\(\int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx\) [186]

Optimal result

Integrand size = 23, antiderivative size = 551 \[ \int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}\right )}{2 d}+\frac {\sqrt {b} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right ) \cos (c+d x) \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}{d \left (\sqrt {b}+\sqrt {a+b}+\sqrt {b} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right ) \cos ^2(c+d x)\right )}-\frac {\sqrt [4]{b} (a+b)^{3/4} \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}+\frac {a \left (\sqrt {b}-\sqrt {a+b}\right ) \left (\sqrt {a+b}+\sqrt {b} \cos ^2(c+d x)\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{\left (\sqrt {a+b}+\sqrt {b} \cos ^2(c+d x)\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{4 \sqrt [4]{b} \sqrt [4]{a+b} \left (\sqrt {b}+\sqrt {a+b}\right ) d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}} \] Output:

-1/2*a^(1/2)*arctanh(a^(1/2)*cos(d*x+c)/(a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c) 
^4)^(1/2))/d+b^(1/2)*(1+b^(1/2)/(a+b)^(1/2))*cos(d*x+c)*(a+b-2*b*cos(d*x+c 
)^2+b*cos(d*x+c)^4)^(1/2)/d/(b^(1/2)+(a+b)^(1/2)+b^(1/2)*(1+b^(1/2)/(a+b)^ 
(1/2))*cos(d*x+c)^2)-b^(1/4)*(a+b)^(3/4)*(1+b^(1/2)*cos(d*x+c)^2/(a+b)^(1/ 
2))*((a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)/(a+b)/(1+b^(1/2)*cos(d*x+c)^2/( 
a+b)^(1/2))^2)^(1/2)*EllipticE(sin(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4) 
)),1/2*(2+2*b^(1/2)/(a+b)^(1/2))^(1/2))/d/(a+b-2*b*cos(d*x+c)^2+b*cos(d*x+ 
c)^4)^(1/2)+1/4*a*(b^(1/2)-(a+b)^(1/2))*((a+b)^(1/2)+b^(1/2)*cos(d*x+c)^2) 
*((a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)/((a+b)^(1/2)+b^(1/2)*cos(d*x+c)^2) 
^2)^(1/2)*EllipticPi(sin(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4))),1/4*(b^ 
(1/2)+(a+b)^(1/2))^2/b^(1/2)/(a+b)^(1/2),1/2*(2+2*b^(1/2)/(a+b)^(1/2))^(1/ 
2))/b^(1/4)/(a+b)^(1/4)/(b^(1/2)+(a+b)^(1/2))/d/(a+b-2*b*cos(d*x+c)^2+b*co 
s(d*x+c)^4)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 24.97 (sec) , antiderivative size = 2045, normalized size of antiderivative = 3.71 \[ \int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\text {Result too large to show} \] Input:

Integrate[Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]^4],x]
 

Output:

