\(\int \frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}} \, dx\) [263]

Optimal result

Integrand size = 25, antiderivative size = 203 \[ \int \frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}} \, dx=-\frac {\sqrt {c} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} b \sqrt {d}}+\frac {\sqrt {c} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} b \sqrt {d}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)} \left (\sqrt {c}+\sqrt {c} \tan (a+b x)\right )}\right )}{\sqrt {2} b \sqrt {d}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}} \, dx=\frac {2 \cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{4},\frac {7}{4},\sin ^2(a+b x)\right ) \sqrt {c \sin (a+b x)} \tan (a+b x)}{3 b \sqrt {d \cos (a+b x)}} \] Input:

Integrate[Sqrt[c*Sin[a + b*x]]/Sqrt[d*Cos[a + b*x]],x]
 

Output:

(2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[3/4, 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]])
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.40, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {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.

\(\displaystyle \int \frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}dx\)

\(\Big \downarrow \) 3054

\(\displaystyle \frac {2 c d \int \frac {c \tan (a+b x)}{d \left (\tan ^2(a+b x) c^2+c^2\right )}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{b}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {2 c d \left (\frac {\int \frac {\tan (a+b x) c+c}{\tan ^2(a+b x) c^2+c^2}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 d}-\frac {\int \frac {c-c \tan (a+b x)}{\tan ^2(a+b x) c^2+c^2}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 d}\right )}{b}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {2 c d \left (\frac {\frac {\int \frac {1}{\frac {\tan (a+b x) c}{d}+\frac {c}{d}-\frac {\sqrt {2} \sqrt {c \sin (a+b x)} \sqrt {c}}{\sqrt {d} \sqrt {d \cos (a+b x)}}}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 d}+\frac {\int \frac {1}{\frac {\tan (a+b x) c}{d}+\frac {c}{d}+\frac {\sqrt {2} \sqrt {c \sin (a+b x)} \sqrt {c}}{\sqrt {d} \sqrt {d \cos (a+b x)}}}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 d}}{2 d}-\frac {\int \frac {c-c \tan (a+b x)}{\tan ^2(a+b x) c^2+c^2}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 d}\right )}{b}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {2 c d \left (\frac {\frac {\int \frac {1}{-\frac {c \tan (a+b x)}{d}-1}d\left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\int \frac {1}{-\frac {c \tan (a+b x)}{d}-1}d\left (\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 d}-\frac {\int \frac {c-c \tan (a+b x)}{\tan ^2(a+b x) c^2+c^2}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 d}\right )}{b}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {2 c d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 d}-\frac {\int \frac {c-c \tan (a+b x)}{\tan ^2(a+b x) c^2+c^2}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 d}\right )}{b}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {2 c d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 d}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {c}-\frac {2 \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{\sqrt {d} \left (\frac {\tan (a+b x) c}{d}+\frac {c}{d}-\frac {\sqrt {2} \sqrt {c \sin (a+b x)} \sqrt {c}}{\sqrt {d} \sqrt {d \cos (a+b x)}}\right )}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {c}+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{\sqrt {d} \left (\frac {\tan (a+b x) c}{d}+\frac {c}{d}+\frac {\sqrt {2} \sqrt {c \sin (a+b x)} \sqrt {c}}{\sqrt {d} \sqrt {d \cos (a+b x)}}\right )}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}}{2 d}\right )}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 c d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 d}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {c}-\frac {2 \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{\sqrt {d} \left (\frac {\tan (a+b x) c}{d}+\frac {c}{d}-\frac {\sqrt {2} \sqrt {c \sin (a+b x)} \sqrt {c}}{\sqrt {d} \sqrt {d \cos (a+b x)}}\right )}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {c}+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{\sqrt {d} \left (\frac {\tan (a+b x) c}{d}+\frac {c}{d}+\frac {\sqrt {2} \sqrt {c \sin (a+b x)} \sqrt {c}}{\sqrt {d} \sqrt {d \cos (a+b x)}}\right )}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}}{2 d}\right )}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 c d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}+1\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} \sqrt {c} \sqrt {d}}}{2 d}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {c}-\frac {2 \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{\frac {\tan (a+b x) c}{d}+\frac {c}{d}-\frac {\sqrt {2} \sqrt {c \sin (a+b x)} \sqrt {c}}{\sqrt {d} \sqrt {d \cos (a+b x)}}}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 \sqrt {2} \sqrt {c} d}+\frac {\int \frac {\sqrt {c}+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{\frac {\tan (a+b x) c}{d}+\frac {c}{d}+\frac {\sqrt {2} \sqrt {c \sin (a+b x)} \sqrt {c}}{\sqrt {d} \sqrt {d \cos (a+b x)}}}d\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}}{2 \sqrt {c} d}}{2 d}\right )}{b}\)

\(\Big \downarrow \) 1103

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

Input:

Int[Sqrt[c*Sin[a + b*x]]/Sqrt[d*Cos[a + b*x]],x]
 

Output:

(2*c*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]*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)))/b
 

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 3054

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]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(362\) vs. \(2(154)=308\).

