\(\int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx\) [271]

Optimal result

Integrand size = 25, antiderivative size = 245 \[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx=\frac {c^{3/2} \sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{4 \sqrt {2} b}-\frac {c^{3/2} \sqrt {d} \arctan \left (1+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{4 \sqrt {2} b}+\frac {c^{3/2} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\left (\sqrt {d}+\sqrt {d} \cot (a+b x)\right ) \sqrt {c \sin (a+b x)}}\right )}{4 \sqrt {2} b}-\frac {c (d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)}}{2 b d} \] Output:

-1/8*c^(3/2)*d^(1/2)*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))*2^(1/2)/b-1/8*c^(3/2)*d^(1/2)*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))*2^(1/2)/b+1/8*c^(3 
/2)*d^(1/2)*arctanh(2^(1/2)*c^(1/2)*(d*cos(b*x+a))^(1/2)/(d^(1/2)+d^(1/2)* 
cot(b*x+a))/(c*sin(b*x+a))^(1/2))*2^(1/2)/b-1/2*c*(d*cos(b*x+a))^(3/2)*(c* 
sin(b*x+a))^(1/2)/b/d
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.27 \[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx=\frac {2 \sqrt {d \cos (a+b x)} \sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {5}{4},\frac {9}{4},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{3/2} \tan (a+b x)}{5 b} \] Input:

Integrate[Sqrt[d*Cos[a + b*x]]*(c*Sin[a + b*x])^(3/2),x]
 

Output:

(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)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.33, 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

\(\displaystyle -\frac {c^3 d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}+d \cot (a+b x)+d\right )}{2 \sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}+d \cot (a+b x)+d\right )}{2 \sqrt {2} \sqrt {c} \sqrt {d}}}{2 c}\right )}{2 b}-\frac {c \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2}}{2 b d}\)

Input:

Int[Sqrt[d*Cos[a + b*x]]*(c*Sin[a + b*x])^(3/2),x]
 

Output:

-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)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 826

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]]))
 

rule 1082

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]
 

rule 1103

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]
 

rule 1476

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]
 

rule 1479

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3048

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]
 

rule 3055

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]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(427\) vs. \(2(182)=364\).

Time = 21.86 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.75

method	result	size
default	\(-\frac {\sqrt {2}\, c \left (\ln \left (-\frac {\cos \left (b x +a \right ) \cot \left (b x +a \right )-2 \cot \left (b x +a \right )+2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{\cos \left (b x +a \right )-1}\right )-\ln \left (\frac {2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-\cos \left (b x +a \right ) \cot \left (b x +a \right )+2 \cos \left (b x +a \right )+\sin \left (b x +a \right )-\csc \left (b x +a \right )+2 \cot \left (b x +a \right )-2}{\cos \left (b x +a \right )-1}\right )+\cos \left (b x +a \right ) \left (4 \cos \left (b x +a \right )+4\right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+2 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-\cos \left (b x +a \right )+1}{\cos \left (b x +a \right )-1}\right )+2 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right )\right ) \sqrt {c \sin \left (b x +a \right )}\, \sqrt {d \cos \left (b x +a \right )}\, \sin \left (b x +a \right )^{3} \sec \left (\frac {b x}{2}+\frac {a}{2}\right )^{5} \csc \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{256 b \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}}\)	\(428\)

Input:

int((d*cos(b*x+a))^(1/2)*(c*sin(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)+c 
sc(b*x+a)+2)/(cos(b*x+a)-1))-ln((2*(-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)*cot(b*x+a)+2*cos(b*x+a)+sin(b*x+a)-csc(b* 
x+a)+2*cot(b*x+a)-2)/(cos(b*x+a)-1))+cos(b*x+a)*(4*cos(b*x+a)+4)*(-2*sin(b 
*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+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*arct 
an(((-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)))*(c*sin(b*x+a))^(1/2)*(d*cos(b*x+a))^(1/2)*sin(b*x+a) 
^3/(-sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*sec(1/2*b*x+1/2*a)^5*cs 
c(1/2*b*x+1/2*a)^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 512 vs. \(2 (182) = 364\).

