Integrand size = 29, antiderivative size = 126 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\text {arctanh}(\cos (c+d x))}{a d}-\frac {\cot (c+d x)}{a d}-\frac {\sec (c+d x)}{a d}-\frac {\sec ^3(c+d x)}{3 a d}-\frac {\sec ^5(c+d x)}{5 a d}+\frac {3 \tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{a d}+\frac {\tan ^5(c+d x)}{5 a d} \] Output:
arctanh(cos(d*x+c))/a/d-cot(d*x+c)/a/d-sec(d*x+c)/a/d-1/3*sec(d*x+c)^3/a/d -1/5*sec(d*x+c)^5/a/d+3*tan(d*x+c)/a/d+tan(d*x+c)^3/a/d+1/5*tan(d*x+c)^5/a /d
Leaf count is larger than twice the leaf count of optimal. \(341\) vs. \(2(126)=252\).
Time = 1.88 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.71 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (176+1216 \cos (2 (c+d x))+149 \cos (3 (c+d x))+528 \cos (4 (c+d x))+149 \cos (5 (c+d x))+120 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+120 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-120 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-120 \cos (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\cos (c+d x) \left (-298-240 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+240 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+352 \sin (c+d x)-596 \sin (2 (c+d x))-480 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (2 (c+d x))+480 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (2 (c+d x))+864 \sin (3 (c+d x))-298 \sin (4 (c+d x))-240 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (4 (c+d x))+240 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (4 (c+d x))+384 \sin (5 (c+d x))\right )}{3840 a d (1+\sin (c+d x))} \] Input:
Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^4)/(a + a*Sin[c + d*x]),x]
Output:
-1/3840*(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*Sec[c + d*x]^3*(176 + 1216*Cos[ 2*(c + d*x)] + 149*Cos[3*(c + d*x)] + 528*Cos[4*(c + d*x)] + 149*Cos[5*(c + d*x)] + 120*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 120*Cos[5*(c + d*x) ]*Log[Cos[(c + d*x)/2]] - 120*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 120 *Cos[5*(c + d*x)]*Log[Sin[(c + d*x)/2]] + Cos[c + d*x]*(-298 - 240*Log[Cos [(c + d*x)/2]] + 240*Log[Sin[(c + d*x)/2]]) + 352*Sin[c + d*x] - 596*Sin[2 *(c + d*x)] - 480*Log[Cos[(c + d*x)/2]]*Sin[2*(c + d*x)] + 480*Log[Sin[(c + d*x)/2]]*Sin[2*(c + d*x)] + 864*Sin[3*(c + d*x)] - 298*Sin[4*(c + d*x)] - 240*Log[Cos[(c + d*x)/2]]*Sin[4*(c + d*x)] + 240*Log[Sin[(c + d*x)/2]]*S in[4*(c + d*x)] + 384*Sin[5*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))
Time = 0.48 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.74, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {3042, 3318, 3042, 3100, 244, 2009, 3102, 25, 254, 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.
