Integrand size = 23, antiderivative size = 521 \[ \int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-a} \cos (c+d x)}{\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}\right )}{2 d}+\frac {\sqrt {b} \cos (c+d x) \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}{\sqrt {a+b} d \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )}-\frac {\sqrt [4]{b} (a+b)^{3/4} \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}-\frac {\sqrt [4]{a+b} \left (\sqrt {b}-\sqrt {a+b}\right )^2 \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{4 \sqrt [4]{b} d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}} \]
1/2*arctan(cos(d*x+c)*(-a)^(1/2)/(a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)^(1/ 2))*(-a)^(1/2)/d+cos(d*x+c)*b^(1/2)*(a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)^ (1/2)/d/(1+cos(d*x+c)^2*b^(1/2)/(a+b)^(1/2))/(a+b)^(1/2)-b^(1/4)*(a+b)^(3/ 4)*(cos(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4)))^2)^(1/2)/cos(2*arctan(b^ (1/4)*cos(d*x+c)/(a+b)^(1/4)))*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))*(1+cos(d*x+c)^2*b^(1/2)/ (a+b)^(1/2))*((a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)/(a+b)/(1+cos(d*x+c)^2* b^(1/2)/(a+b)^(1/2))^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)^(1/4)*(cos(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4)))^2)^(1/2)/ cos(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4)))*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))*(1+cos(d*x+c)^2*b^(1/2)/(a+b)^(1/2 ))*(b^(1/2)-(a+b)^(1/2))^2*((a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)/(a+b)/(1 +cos(d*x+c)^2*b^(1/2)/(a+b)^(1/2))^2)^(1/2)/b^(1/4)/d/(a+b-2*b*cos(d*x+c)^ 2+b*cos(d*x+c)^4)^(1/2)
Result contains complex when optimal does not.
Time = 24.44 (sec) , antiderivative size = 2045, normalized size of antiderivative = 3.93 \[ \int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\text {Result too large to show} \]
(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...
Time = 0.84 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.13, 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 |
\(\displaystyle -\frac {\frac {a \left (\frac {\sqrt [4]{a+b} \left (1-\frac {\sqrt {b}}{\sqrt {a+b}}\right ) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{4 \sqrt [4]{b} \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}+\frac {\left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}\right )}{2 \sqrt {a}}\right )}{\frac {\sqrt {b}}{\sqrt {a+b}}+1}+\frac {\sqrt {b} \left (\frac {\sqrt [4]{a+b} \left (\sqrt {a+b}+\sqrt {b}\right ) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\sqrt [4]{b} \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac {\left (\sqrt {a+b}+\sqrt {b}\right ) \cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )}\right )}{\frac {\sqrt {b}}{\sqrt {a+b}}+1}}{d}\) |
-(((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)
3.3.40.3.1 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[((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]
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]
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)]
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]
\[\int \csc \left (d x +c \right ) \sqrt {a +b \left (\sin ^{4}\left (d x +c \right )\right )}d x\]
\[ \int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\int { \sqrt {b \sin \left (d x + c\right )^{4} + a} \csc \left (d x + c\right ) \,d x } \]
\[ \int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\int \sqrt {a + b \sin ^{4}{\left (c + d x \right )}} \csc {\left (c + d x \right )}\, dx \]
\[ \int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\int { \sqrt {b \sin \left (d x + c\right )^{4} + a} \csc \left (d x + c\right ) \,d x } \]
\[ \int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\int { \sqrt {b \sin \left (d x + c\right )^{4} + a} \csc \left (d x + c\right ) \,d x } \]
Timed out. \[ \int \csc (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\int \frac {\sqrt {b\,{\sin \left (c+d\,x\right )}^4+a}}{\sin \left (c+d\,x\right )} \,d x \]