(b*Cos[c + d*x]*Sqrt[8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)] 
]*Cot[c + d*x]^2*(1 + Cot[c + d*x]^2)^2*Sqrt[(a + b + 2*a*Cot[c + d*x]^2 + 
 a*Cot[c + d*x]^4)/(1 + Cot[c + d*x]^2)^2]*Sqrt[(I*(I*Sqrt[a] + Sqrt[b])*( 
(-I)*Sqrt[a] + Sqrt[b] - I*Sqrt[a]*Cot[c + d*x]^2)*Tan[c + d*x]^2)/(Sqrt[a 
]*Sqrt[b])]*Sqrt[((-I)*((-I)*Sqrt[a] + Sqrt[b])*(I*Sqrt[a] + Sqrt[b] + I*S 
qrt[a]*Cot[c + d*x]^2)*Tan[c + d*x]^2)/(Sqrt[a]*Sqrt[b])]*Sqrt[((a + b)*(a 
 + b + 2*a*Cot[c + d*x]^2 + a*Cot[c + d*x]^4)*Tan[c + d*x]^4)/(a*b)]*(1 + 
Tan[c + d*x]^2)*Sqrt[(b*Tan[c + d*x]^4 + a*(1 + Tan[c + d*x]^2)^2)/(1 + Ta 
n[c + d*x]^2)^2]*(1 + (b*(Sqrt[b]*EllipticE[ArcSin[Sqrt[(I*(a - I*Sqrt[a]* 
Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])]/Sqrt[2] 
], (2*Sqrt[a])/(Sqrt[a] - I*Sqrt[b])] + I*(Sqrt[a] + I*Sqrt[b])*EllipticF[ 
ArcSin[Sqrt[(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^ 
2))/(Sqrt[a]*Sqrt[b])]/Sqrt[2]], (2*Sqrt[a])/(Sqrt[a] - I*Sqrt[b])])*Sqrt[ 
(((-I)*Sqrt[a] + Sqrt[b])*(1 + Tan[c + d*x]^2))/Sqrt[b]]*Sqrt[(I*(a - I*Sq 
rt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])])/ 
((a + b)*Sqrt[((-I)*(a + I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + 
d*x]^2))/(Sqrt[a]*Sqrt[b])]*(Sqrt[b]*Tan[c + d*x]^2 - I*Sqrt[a]*(1 + Tan[c 
 + d*x]^2))) - (a*((-I)*Sqrt[b]*EllipticE[ArcSin[Sqrt[(I*(a - I*Sqrt[a]*Sq 
rt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])]/Sqrt[2]], 
 (2*Sqrt[a])/(Sqrt[a] - I*Sqrt[b])] + (Sqrt[a] + I*Sqrt[b])*EllipticF[A...
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 3694, 1523, 25, 27, 1509, 2222}

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.

\(\displaystyle \int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {a+b \sin (c+d x)^4}}{\sin (c+d x)}dx\)

\(\Big \downarrow \) 3694

\(\displaystyle -\frac {\int \frac {\sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}{1-\cos ^2(c+d x)}d\cos (c+d x)}{d}\)

\(\Big \downarrow \) 1523

\(\displaystyle -\frac {\frac {a \int \frac {\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{\frac {\sqrt {b}}{\sqrt {a+b}}+1}-\frac {\int -\frac {\sqrt {b} \left (-\sqrt {b} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right ) \cos ^2(c+d x)+\sqrt {b}+\sqrt {a+b}\right )}{\sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{\frac {\sqrt {b}}{\sqrt {a+b}}+1}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {a \int \frac {\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{\frac {\sqrt {b}}{\sqrt {a+b}}+1}+\frac {\int \frac {\sqrt {b} \left (-\sqrt {b} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right ) \cos ^2(c+d x)+\sqrt {b}+\sqrt {a+b}\right )}{\sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{\frac {\sqrt {b}}{\sqrt {a+b}}+1}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {a \int \frac {\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{\frac {\sqrt {b}}{\sqrt {a+b}}+1}+\frac {\sqrt {b} \int \frac {-\sqrt {b} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right ) \cos ^2(c+d x)+\sqrt {b}+\sqrt {a+b}}{\sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{\frac {\sqrt {b}}{\sqrt {a+b}}+1}}{d}\)

\(\Big \downarrow \) 1509

\(\displaystyle -\frac {\frac {a \int \frac {\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{\frac {\sqrt {b}}{\sqrt {a+b}}+1}+\frac {\sqrt {b} \left (\frac {\sqrt [4]{a+b} \left (\sqrt {a+b}+\sqrt {b}\right ) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\sqrt [4]{b} \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac {\left (\sqrt {a+b}+\sqrt {b}\right ) \cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )}\right )}{\frac {\sqrt {b}}{\sqrt {a+b}}+1}}{d}\)

\(\Big \downarrow \) 2222

\(\displaystyle -\frac {\frac {a \left (\frac {\sqrt [4]{a+b} \left (1-\frac {\sqrt {b}}{\sqrt {a+b}}\right ) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{4 \sqrt [4]{b} \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}+\frac {\left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}\right )}{2 \sqrt {a}}\right )}{\frac {\sqrt {b}}{\sqrt {a+b}}+1}+\frac {\sqrt {b} \left (\frac {\sqrt [4]{a+b} \left (\sqrt {a+b}+\sqrt {b}\right ) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\sqrt [4]{b} \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac {\left (\sqrt {a+b}+\sqrt {b}\right ) \cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )}\right )}{\frac {\sqrt {b}}{\sqrt {a+b}}+1}}{d}\)

