Integrand size = 22, antiderivative size = 136 \[ \int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\text {arctanh}(\cos (c+d x))}{a d}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a \sqrt {\sqrt {a}+\sqrt {b}} d} \]
-arctanh(cos(d*x+c))/a/d-1/2*b^(1/4)*arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^ (1/2))^(1/2))/a/d/(a^(1/2)-b^(1/2))^(1/2)+1/2*b^(1/4)*arctanh(b^(1/4)*cos( d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/a/d/(a^(1/2)+b^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.69 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.34 \[ \int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {8 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-8 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+i b \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+6 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-3 i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-6 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+3 i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{8 a d} \]
-1/8*(8*Log[Cos[(c + d*x)/2]] - 8*Log[Sin[(c + d*x)/2]] + I*b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 6*ArcTa n[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 6*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + (3*I)*Lo g[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1 ^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a*d)
Time = 0.34 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3042, 3694, 1484, 2009}
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.
| \(\displaystyle \int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x) \left (a-b \sin (c+d x)^4\right )}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {1}{\left (1-\cos ^2(c+d x)\right ) \left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 1484 |
| \(\displaystyle -\frac {\int \left (\frac {b-b \cos ^2(c+d x)}{a \left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )}-\frac {1}{a \left (\cos ^2(c+d x)-1\right )}\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\text {arctanh}(\cos (c+d x))}{a}}{d}\) |
-(((b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a*S qrt[Sqrt[a] - Sqrt[b]]) + ArcTanh[Cos[c + d*x]]/a - (b^(1/4)*ArcTanh[(b^(1 /4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a*Sqrt[Sqrt[a] + Sqrt[b]])) /d)
3.2.100.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symb ol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; Fre eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]
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]
Time = 0.94 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.88
| method | result | size |
| derivativedivides | \(\frac {-\frac {b^{2} \left (\frac {\arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 b \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 b \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{a}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2 a}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2 a}}{d}\) | \(119\) |
| default | \(\frac {-\frac {b^{2} \left (\frac {\arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 b \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 b \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{a}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2 a}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2 a}}{d}\) | \(119\) |
| risch | \(\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d a}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d a}+2 i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4096 a^{5} d^{4}-4096 a^{4} b \,d^{4}\right ) \textit {\_Z}^{4}-128 a^{2} b \,d^{2} \textit {\_Z}^{2}-b \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {1024 i d^{3} a^{4}}{b}+1024 i a^{3} d^{3}\right ) \textit {\_R}^{3}+32 i a d \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}+1\right )\right )\) | \(143\) |
1/d*(-1/a*b^2*(1/2/b/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b) ^(1/2)-b)*b)^(1/2))-1/2/b/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/( ((a*b)^(1/2)+b)*b)^(1/2)))+1/2/a*ln(cos(d*x+c)-1)-1/2/a*ln(1+cos(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 773 vs. \(2 (100) = 200\).
Time = 0.37 (sec) , antiderivative size = 773, normalized size of antiderivative = 5.68 \[ \int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {a d \sqrt {-\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} \log \left (b \cos \left (d x + c\right ) - {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - a b d\right )} \sqrt {-\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b}{{\left (a^{3} - a^{2} b\right )} d^{2}}}\right ) - a d \sqrt {\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} \log \left (b \cos \left (d x + c\right ) - {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + a b d\right )} \sqrt {\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - b}{{\left (a^{3} - a^{2} b\right )} d^{2}}}\right ) - a d \sqrt {-\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} \log \left (-b \cos \left (d x + c\right ) - {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - a b d\right )} \sqrt {-\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b}{{\left (a^{3} - a^{2} b\right )} d^{2}}}\right ) + a d \sqrt {\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} \log \left (-b \cos \left (d x + c\right ) - {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + a b d\right )} \sqrt {\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - b}{{\left (a^{3} - a^{2} b\right )} d^{2}}}\right ) - 2 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, a d} \]
