Integrand size = 29, antiderivative size = 332 \[ \int \frac {\csc ^3(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 b^7 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{5/2} d}-\frac {5 \text {arctanh}(\cos (c+d x))}{2 a d}-\frac {b^2 \text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {b \cot (c+d x)}{a^2 d}+\frac {5 \sec (c+d x)}{2 a d}+\frac {b^2 \sec (c+d x)}{a^3 d}+\frac {5 \sec ^3(c+d x)}{6 a d}+\frac {b^2 \sec ^3(c+d x)}{3 a^3 d}-\frac {\csc ^2(c+d x) \sec ^3(c+d x)}{2 a d}+\frac {b^3 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^3 \left (a^2-b^2\right ) d}-\frac {b^3 \sec (c+d x) \left (3 b^3+a \left (2 a^2-5 b^2\right ) \sin (c+d x)\right )}{3 a^3 \left (a^2-b^2\right )^2 d}-\frac {2 b \tan (c+d x)}{a^2 d}-\frac {b \tan ^3(c+d x)}{3 a^2 d} \]
-2*b^7*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^3/(a^2-b^2)^(5/2 )/d-5/2*arctanh(cos(d*x+c))/a/d-b^2*arctanh(cos(d*x+c))/a^3/d+b*cot(d*x+c) /a^2/d+5/2*sec(d*x+c)/a/d+b^2*sec(d*x+c)/a^3/d+5/6*sec(d*x+c)^3/a/d+1/3*b^ 2*sec(d*x+c)^3/a^3/d-1/2*csc(d*x+c)^2*sec(d*x+c)^3/a/d+1/3*b^3*sec(d*x+c)^ 3*(b-a*sin(d*x+c))/a^3/(a^2-b^2)/d-1/3*b^3*sec(d*x+c)*(3*b^3+a*(2*a^2-5*b^ 2)*sin(d*x+c))/a^3/(a^2-b^2)^2/d-2*b*tan(d*x+c)/a^2/d-1/3*b*tan(d*x+c)^3/a ^2/d
Leaf count is larger than twice the leaf count of optimal. \(947\) vs. \(2(332)=664\).
Time = 6.82 (sec) , antiderivative size = 947, normalized size of antiderivative = 2.85 \[ \int \frac {\csc ^3(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx=16 \left (\frac {a \left (13 a^2-19 b^2\right ) \csc (c+d x) (a+b \sin (c+d x))}{96 \left (a^2-b^2\right )^2 d (b+a \csc (c+d x))}-\frac {b^7 \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right ) \csc (c+d x) (a+b \sin (c+d x))}{8 a^3 \left (a^2-b^2\right )^{5/2} d (b+a \csc (c+d x))}+\frac {b \cot \left (\frac {1}{2} (c+d x)\right ) \csc (c+d x) (a+b \sin (c+d x))}{32 a^2 d (b+a \csc (c+d x))}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right ) \csc (c+d x) (a+b \sin (c+d x))}{128 a d (b+a \csc (c+d x))}+\frac {\left (-5 a^2-2 b^2\right ) \csc (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sin (c+d x))}{32 a^3 d (b+a \csc (c+d x))}+\frac {\left (5 a^2+2 b^2\right ) \csc (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sin (c+d x))}{32 a^3 d (b+a \csc (c+d x))}+\frac {\csc (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+b \sin (c+d x))}{128 a d (b+a \csc (c+d x))}+\frac {\csc (c+d x) (a+b \sin (c+d x))}{192 (a+b) d (b+a \csc (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {\csc (c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a+b \sin (c+d x))}{96 (a+b) d (b+a \csc (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {\csc (c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a+b \sin (c+d x))}{96 (a-b) d (b+a \csc (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {\csc (c+d x) (a+b \sin (c+d x))}{192 (a-b) d (b+a \csc (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {\csc (c+d x) \left (-13 a \sin \left (\frac {1}{2} (c+d x)\right )+16 b \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sin (c+d x))}{96 (a-b)^2 d (b+a \csc (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {\csc (c+d x) \left (13 a \sin \left (\frac {1}{2} (c+d x)\right )+16 b \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sin (c+d x))}{96 (a+b)^2 d (b+a \csc (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {b \csc (c+d x) (a+b \sin (c+d x)) \tan \left (\frac {1}{2} (c+d x)\right )}{32 a^2 d (b+a \csc (c+d x))}\right ) \]
16*((a*(13*a^2 - 19*b^2)*Csc[c + d*x]*(a + b*Sin[c + d*x]))/(96*(a^2 - b^2 )^2*d*(b + a*Csc[c + d*x])) - (b^7*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d* x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]]*Csc[c + d*x]*(a + b*Sin[c + d*x]))/(8*a^3*(a^2 - b^2)^(5/2)*d*(b + a*Csc[c + d*x])) + (b*Cot[(c + d*x) /2]*Csc[c + d*x]*(a + b*Sin[c + d*x]))/(32*a^2*d*(b + a*Csc[c + d*x])) - ( Csc[(c + d*x)/2]^2*Csc[c + d*x]*(a + b*Sin[c + d*x]))/(128*a*d*(b + a*Csc[ c + d*x])) + ((-5*a^2 - 2*b^2)*Csc[c + d*x]*Log[Cos[(c + d*x)/2]]*(a + b*S in[c + d*x]))/(32*a^3*d*(b + a*Csc[c + d*x])) + ((5*a^2 + 2*b^2)*Csc[c + d *x]*Log[Sin[(c + d*x)/2]]*(a + b*Sin[c + d*x]))/(32*a^3*d*(b + a*Csc[c + d *x])) + (Csc[c + d*x]*Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x]))/(128*a*d*(b + a*Csc[c + d*x])) + (Csc[c + d*x]*(a + b*Sin[c + d*x]))/(192*(a + b)*d*( b + a*Csc[c + d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (Csc[c + d* x]*Sin[(c + d*x)/2]*(a + b*Sin[c + d*x]))/(96*(a + b)*d*(b + a*Csc[c + d*x ])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) - (Csc[c + d*x]*Sin[(c + d*x)/ 2]*(a + b*Sin[c + d*x]))/(96*(a - b)*d*(b + a*Csc[c + d*x])*(Cos[(c + d*x) /2] + Sin[(c + d*x)/2])^3) + (Csc[c + d*x]*(a + b*Sin[c + d*x]))/(192*(a - b)*d*(b + a*Csc[c + d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (Csc [c + d*x]*(-13*a*Sin[(c + d*x)/2] + 16*b*Sin[(c + d*x)/2])*(a + b*Sin[c + d*x]))/(96*(a - b)^2*d*(b + a*Csc[c + d*x])*(Cos[(c + d*x)/2] + Sin[(c + d *x)/2])) + (Csc[c + d*x]*(13*a*Sin[(c + d*x)/2] + 16*b*Sin[(c + d*x)/2]...
Time = 0.75 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3377, 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.