Time = 0.14 (sec) , antiderivative size = 512, normalized size of antiderivative = 2.09 \[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx=-\frac {\sqrt {2} \sqrt {c d} c \arctan \left (\frac {2 \, c d \cos \left (b x + a\right )^{3} - 2 \, c d \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) - 2 \, c d \cos \left (b x + a\right ) + \sqrt {2} \sqrt {c d} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )}}{2 \, {\left (c d \cos \left (b x + a\right )^{3} + c d \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) - c d \cos \left (b x + a\right )\right )}}\right ) + \sqrt {2} \sqrt {c d} c \arctan \left (-\frac {2 \, c d \cos \left (b x + a\right )^{3} - 2 \, c d \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) - 2 \, c d \cos \left (b x + a\right ) - \sqrt {2} \sqrt {c d} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )}}{2 \, {\left (c d \cos \left (b x + a\right )^{3} + c d \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) - c d \cos \left (b x + a\right )\right )}}\right ) - 2 \, \sqrt {2} \sqrt {c d} c \arctan \left (-\frac {\sqrt {2} \sqrt {c d} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )}}{2 \, c d \cos \left (b x + a\right ) \sin \left (b x + a\right )}\right ) + 16 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} c \cos \left (b x + a\right ) - \sqrt {2} \sqrt {c d} c \log \left (4 \, c d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 2 \, \sqrt {2} \sqrt {c d} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} {\left (\cos \left (b x + a\right ) + \sin \left (b x + a\right )\right )} + c d\right ) + \sqrt {2} \sqrt {c d} c \log \left (4 \, c d \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 2 \, \sqrt {2} \sqrt {c d} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} {\left (\cos \left (b x + a\right ) + \sin \left (b x + a\right )\right )} + c d\right )}{32 \, b} \] Input:

integrate((d*cos(b*x+a))^(1/2)*(c*sin(b*x+a))^(3/2),x, algorithm="fricas")
 

Output:

-1/32*(sqrt(2)*sqrt(c*d)*c*arctan(1/2*(2*c*d*cos(b*x + a)^3 - 2*c*d*cos(b* 
x + a)^2*sin(b*x + a) - 2*c*d*cos(b*x + a) + sqrt(2)*sqrt(c*d)*sqrt(d*cos( 
b*x + a))*sqrt(c*sin(b*x + a)))/(c*d*cos(b*x + a)^3 + c*d*cos(b*x + a)^2*s 
in(b*x + a) - c*d*cos(b*x + a))) + sqrt(2)*sqrt(c*d)*c*arctan(-1/2*(2*c*d* 
cos(b*x + a)^3 - 2*c*d*cos(b*x + a)^2*sin(b*x + a) - 2*c*d*cos(b*x + a) - 
sqrt(2)*sqrt(c*d)*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a)))/(c*d*cos(b*x 
+ a)^3 + c*d*cos(b*x + a)^2*sin(b*x + a) - c*d*cos(b*x + a))) - 2*sqrt(2)* 
sqrt(c*d)*c*arctan(-1/2*sqrt(2)*sqrt(c*d)*sqrt(d*cos(b*x + a))*sqrt(c*sin( 
b*x + a))*(cos(b*x + a) - sin(b*x + a))/(c*d*cos(b*x + a)*sin(b*x + a))) + 
 16*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))*c*cos(b*x + a) - sqrt(2)*sqr 
t(c*d)*c*log(4*c*d*cos(b*x + a)*sin(b*x + a) + 2*sqrt(2)*sqrt(c*d)*sqrt(d* 
cos(b*x + a))*sqrt(c*sin(b*x + a))*(cos(b*x + a) + sin(b*x + a)) + c*d) + 
sqrt(2)*sqrt(c*d)*c*log(4*c*d*cos(b*x + a)*sin(b*x + a) - 2*sqrt(2)*sqrt(c 
*d)*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))*(cos(b*x + a) + sin(b*x + a) 
) + c*d))/b
 

Sympy [F]

\[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx=\int \left (c \sin {\left (a + b x \right )}\right )^{\frac {3}{2}} \sqrt {d \cos {\left (a + b x \right )}}\, dx \] Input:

integrate((d*cos(b*x+a))**(1/2)*(c*sin(b*x+a))**(3/2),x)
 

Output:

Integral((c*sin(a + b*x))**(3/2)*sqrt(d*cos(a + b*x)), x)
 

Maxima [F]

\[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}} \,d x } \] Input:

integrate((d*cos(b*x+a))^(1/2)*(c*sin(b*x+a))^(3/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(d*cos(b*x + a))*(c*sin(b*x + a))^(3/2), x)
 

Giac [F]

\[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}} \,d x } \] Input:

integrate((d*cos(b*x+a))^(1/2)*(c*sin(b*x+a))^(3/2),x, algorithm="giac")
 

Output:

integrate(sqrt(d*cos(b*x + a))*(c*sin(b*x + a))^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx=\int \sqrt {d\,\cos \left (a+b\,x\right )}\,{\left (c\,\sin \left (a+b\,x\right )\right )}^{3/2} \,d x \] Input:

int((d*cos(a + b*x))^(1/2)*(c*sin(a + b*x))^(3/2),x)
 

Output:

int((d*cos(a + b*x))^(1/2)*(c*sin(a + b*x))^(3/2), x)
 

Reduce [F]

\[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx=\sqrt {d}\, \sqrt {c}\, \left (\int \sqrt {\sin \left (b x +a \right )}\, \sqrt {\cos \left (b x +a \right )}\, \sin \left (b x +a \right )d x \right ) c \] Input:

int((d*cos(b*x+a))^(1/2)*(c*sin(b*x+a))^(3/2),x)
 

Output:

sqrt(d)*sqrt(c)*int(sqrt(sin(a + b*x))*sqrt(cos(a + b*x))*sin(a + b*x),x)* 
c