Integrand size = 25, antiderivative size = 320 \[ \int (d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)} \, dx=-\frac {\sqrt {c} d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{4 \sqrt {2} b}+\frac {\sqrt {c} d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{4 \sqrt {2} b}+\frac {\sqrt {c} d^{3/2} \log \left (\sqrt {c}-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)\right )}{8 \sqrt {2} b}-\frac {\sqrt {c} d^{3/2} \log \left (\sqrt {c}+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)\right )}{8 \sqrt {2} b}+\frac {d \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2}}{2 b c} \]
-1/8*d^(3/2)*arctan(1-2^(1/2)*d^(1/2)*(c*sin(b*x+a))^(1/2)/c^(1/2)/(d*cos( b*x+a))^(1/2))*c^(1/2)/b*2^(1/2)+1/8*d^(3/2)*arctan(1+2^(1/2)*d^(1/2)*(c*s in(b*x+a))^(1/2)/c^(1/2)/(d*cos(b*x+a))^(1/2))*c^(1/2)/b*2^(1/2)+1/16*d^(3 /2)*ln(c^(1/2)-2^(1/2)*d^(1/2)*(c*sin(b*x+a))^(1/2)/(d*cos(b*x+a))^(1/2)+c ^(1/2)*tan(b*x+a))*c^(1/2)/b*2^(1/2)-1/16*d^(3/2)*ln(c^(1/2)+2^(1/2)*d^(1/ 2)*(c*sin(b*x+a))^(1/2)/(d*cos(b*x+a))^(1/2)+c^(1/2)*tan(b*x+a))*c^(1/2)/b *2^(1/2)+1/2*d*(c*sin(b*x+a))^(3/2)*(d*cos(b*x+a))^(1/2)/b/c
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.22 \[ \int (d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)} \, dx=\frac {2 d^2 \cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{4},\frac {7}{4},\sin ^2(a+b x)\right ) \sqrt {c \sin (a+b x)} \tan (a+b x)}{3 b \sqrt {d \cos (a+b x)}} \]
(2*d^2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-1/4, 3/4, 7/4, Sin[a + b* x]^2]*Sqrt[c*Sin[a + b*x]]*Tan[a + b*x])/(3*b*Sqrt[d*Cos[a + b*x]])
Time = 0.61 (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, 3049, 3042, 3054, 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.
| \(\displaystyle \int \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2}dx\) |
| \(\Big \downarrow \) 3049 |
| \(\displaystyle \frac {1}{4} d^2 \int \frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}dx+\frac {d (c \sin (a+b x))^{3/2} \sqrt {d \cos (a+b x)}}{2 b c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} d^2 \int \frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}dx+\frac {d (c \sin (a+b x))^{3/2} \sqrt {d \cos (a+b x)}}{2 b c}\) |
| \(\Big \downarrow \) 3054 |
| \(\displaystyle \frac {c d^3 \int \frac {c \tan (a+b x)}{d \left (\tan ^2(a+b x) c^2+c^2\right )}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 b}+\frac {d (c \sin (a+b x))^{3/2} \sqrt {d \cos (a+b x)}}{2 b c}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {c d^3 \left (\frac {\int \frac {\tan (a+b x) c+c}{\tan ^2(a+b x) c^2+c^2}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 d}-\frac {\int \frac {c-c \tan (a+b x)}{\tan ^2(a+b x) c^2+c^2}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 d}\right )}{2 b}+\frac {d (c \sin (a+b x))^{3/2} \sqrt {d \cos (a+b x)}}{2 b c}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {c d^3 \left (\frac {\frac {\int \frac {1}{\frac {\tan (a+b x) c}{d}+\frac {c}{d}-\frac {\sqrt {2} \sqrt {c \sin (a+b x)} \sqrt {c}}{\sqrt {d} \sqrt {d \cos (a+b x)}}}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 d}+\frac {\int \frac {1}{\frac {\tan (a+b x) c}{d}+\frac {c}{d}+\frac {\sqrt {2} \sqrt {c \sin (a+b x)} \sqrt {c}}{\sqrt {d} \sqrt {d \cos (a+b x)}}}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 d}}{2 d}-\frac {\int \frac {c-c \tan (a+b x)}{\tan ^2(a+b x) c^2+c^2}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 d}\right )}{2 b}+\frac {d (c \sin (a+b x))^{3/2} \sqrt {d \cos (a+b x)}}{2 b c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {c d^3 \left (\frac {\frac {\int \frac {1}{-\frac {c \tan (a+b x)}{d}-1}d\left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\int \frac {1}{-\frac {c \tan (a+b x)}{d}-1}d\left (\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 d}-\frac {\int \frac {c-c \tan (a+b x)}{\tan ^2(a+b x) c^2+c^2}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 d}\right )}{2 b}+\frac {d (c \sin (a+b x))^{3/2} \sqrt {d \cos (a+b x)}}{2 b c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {c d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 d}-\frac {\int \frac {c-c \tan (a+b x)}{\tan ^2(a+b x) c^2+c^2}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 d}\right )}{2 b}+\frac {d (c \sin (a+b x))^{3/2} \sqrt {d \cos (a+b x)}}{2 b c}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {c d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 d}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {c}-\frac {2 \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{\sqrt {d} \left (\frac {\tan (a+b x) c}{d}+\frac {c}{d}-\frac {\sqrt {2} \sqrt {c \sin (a+b x)} \sqrt {c}}{\sqrt {d} \sqrt {d \cos (a+b x)}}\right )}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {c}+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{\sqrt {d} \left (\frac {\tan (a+b x) c}{d}+\frac {c}{d}+\frac {\sqrt {2} \sqrt {c \sin (a+b x)} \sqrt {c}}{\sqrt {d} \sqrt {d \cos (a+b x)}}\right )}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}}{2 d}\right )}{2 b}+\frac {d (c \sin (a+b x))^{3/2} \sqrt {d \cos (a+b x)}}{2 b c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 d}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {c}-\frac {2 \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{\sqrt {d} \left (\frac {\tan (a+b x) c}{d}+\frac {c}{d}-\frac {\sqrt {2} \sqrt {c \sin (a+b x)} \sqrt {c}}{\sqrt {d} \sqrt {d \cos (a+b x)}}\right )}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {c}+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{\sqrt {d} \left (\frac {\tan (a+b x) c}{d}+\frac {c}{d}+\frac {\sqrt {2} \sqrt {c \sin (a+b x)} \sqrt {c}}{\sqrt {d} \sqrt {d \cos (a+b x)}}\right )}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}}{2 d}\right )}{2 b}+\frac {d (c \sin (a+b x))^{3/2} \sqrt {d \cos (a+b x)}}{2 b c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 d}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {c}-\frac {2 \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{\frac {\tan (a+b x) c}{d}+\frac {c}{d}-\frac {\sqrt {2} \sqrt {c \sin (a+b x)} \sqrt {c}}{\sqrt {d} \sqrt {d \cos (a+b x)}}}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 \sqrt {2} \sqrt {c} d}+\frac {\int \frac {\sqrt {c}+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{\frac {\tan (a+b x) c}{d}+\frac {c}{d}+\frac {\sqrt {2} \sqrt {c \sin (a+b x)} \sqrt {c}}{\sqrt {d} \sqrt {d \cos (a+b x)}}}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 \sqrt {c} d}}{2 d}\right )}{2 b}+\frac {d (c \sin (a+b x))^{3/2} \sqrt {d \cos (a+b x)}}{2 b c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {c d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+c \tan (a+b x)+c\right )}{2 \sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+c \tan (a+b x)+c\right )}{2 \sqrt {2} \sqrt {c} \sqrt {d}}}{2 d}\right )}{2 b}+\frac {d (c \sin (a+b x))^{3/2} \sqrt {d \cos (a+b x)}}{2 b c}\) |
(c*d^3*((-(ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/(Sqrt[c]*Sqrt [d*Cos[a + b*x]])]/(Sqrt[2]*Sqrt[c]*Sqrt[d])) + ArcTan[1 + (Sqrt[2]*Sqrt[d ]*Sqrt[c*Sin[a + b*x]])/(Sqrt[c]*Sqrt[d*Cos[a + b*x]])]/(Sqrt[2]*Sqrt[c]*S qrt[d]))/(2*d) - (-1/2*Log[c - (Sqrt[2]*Sqrt[c]*Sqrt[d]*Sqrt[c*Sin[a + b*x ]])/Sqrt[d*Cos[a + b*x]] + c*Tan[a + b*x]]/(Sqrt[2]*Sqrt[c]*Sqrt[d]) + Log [c + (Sqrt[2]*Sqrt[c]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/Sqrt[d*Cos[a + b*x]] + c*Tan[a + b*x]]/(2*Sqrt[2]*Sqrt[c]*Sqrt[d]))/(2*d)))/(2*b) + (d*Sqrt[d*Co s[a + b*x]]*(c*Sin[a + b*x])^(3/2))/(2*b*c)
3.3.62.3.1 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_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[a*(b*Sin[e + f*x])^(n + 1)*((a*Cos[e + f*x])^(m - 1)/ (b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Sin[e + f*x])^n*(a *Cos[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_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), 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*Sin[e + f*x])^(1/k)/(b*Cos[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
Time = 2.83 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.37