Time = 217.15 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.79

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {c \sin \left (b x +a \right )}\, \left (\left (1-\cos \left (b x +a \right )\right )^{2} \csc \left (b x +a \right )^{2}-1\right ) \left (\ln \left (\frac {\left (1-\cos \left (b x +a \right )\right )^{2} \csc \left (b x +a \right )+2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )+2-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )}{1-\cos \left (b x +a \right )}\right )-2 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-\cos \left (b x +a \right )+1}{\cos \left (b x +a \right )-1}\right )-\ln \left (-\frac {-\left (1-\cos \left (b x +a \right )\right )^{2} \csc \left (b x +a \right )+2 \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-2+2 \cos \left (b x +a \right )+\sin \left (b x +a \right )}{1-\cos \left (b x +a \right )}\right )-2 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right )\right )}{8 b \sqrt {d \cos \left (b x +a \right )}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}}\)	\(363\)

Input:

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

Output:

1/8/b*2^(1/2)*(c*sin(b*x+a))^(1/2)*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*(ln(1 
/(1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(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)+2-2*cos(b*x+a)-sin(b*x+a)))-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))-ln(-1/(1-cos(b*x+a))*(-(1-cos(b*x+a))^2*csc(b*x+a)+2*(-2*si 
n(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*sin(b*x+a)-2+2*cos(b*x+a)+sin( 
b*x+a)))-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)))/(d*cos(b*x+a))^(1/2)/(-sin(b*x+a)*cos 
(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (154) = 308\).

Time = 0.13 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.35 \[ \int \frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}} \, dx=-\frac {\sqrt {2} \sqrt {\frac {c}{d}} \arctan \left (\frac {2 \, c \cos \left (b x + a\right )^{3} - 2 \, c \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) + \sqrt {2} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} \sqrt {\frac {c}{d}} - 2 \, c \cos \left (b x + a\right )}{2 \, {\left (c \cos \left (b x + a\right )^{3} + c \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) - c \cos \left (b x + a\right )\right )}}\right ) + \sqrt {2} \sqrt {\frac {c}{d}} \arctan \left (-\frac {2 \, c \cos \left (b x + a\right )^{3} - 2 \, c \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) - \sqrt {2} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} \sqrt {\frac {c}{d}} - 2 \, c \cos \left (b x + a\right )}{2 \, {\left (c \cos \left (b x + a\right )^{3} + c \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) - c \cos \left (b x + a\right )\right )}}\right ) - 2 \, \sqrt {2} \sqrt {\frac {c}{d}} \arctan \left (-\frac {\sqrt {2} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} \sqrt {\frac {c}{d}} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )}}{2 \, c \cos \left (b x + a\right ) \sin \left (b x + a\right )}\right ) + \sqrt {2} \sqrt {\frac {c}{d}} \log \left (2 \, \sqrt {2} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} \sqrt {\frac {c}{d}} {\left (\cos \left (b x + a\right ) + \sin \left (b x + a\right )\right )} + 4 \, c \cos \left (b x + a\right ) \sin \left (b x + a\right ) + c\right ) - \sqrt {2} \sqrt {\frac {c}{d}} \log \left (-2 \, \sqrt {2} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} \sqrt {\frac {c}{d}} {\left (\cos \left (b x + a\right ) + \sin \left (b x + a\right )\right )} + 4 \, c \cos \left (b x + a\right ) \sin \left (b x + a\right ) + c\right )}{8 \, b} \] Input:

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

Output:

-1/8*(sqrt(2)*sqrt(c/d)*arctan(1/2*(2*c*cos(b*x + a)^3 - 2*c*cos(b*x + a)^ 
2*sin(b*x + a) + sqrt(2)*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))*sqrt(c/ 
d) - 2*c*cos(b*x + a))/(c*cos(b*x + a)^3 + c*cos(b*x + a)^2*sin(b*x + a) - 
 c*cos(b*x + a))) + sqrt(2)*sqrt(c/d)*arctan(-1/2*(2*c*cos(b*x + a)^3 - 2* 
c*cos(b*x + a)^2*sin(b*x + a) - sqrt(2)*sqrt(d*cos(b*x + a))*sqrt(c*sin(b* 
x + a))*sqrt(c/d) - 2*c*cos(b*x + a))/(c*cos(b*x + a)^3 + c*cos(b*x + a)^2 
*sin(b*x + a) - c*cos(b*x + a))) - 2*sqrt(2)*sqrt(c/d)*arctan(-1/2*sqrt(2) 
*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))*sqrt(c/d)*(cos(b*x + a) - sin(b 
*x + a))/(c*cos(b*x + a)*sin(b*x + a))) + sqrt(2)*sqrt(c/d)*log(2*sqrt(2)* 
sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))*sqrt(c/d)*(cos(b*x + a) + sin(b* 
x + a)) + 4*c*cos(b*x + a)*sin(b*x + a) + c) - sqrt(2)*sqrt(c/d)*log(-2*sq 
rt(2)*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))*sqrt(c/d)*(cos(b*x + a) + 
sin(b*x + a)) + 4*c*cos(b*x + a)*sin(b*x + a) + c))/b
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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