| \(\displaystyle \int \frac {\csc ^3(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^3 \cos (c+d x)^4 (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3377 |
| \(\displaystyle \int \left (-\frac {b^3 \sec ^4(c+d x)}{a^3 (a+b \sin (c+d x))}+\frac {b^2 \csc (c+d x) \sec ^4(c+d x)}{a^3}-\frac {b \csc ^2(c+d x) \sec ^4(c+d x)}{a^2}+\frac {\csc ^3(c+d x) \sec ^4(c+d x)}{a}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^2 \text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {b^2 \sec ^3(c+d x)}{3 a^3 d}+\frac {b^2 \sec (c+d x)}{a^3 d}-\frac {b \tan ^3(c+d x)}{3 a^2 d}-\frac {2 b \tan (c+d x)}{a^2 d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {2 b^7 \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 d \left (a^2-b^2\right )^{5/2}}+\frac {b^3 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^3 d \left (a^2-b^2\right )}-\frac {b^3 \sec (c+d x) \left (a \left (2 a^2-5 b^2\right ) \sin (c+d x)+3 b^3\right )}{3 a^3 d \left (a^2-b^2\right )^2}-\frac {5 \text {arctanh}(\cos (c+d x))}{2 a d}+\frac {5 \sec ^3(c+d x)}{6 a d}+\frac {5 \sec (c+d x)}{2 a d}-\frac {\csc ^2(c+d x) \sec ^3(c+d x)}{2 a d}\) |
(-2*b^7*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2) ^(5/2)*d) - (5*ArcTanh[Cos[c + d*x]])/(2*a*d) - (b^2*ArcTanh[Cos[c + d*x]] )/(a^3*d) + (b*Cot[c + d*x])/(a^2*d) + (5*Sec[c + d*x])/(2*a*d) + (b^2*Sec [c + d*x])/(a^3*d) + (5*Sec[c + d*x]^3)/(6*a*d) + (b^2*Sec[c + d*x]^3)/(3* a^3*d) - (Csc[c + d*x]^2*Sec[c + d*x]^3)/(2*a*d) + (b^3*Sec[c + d*x]^3*(b - a*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)*d) - (b^3*Sec[c + d*x]*(3*b^3 + a*(2 *a^2 - 5*b^2)*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*d) - (2*b*Tan[c + d*x])/ (a^2*d) - (b*Tan[c + d*x]^3)/(3*a^2*d)
3.14.58.3.1 Defintions of rubi rules used
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*sin[(e_.) + (f_.)*(x_)]^(n_))/((a _) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, sin[e + f*x]^n/(a + b*sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[n] && (LtQ[n, 0] || IGtQ[p + 1/ 2, 0])
Time = 1.19 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.90
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {2 b^{7} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{3} \left (a +b \right )^{2} \left (a -b \right )^{2} \sqrt {a^{2}-b^{2}}}-\frac {10 a +12 b}{4 \left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{3 \left (a +b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-10 a +12 b}{4 \left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {1}{3 \left (a -b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 \left (a -b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (10 a^{2}+4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(298\) |
| default | \(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {2 b^{7} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{3} \left (a +b \right )^{2} \left (a -b \right )^{2} \sqrt {a^{2}-b^{2}}}-\frac {10 a +12 b}{4 \left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{3 \left (a +b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-10 a +12 b}{4 \left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {1}{3 \left (a -b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 \left (a -b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (10 a^{2}+4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(298\) |
