^6*d^2/b^4)^(1/4)*b*c^3*d*sin(b*x + a) + (-c^6*d^2/b^4)^(3/4)*b^ 3*cos(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))) + I*(-c^6*d^2/b ^4)^(1/4)*b*log(-2*c^5*d^2*cos(b*x + a)^2 - 2*sqrt(-c^6*d^2/b^4)*b^2*c^2*d *cos(b*x + a)*sin(b*x + a) + c^5*d^2 - 2*(I*(-c^6*d^2/b^4)^(1/4)*b*c^3*d*s in(b*x + a) - I*(-c^6*d^2/b^4)^(3/4)*b^3*cos(b*x + a))*sqrt(d*cos(b*x + a) )*sqrt(c*sin(b*x + a))) - I*(-c^6*d^2/b^4)^(1/4)*b*log(-2*c^5*d^2*cos(b*x + a)^2 - 2*sqrt(-c^6*d^2/b^4)*b^2*c^2*d*cos(b*x + a)*sin(b*x + a) + c^5*d^ 2 - 2*(-I*(-c^6*d^2/b^4)^(1/4)*b*c^3*d*sin(b*x + a) + I*(-c^6*d^2/b^4)^(3/ 4)*b^3*cos(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))) + (-c^6*d^ 2/b^4)^(1/4)*b*log(-c^5*d^2 + 2*((-c^6*d^2/b^4)^(1/4)*b*c^3*d*sin(b*x + a) - (-c^6*d^2/b^4)^(3/4)*b^3*cos(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin( b*x + a))) - (-c^6*d^2/b^4)^(1/4)*b*log(-c^5*d^2 - 2*((-c^6*d^2/b^4)^(1/4) *b*c^3*d*sin(b*x + a) - (-c^6*d^2/b^4)^(3/4)*b^3*cos(b*x + a))*sqrt(d*cos( b*x + a))*sqrt(c*sin(b*x + a))) - I*(-c^6*d^2/b^4)^(1/4)*b*log(-c^5*d^2...
\[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx=\int \left (c \sin {\left (a + b x \right )}\right )^{\frac {3}{2}} \sqrt {d \cos {\left (a + b x \right )}}\, dx \]
\[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}} \,d x } \]
\[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx=\int \sqrt {d\,\cos \left (a+b\,x\right )}\,{\left (c\,\sin \left (a+b\,x\right )\right )}^{3/2} \,d x \]