Input:

Int[Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]^4],x]
 

Output:

-(((Sqrt[b]*(-(((Sqrt[b] + Sqrt[a + b])*Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[ 
c + d*x]^2 + b*Cos[c + d*x]^4])/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqr 
t[a + b]))) + ((a + b)^(1/4)*(Sqrt[b] + Sqrt[a + b])*(1 + (Sqrt[b]*Cos[c + 
 d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4) 
/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticE[2*ArcTa 
n[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(b^ 
(1/4)*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])))/(1 + Sqrt[b]/ 
Sqrt[a + b]) + (a*(((1 + Sqrt[b]/Sqrt[a + b])*ArcTanh[(Sqrt[a]*Cos[c + d*x 
])/Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]])/(2*Sqrt[a]) + ((a 
 + b)^(1/4)*(1 - Sqrt[b]/Sqrt[a + b])*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a 
 + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + 
(Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticPi[(Sqrt[b] + Sqrt[a + b 
])^2/(4*Sqrt[b]*Sqrt[a + b]), 2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4 
)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(4*b^(1/4)*Sqrt[a + b - 2*b*Cos[c + d*x] 
^2 + b*Cos[c + d*x]^4])))/(1 + Sqrt[b]/Sqrt[a + 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 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1523

Int[Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(c*d^2 - b*d*e + a*e^2)/(e*(e - d*q))   I 
nt[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] - Simp[1/(e*(e 
 - d*q))   Int[(c*d - b*e + a*e*q - (c*e - a*d*q^3)*x^2)/Sqrt[a + b*x^2 + c 
*x^4], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c 
*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
 

rule 2222

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A 
rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ 
b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + 
 b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell 
ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] 
/; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && 
 EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
 

rule 3042

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

rule 3694

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^ 
(p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f 
Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4)^p, 
 x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 
 1)/2]
 

Maple [F]

\[\int \csc \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )^{4}}d x\]

Input:

int(csc(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x)
 

Output:

int(csc(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x)
 

Fricas [F]

\[ \int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\int { \sqrt {b \sin \left (d x + c\right )^{4} + a} \csc \left (d x + c\right ) \,d x } \] Input:

integrate(csc(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x, algorithm="fricas")
 

Output:

integral(sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*csc(d*x + c), 
 x)
 

Sympy [F]

\[ \int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\int \sqrt {a + b \sin ^{4}{\left (c + d x \right )}} \csc {\left (c + d x \right )}\, dx \] Input:

integrate(csc(d*x+c)*(a+b*sin(d*x+c)**4)**(1/2),x)
 

Output:

Integral(sqrt(a + b*sin(c + d*x)**4)*csc(c + d*x), x)
 

Maxima [F]

\[ \int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\int { \sqrt {b \sin \left (d x + c\right )^{4} + a} \csc \left (d x + c\right ) \,d x } \] Input:

integrate(csc(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(b*sin(d*x + c)^4 + a)*csc(d*x + c), x)
 

Giac [F]

\[ \int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\int { \sqrt {b \sin \left (d x + c\right )^{4} + a} \csc \left (d x + c\right ) \,d x } \] Input:

integrate(csc(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(b*sin(d*x + c)^4 + a)*csc(d*x + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\int \frac {\sqrt {b\,{\sin \left (c+d\,x\right )}^4+a}}{\sin \left (c+d\,x\right )} \,d x \] Input:

int((a + b*sin(c + d*x)^4)^(1/2)/sin(c + d*x),x)
 

Output:

int((a + b*sin(c + d*x)^4)^(1/2)/sin(c + d*x), x)
 

Reduce [F]

\[ \int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\int \sqrt {\sin \left (d x +c \right )^{4} b +a}\, \csc \left (d x +c \right )d x \] Input:

int(csc(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x)
 

Output:

int(sqrt(sin(c + d*x)**4*b + a)*csc(c + d*x),x)