| risch | \(\frac {i \left (-44 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-20 i a^{5} {\mathrm e}^{7 i \left (d x +c \right )}-20 i a^{5} {\mathrm e}^{3 i \left (d x +c \right )}+22 i a^{5} {\mathrm e}^{5 i \left (d x +c \right )}-15 i a^{5} {\mathrm e}^{9 i \left (d x +c \right )}-16 \,{\mathrm e}^{2 i \left (d x +c \right )} a^{4} b +28 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+32 a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}-12 a^{2} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+28 a^{2} b^{3}-6 b^{5}-16 a^{4} b -15 i a^{5} {\mathrm e}^{i \left (d x +c \right )}+12 b^{5} {\mathrm e}^{6 i \left (d x +c \right )}-12 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+6 b^{5} {\mathrm e}^{8 i \left (d x +c \right )}-3 i a \,b^{4} {\mathrm e}^{i \left (d x +c \right )}+24 i a^{3} b^{2} {\mathrm e}^{i \left (d x +c \right )}-3 i a \,b^{4} {\mathrm e}^{9 i \left (d x +c \right )}-18 i a \,b^{4} {\mathrm e}^{5 i \left (d x +c \right )}-12 i a \,b^{4} {\mathrm e}^{3 i \left (d x +c \right )}+24 i a^{3} b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+32 i a^{3} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-12 i a \,b^{4} {\mathrm e}^{7 i \left (d x +c \right )}-16 i a^{3} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+32 i a^{3} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}\right )}{3 \left (-a^{2}+b^{2}\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} a^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {i b^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{3}}+\frac {i b^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{3}}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d a}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{a^{3} d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d a}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{a^{3} d}\) | \(691\) |
1/d*(1/4/a^2*(1/2*tan(1/2*d*x+1/2*c)^2*a-2*b*tan(1/2*d*x+1/2*c))-2/a^3/(a+ b)^2/(a-b)^2*b^7/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/( a^2-b^2)^(1/2))-1/4/(a+b)^2*(10*a+12*b)/(tan(1/2*d*x+1/2*c)-1)-1/3/(a+b)/( tan(1/2*d*x+1/2*c)-1)^3-1/2/(a+b)/(tan(1/2*d*x+1/2*c)-1)^2-1/4/(a-b)^2*(-1 0*a+12*b)/(tan(1/2*d*x+1/2*c)+1)+1/3/(a-b)/(tan(1/2*d*x+1/2*c)+1)^3-1/2/(a -b)/(tan(1/2*d*x+1/2*c)+1)^2-1/8/a/tan(1/2*d*x+1/2*c)^2+1/4/a^3*(10*a^2+4* b^2)*ln(tan(1/2*d*x+1/2*c))+1/2*b/a^2/tan(1/2*d*x+1/2*c))
Time = 1.42 (sec) , antiderivative size = 1182, normalized size of antiderivative = 3.56 \[ \int \frac {\csc ^3(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
[-1/12*(4*a^8 - 8*a^6*b^2 + 4*a^4*b^4 - 6*(5*a^8 - 13*a^6*b^2 + 9*a^4*b^4 - a^2*b^6)*cos(d*x + c)^4 + 4*(5*a^8 - 13*a^6*b^2 + 8*a^4*b^4)*cos(d*x + c )^2 + 6*(b^7*cos(d*x + c)^5 - b^7*cos(d*x + c)^3)*sqrt(-a^2 + b^2)*log(-(( 2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d* x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^ 2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) + 3*((5*a^8 - 13*a^6*b^2 + 9*a^4*b^4 + a^2*b^6 - 2*b^8)*cos(d*x + c)^5 - (5*a^8 - 13*a^6*b^2 + 9*a^4*b^4 + a^2* b^6 - 2*b^8)*cos(d*x + c)^3)*log(1/2*cos(d*x + c) + 1/2) - 3*((5*a^8 - 13* a^6*b^2 + 9*a^4*b^4 + a^2*b^6 - 2*b^8)*cos(d*x + c)^5 - (5*a^8 - 13*a^6*b^ 2 + 9*a^4*b^4 + a^2*b^6 - 2*b^8)*cos(d*x + c)^3)*log(-1/2*cos(d*x + c) + 1 /2) - 4*(a^7*b - 2*a^5*b^3 + a^3*b^5 - (8*a^7*b - 22*a^5*b^3 + 17*a^3*b^5 - 3*a*b^7)*cos(d*x + c)^4 + (4*a^7*b - 11*a^5*b^3 + 7*a^3*b^5)*cos(d*x + c )^2)*sin(d*x + c))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c) ^5 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c)^3), -1/12*(4*a ^8 - 8*a^6*b^2 + 4*a^4*b^4 - 6*(5*a^8 - 13*a^6*b^2 + 9*a^4*b^4 - a^2*b^6)* cos(d*x + c)^4 + 4*(5*a^8 - 13*a^6*b^2 + 8*a^4*b^4)*cos(d*x + c)^2 - 12*(b ^7*cos(d*x + c)^5 - b^7*cos(d*x + c)^3)*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) + 3*((5*a^8 - 13*a^6*b^2 + 9*a^ 4*b^4 + a^2*b^6 - 2*b^8)*cos(d*x + c)^5 - (5*a^8 - 13*a^6*b^2 + 9*a^4*b^4 + a^2*b^6 - 2*b^8)*cos(d*x + c)^3)*log(1/2*cos(d*x + c) + 1/2) - 3*((5*...