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (4 \sin \left (b x +a \right ) \sqrt {2}\, \cos \left (b x +a \right ) \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}+4 \sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}+\ln \left (-2 \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \cot \left (b x +a \right )-2 \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \csc \left (b x +a \right )+2-2 \cot \left (b x +a \right )\right )-\ln \left (2 \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \cot \left (b x +a \right )+2 \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \csc \left (b x +a \right )+2-2 \cot \left (b x +a \right )\right )+2 \arctan \left (\frac {\sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}-\cos \left (b x +a \right )+1}{\cos \left (b x +a \right )-1}\right )+2 \arctan \left (\frac {\sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right )\right ) \sqrt {c \sin \left (b x +a \right )}\, \sqrt {d \cos \left (b x +a \right )}\, d}{16 b \left (1+\cos \left (b x +a \right )\right ) \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}}\) | \(437\) |
1/16/b*2^(1/2)*(4*sin(b*x+a)*2^(1/2)*cos(b*x+a)*(-sin(b*x+a)*cos(b*x+a)/(1 +cos(b*x+a))^2)^(1/2)+4*sin(b*x+a)*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos( b*x+a))^2)^(1/2)+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)*cos(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))+2*arc tan((sin(b*x+a)*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)+co s(b*x+a)-1)/(cos(b*x+a)-1)))*(c*sin(b*x+a))^(1/2)*(d*cos(b*x+a))^(1/2)*d/( 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 = 0.47 (sec) , antiderivative size = 1033, normalized size of antiderivative = 3.23 \[ \int (d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)} \, dx=\text {Too large to display} \]
1/32*(16*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))*d*sin(b*x + a) - (-c^2* d^6/b^4)^(1/4)*b*log(1/2*c^2*d^5*cos(b*x + a)*sin(b*x + a) + 1/2*((-c^2*d^ 6/b^4)^(1/4)*b*c*d^3*sin(b*x + a) - (-c^2*d^6/b^4)^(3/4)*b^3*cos(b*x + a)) *sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a)) - 1/4*(2*b^2*c*d^2*cos(b*x + a) ^2 - b^2*c*d^2)*sqrt(-c^2*d^6/b^4)) + (-c^2*d^6/b^4)^(1/4)*b*log(1/2*c^2*d ^5*cos(b*x + a)*sin(b*x + a) - 1/2*((-c^2*d^6/b^4)^(1/4)*b*c*d^3*sin(b*x + a) - (-c^2*d^6/b^4)^(3/4)*b^3*cos(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*s in(b*x + a)) - 1/4*(2*b^2*c*d^2*cos(b*x + a)^2 - b^2*c*d^2)*sqrt(-c^2*d^6/ b^4)) - I*(-c^2*d^6/b^4)^(1/4)*b*log(1/2*c^2*d^5*cos(b*x + a)*sin(b*x + a) + 1/2*(I*(-c^2*d^6/b^4)^(1/4)*b*c*d^3*sin(b*x + a) + I*(-c^2*d^6/b^4)^(3/ 4)*b^3*cos(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a)) + 1/4*(2*b^ 2*c*d^2*cos(b*x + a)^2 - b^2*c*d^2)*sqrt(-c^2*d^6/b^4)) + I*(-c^2*d^6/b^4) ^(1/4)*b*log(1/2*c^2*d^5*cos(b*x + a)*sin(b*x + a) + 1/2*(-I*(-c^2*d^6/b^4 )^(1/4)*b*c*d^3*sin(b*x + a) - I*(-c^2*d^6/b^4)^(3/4)*b^3*cos(b*x + a))*sq rt(d*cos(b*x + a))*sqrt(c*sin(b*x + a)) + 1/4*(2*b^2*c*d^2*cos(b*x + a)^2 - b^2*c*d^2)*sqrt(-c^2*d^6/b^4)) - (-c^2*d^6/b^4)^(1/4)*b*log(c^2*d^5 + 2* ((-c^2*d^6/b^4)^(1/4)*b*c*d^3*cos(b*x + a) - (-c^2*d^6/b^4)^(3/4)*b^3*sin( b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))) + (-c^2*d^6/b^4)^(1/4 )*b*log(c^2*d^5 - 2*((-c^2*d^6/b^4)^(1/4)*b*c*d^3*cos(b*x + a) - (-c^2*d^6 /b^4)^(3/4)*b^3*sin(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))...
\[ \int (d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)} \, dx=\int \sqrt {c \sin {\left (a + b x \right )}} \left (d \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int (d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)} \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}} \sqrt {c \sin \left (b x + a\right )} \,d x } \]
\[ \int (d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)} \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}} \sqrt {c \sin \left (b x + a\right )} \,d x } \]
Timed out. \[ \int (d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)} \, dx=\int {\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}\,\sqrt {c\,\sin \left (a+b\,x\right )} \,d x \]