1/4*(a*d*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))*log(b*cos(d*x + c) - ((a^4 - a^3*b)*d^3*sqrt(b/( (a^5 - 2*a^4*b + a^3*b^2)*d^4)) - a*b*d)*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b/( (a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))) - a*d*sqrt(((a^ 3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - b)/((a^3 - a^2*b) *d^2))*log(b*cos(d*x + c) - ((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*a^4*b + a^ 3*b^2)*d^4)) + a*b*d)*sqrt(((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3 *b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2))) - a*d*sqrt(-((a^3 - a^2*b)*d^2*sqrt (b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))*log(-b*cos(d *x + c) - ((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - a*b *d)*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b)/ ((a^3 - a^2*b)*d^2))) + a*d*sqrt(((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2))*log(-b*cos(d*x + c) - ((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + a*b*d)*sqrt(((a^3 - a ^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2) )) - 2*log(1/2*cos(d*x + c) + 1/2) + 2*log(-1/2*cos(d*x + c) + 1/2))/(a*d)
\[ \int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int \frac {\csc {\left (c + d x \right )}}{a - b \sin ^{4}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\csc \left (d x + c\right )}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]
-1/2*(2*a*d*integrate(-2*(12*b^2*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) - 4*b^2 *cos(d*x + c)*sin(2*d*x + 2*c) + 4*b^2*cos(2*d*x + 2*c)*sin(d*x + c) - b^2 *sin(d*x + c) + (b^2*sin(7*d*x + 7*c) - 3*b^2*sin(5*d*x + 5*c) + 3*b^2*sin (3*d*x + 3*c) - b^2*sin(d*x + c))*cos(8*d*x + 8*c) + 2*(2*b^2*sin(6*d*x + 6*c) + 2*b^2*sin(2*d*x + 2*c) + (8*a*b - 3*b^2)*sin(4*d*x + 4*c))*cos(7*d* x + 7*c) + 4*(3*b^2*sin(5*d*x + 5*c) - 3*b^2*sin(3*d*x + 3*c) + b^2*sin(d* x + c))*cos(6*d*x + 6*c) - 6*(2*b^2*sin(2*d*x + 2*c) + (8*a*b - 3*b^2)*sin (4*d*x + 4*c))*cos(5*d*x + 5*c) - 2*(3*(8*a*b - 3*b^2)*sin(3*d*x + 3*c) - (8*a*b - 3*b^2)*sin(d*x + c))*cos(4*d*x + 4*c) - (b^2*cos(7*d*x + 7*c) - 3 *b^2*cos(5*d*x + 5*c) + 3*b^2*cos(3*d*x + 3*c) - b^2*cos(d*x + c))*sin(8*d *x + 8*c) - (4*b^2*cos(6*d*x + 6*c) + 4*b^2*cos(2*d*x + 2*c) - b^2 + 2*(8* a*b - 3*b^2)*cos(4*d*x + 4*c))*sin(7*d*x + 7*c) - 4*(3*b^2*cos(5*d*x + 5*c ) - 3*b^2*cos(3*d*x + 3*c) + b^2*cos(d*x + c))*sin(6*d*x + 6*c) + 3*(4*b^2 *cos(2*d*x + 2*c) - b^2 + 2*(8*a*b - 3*b^2)*cos(4*d*x + 4*c))*sin(5*d*x + 5*c) + 2*(3*(8*a*b - 3*b^2)*cos(3*d*x + 3*c) - (8*a*b - 3*b^2)*cos(d*x + c ))*sin(4*d*x + 4*c) - 3*(4*b^2*cos(2*d*x + 2*c) - b^2)*sin(3*d*x + 3*c))/( a*b^2*cos(8*d*x + 8*c)^2 + 16*a*b^2*cos(6*d*x + 6*c)^2 + 16*a*b^2*cos(2*d* x + 2*c)^2 + a*b^2*sin(8*d*x + 8*c)^2 + 16*a*b^2*sin(6*d*x + 6*c)^2 + 16*a *b^2*sin(2*d*x + 2*c)^2 - 8*a*b^2*cos(2*d*x + 2*c) + a*b^2 + 4*(64*a^3 - 4 8*a^2*b + 9*a*b^2)*cos(4*d*x + 4*c)^2 + 4*(64*a^3 - 48*a^2*b + 9*a*b^2)...
\[ \int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\csc \left (d x + c\right )}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]
Time = 15.64 (sec) , antiderivative size = 2031, normalized size of antiderivative = 14.93 \[ \int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]
- (atan(((((((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)*(256*a^4*b^ 4 - 192*a^3*b^5 + cos(c + d*x)*(768*a^4*b^5 - 512*a^5*b^4)*((a^2*b + (a^5* b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)) - 144*a^2*b^5*cos(c + d*x))*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) + 12*a*b^5)*((a^2*b + (a^5*b)^(1 /2))/(16*(a^4*b - a^5)))^(1/2) + 6*b^5*cos(c + d*x))*((a^2*b + (a^5*b)^(1/ 2))/(16*(a^4*b - a^5)))^(1/2)*1i + (((((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)*(192*a^3*b^5 - 256*a^4*b^4 + cos(c + d*x)*(768*a^4*b^5 - 51 2*a^5*b^4)*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)) - 144*a^2*b ^5*cos(c + d*x))*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) - 12*a *b^5)*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) + 6*b^5*cos(c + d *x))*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)*1i)/((((((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)*(256*a^4*b^4 - 192*a^3*b^5 + cos( c + d*x)*(768*a^4*b^5 - 512*a^5*b^4)*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)) - 144*a^2*b^5*cos(c + d*x))*((a^2*b + (a^5*b)^(1/2))/(16*(a ^4*b - a^5)))^(1/2) + 12*a*b^5)*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5) ))^(1/2) + 6*b^5*cos(c + d*x))*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)) )^(1/2) - (((((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)*(192*a^3*b ^5 - 256*a^4*b^4 + cos(c + d*x)*(768*a^4*b^5 - 512*a^5*b^4)*((a^2*b + (a^5 *b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)) - 144*a^2*b^5*cos(c + d*x))*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) - 12*a*b^5)*((a^2*b + (a^5*b...