Timed out. \[ \int \frac {\csc ^3(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\csc ^3(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.46 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.26 \[ \int \frac {\csc ^3(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {48 \, {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} b^{7}}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {a^{2} - b^{2}}} - \frac {3 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{2}} - \frac {12 \, {\left (5 \, a^{2} + 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {16 \, {\left (6 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 14 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 18 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7 \, a^{3} + 10 \, a b^{2}\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}} + \frac {3 \, {\left (30 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{24 \, d} \]
-1/24*(48*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))*b^7/((a^7 - 2*a^5*b^2 + a^3*b^4)*sqrt(a^2 - b^2)) - 3*(a*tan(1/2*d*x + 1/2*c)^2 - 4*b*tan(1/2*d*x + 1/2*c))/a^2 - 1 2*(5*a^2 + 2*b^2)*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - 16*(6*a^2*b*tan(1/2 *d*x + 1/2*c)^5 - 9*b^3*tan(1/2*d*x + 1/2*c)^5 - 9*a^3*tan(1/2*d*x + 1/2*c )^4 + 12*a*b^2*tan(1/2*d*x + 1/2*c)^4 - 8*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 1 4*b^3*tan(1/2*d*x + 1/2*c)^3 + 12*a^3*tan(1/2*d*x + 1/2*c)^2 - 18*a*b^2*ta n(1/2*d*x + 1/2*c)^2 + 6*a^2*b*tan(1/2*d*x + 1/2*c) - 9*b^3*tan(1/2*d*x + 1/2*c) - 7*a^3 + 10*a*b^2)/((a^4 - 2*a^2*b^2 + b^4)*(tan(1/2*d*x + 1/2*c)^ 2 - 1)^3) + 3*(30*a^2*tan(1/2*d*x + 1/2*c)^2 + 12*b^2*tan(1/2*d*x + 1/2*c) ^2 - 4*a*b*tan(1/2*d*x + 1/2*c) + a^2)/(a^3*tan(1/2*d*x + 1/2*c)^2))/d
Time = 18.37 (sec) , antiderivative size = 5035, normalized size of antiderivative = 15.17 \[ \int \frac {\csc ^3(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
((b^9*sin(c + d*x))/24 + (41*b^9*sin(3*c + 3*d*x))/48 + (23*b^9*sin(5*c + 5*d*x))/48)/(d*sin(c + d*x)^2*((3*cos(c + d*x))/4 + cos(3*c + 3*d*x)/4)*(a ^4 + b^4 - 2*a^2*b^2)*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) - (a*((19*b^8)/ 48 - (5*b^8*cos(c + d*x))/12 + (17*b^8*cos(2*c + 2*d*x))/12 + (5*b^8*cos(3 *c + 3*d*x))/24 + (11*b^8*cos(4*c + 4*d*x))/16 + (5*b^8*cos(5*c + 5*d*x))/ 24 - (5*b^8*cos(c + d*x)*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/16 + (5*b^8*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x))/32 + ( 5*b^8*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(5*c + 5*d*x))/32))/(d *sin(c + d*x)^2*((3*cos(c + d*x))/4 + cos(3*c + 3*d*x)/4)*(a^4 + b^4 - 2*a ^2*b^2)*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (a^8*((b*sin(3*c + 3*d*x))/ 6 - (b*sin(c + d*x))/3 + (b*sin(5*c + 5*d*x))/6))/(d*sin(c + d*x)^2*((3*co s(c + d*x))/4 + cos(3*c + 3*d*x)/4)*(a^4 + b^4 - 2*a^2*b^2)*(a^6 - b^6 + 3 *a^2*b^4 - 3*a^4*b^2)) + ((3*b^10)/16 + (b^10*cos(2*c + 2*d*x))/4 + (b^10* cos(4*c + 4*d*x))/16 + (5*b^10*cos(c + d*x)*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/16 - (5*b^10*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos( 3*c + 3*d*x))/32 - (5*b^10*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos( 5*c + 5*d*x))/32)/(a*d*sin(c + d*x)^2*((3*cos(c + d*x))/4 + cos(3*c + 3*d* x)/4)*(a^4 + b^4 - 2*a^2*b^2)*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) - ((b^1 1*sin(c + d*x))/8 + (3*b^11*sin(3*c + 3*d*x))/16 + (b^11*sin(5*c + 5*d*x)) /16)/(a^2*d*sin(c + d*x)^2*((3*cos(c + d*x))/4 + cos(3*c + 3*d*x)/4)*(a...