| \(\displaystyle \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^2 \cos (c+d x)^4 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \csc ^2(c+d x) \sec ^6(c+d x)dx}{a}-\frac {\int \csc (c+d x) \sec ^6(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \csc (c+d x)^2 \sec (c+d x)^6dx}{a}-\frac {\int \csc (c+d x) \sec (c+d x)^6dx}{a}\) |
| \(\Big \downarrow \) 3100 |
| \(\displaystyle \frac {\int \cot ^2(c+d x) \left (\tan ^2(c+d x)+1\right )^3d\tan (c+d x)}{a d}-\frac {\int \csc (c+d x) \sec (c+d x)^6dx}{a}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {\int \left (\tan ^4(c+d x)+3 \tan ^2(c+d x)+\cot ^2(c+d x)+3\right )d\tan (c+d x)}{a d}-\frac {\int \csc (c+d x) \sec (c+d x)^6dx}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{5} \tan ^5(c+d x)+\tan ^3(c+d x)+3 \tan (c+d x)-\cot (c+d x)}{a d}-\frac {\int \csc (c+d x) \sec (c+d x)^6dx}{a}\) |
| \(\Big \downarrow \) 3102 |
| \(\displaystyle \frac {\frac {1}{5} \tan ^5(c+d x)+\tan ^3(c+d x)+3 \tan (c+d x)-\cot (c+d x)}{a d}-\frac {\int -\frac {\sec ^6(c+d x)}{1-\sec ^2(c+d x)}d\sec (c+d x)}{a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\sec ^6(c+d x)}{1-\sec ^2(c+d x)}d\sec (c+d x)}{a d}+\frac {\frac {1}{5} \tan ^5(c+d x)+\tan ^3(c+d x)+3 \tan (c+d x)-\cot (c+d x)}{a d}\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle \frac {\int \left (-\sec ^4(c+d x)-\sec ^2(c+d x)+\frac {1}{1-\sec ^2(c+d x)}-1\right )d\sec (c+d x)}{a d}+\frac {\frac {1}{5} \tan ^5(c+d x)+\tan ^3(c+d x)+3 \tan (c+d x)-\cot (c+d x)}{a d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{5} \tan ^5(c+d x)+\tan ^3(c+d x)+3 \tan (c+d x)-\cot (c+d x)}{a d}-\frac {-\text {arctanh}(\sec (c+d x))+\frac {1}{5} \sec ^5(c+d x)+\frac {1}{3} \sec ^3(c+d x)+\sec (c+d x)}{a d}\) |
Input:
Int[(Csc[c + d*x]^2*Sec[c + d*x]^4)/(a + a*Sin[c + d*x]),x]
Output:
-((-ArcTanh[Sec[c + d*x]] + Sec[c + d*x] + Sec[c + d*x]^3/3 + Sec[c + d*x] ^5/5)/(a*d)) + (-Cot[c + d*x] + 3*Tan[c + d*x] + Tan[c + d*x]^3 + Tan[c + d*x]^5/5)/(a*d)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Simp[1/f Subst[Int[(1 + x^2)^((m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]] , x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]
Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_S ymbol] :> Simp[1/(f*a^n) Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/ 2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1 )/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Time = 1.11 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.30
| method | result | size |
| derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {4}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {14}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {5}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {39}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {9}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}\) | \(164\) |
| default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {4}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {14}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {5}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {39}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {9}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}\) | \(164\) |
| parallelrisch | \(\frac {-30 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-285 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-240 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+510 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+600 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-402 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-544 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+15 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+152}{30 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(188\) |
| risch | \(-\frac {2 \left (20 \,{\mathrm e}^{7 i \left (d x +c \right )}+30 i {\mathrm e}^{8 i \left (d x +c \right )}-22 \,{\mathrm e}^{5 i \left (d x +c \right )}+70 i {\mathrm e}^{6 i \left (d x +c \right )}-172 \,{\mathrm e}^{3 i \left (d x +c \right )}-81 \,{\mathrm e}^{i \left (d x +c \right )}-78 i {\mathrm e}^{2 i \left (d x +c \right )}-48 i+26 i {\mathrm e}^{4 i \left (d x +c \right )}+15 \,{\mathrm e}^{9 i \left (d x +c \right )}\right )}{15 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d a}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d a}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d a}\) | \(196\) |
| norman | \(\frac {\frac {1}{2 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2 a d}+\frac {20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d a}+\frac {19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d a}-\frac {35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{2 d a}-\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 d a}-\frac {133 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{15 d a}-\frac {259 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{30 d a}+\frac {19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{60 d a}+\frac {151 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{10 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}\) | \(236\) |
Input:
int(csc(d*x+c)^2*sec(d*x+c)^4/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/2/d/a*(tan(1/2*d*x+1/2*c)-4/5/(tan(1/2*d*x+1/2*c)+1)^5+2/(tan(1/2*d*x+1/ 2*c)+1)^4-14/3/(tan(1/2*d*x+1/2*c)+1)^3+5/(tan(1/2*d*x+1/2*c)+1)^2-39/4/(t an(1/2*d*x+1/2*c)+1)-1/3/(tan(1/2*d*x+1/2*c)-1)^3-1/2/(tan(1/2*d*x+1/2*c)- 1)^2-9/4/(tan(1/2*d*x+1/2*c)-1)-1/tan(1/2*d*x+1/2*c)-2*ln(tan(1/2*d*x+1/2* c)))
Time = 0.09 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.54 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {66 \, \cos \left (d x + c\right )^{4} - 28 \, \cos \left (d x + c\right )^{2} + 15 \, {\left (\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 15 \, {\left (\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (48 \, \cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - 8}{30 \, {\left (a d \cos \left (d x + c\right )^{5} - a d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - a d \cos \left (d x + c\right )^{3}\right )}} \] Input:
integrate(csc(d*x+c)^2*sec(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/30*(66*cos(d*x + c)^4 - 28*cos(d*x + c)^2 + 15*(cos(d*x + c)^5 - cos(d*x + c)^3*sin(d*x + c) - cos(d*x + c)^3)*log(1/2*cos(d*x + c) + 1/2) - 15*(c os(d*x + c)^5 - cos(d*x + c)^3*sin(d*x + c) - cos(d*x + c)^3)*log(-1/2*cos (d*x + c) + 1/2) + 2*(48*cos(d*x + c)^4 - 9*cos(d*x + c)^2 - 1)*sin(d*x + c) - 8)/(a*d*cos(d*x + c)^5 - a*d*cos(d*x + c)^3*sin(d*x + c) - a*d*cos(d* x + c)^3)
Timed out. \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)**2*sec(d*x+c)**4/(a+a*sin(d*x+c)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (120) = 240\).
Time = 0.05 (sec) , antiderivative size = 379, normalized size of antiderivative = 3.01 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\frac {122 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {26 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {454 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {252 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {510 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {330 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {210 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {195 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 15}{\frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {6 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {2 \, a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {2 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {a \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}} + \frac {30 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {15 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{30 \, d} \] Input:
integrate(csc(d*x+c)^2*sec(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/30*((122*sin(d*x + c)/(cos(d*x + c) + 1) - 26*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 454*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 252*sin(d*x + c)^4/ (cos(d*x + c) + 1)^4 + 510*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 330*sin(d *x + c)^6/(cos(d*x + c) + 1)^6 - 210*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 195*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 15)/(a*sin(d*x + c)/(cos(d*x + c) + 1) + 2*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 2*a*sin(d*x + c)^3/(co s(d*x + c) + 1)^3 - 6*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 6*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 2*a*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 2* a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - a*sin(d*x + c)^9/(cos(d*x + c) + 1 )^9) + 30*log(sin(d*x + c)/(cos(d*x + c) + 1))/a - 15*sin(d*x + c)/(a*(cos (d*x + c) + 1)))/d
Time = 0.14 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.41 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {60 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} - \frac {60 \, {\left (2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {5 \, {\left (27 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 25\right )}}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {585 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2040 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2890 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 493}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{120 \, d} \] Input:
integrate(csc(d*x+c)^2*sec(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/120*(120*log(abs(tan(1/2*d*x + 1/2*c)))/a - 60*tan(1/2*d*x + 1/2*c)/a - 60*(2*tan(1/2*d*x + 1/2*c) - 1)/(a*tan(1/2*d*x + 1/2*c)) + 5*(27*tan(1/2* d*x + 1/2*c)^2 - 48*tan(1/2*d*x + 1/2*c) + 25)/(a*(tan(1/2*d*x + 1/2*c) - 1)^3) + (585*tan(1/2*d*x + 1/2*c)^4 + 2040*tan(1/2*d*x + 1/2*c)^3 + 2890*t an(1/2*d*x + 1/2*c)^2 + 1880*tan(1/2*d*x + 1/2*c) + 493)/(a*(tan(1/2*d*x + 1/2*c) + 1)^5))/d
Time = 34.82 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.04 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-22\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-34\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {454\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{15}+\frac {26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}-\frac {122\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{15}-1}{d\,\left (-2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a\,d} \] Input:
int(1/(cos(c + d*x)^4*sin(c + d*x)^2*(a + a*sin(c + d*x))),x)
Output:
((26*tan(c/2 + (d*x)/2)^2)/15 - (122*tan(c/2 + (d*x)/2))/15 + (454*tan(c/2 + (d*x)/2)^3)/15 + (84*tan(c/2 + (d*x)/2)^4)/5 - 34*tan(c/2 + (d*x)/2)^5 - 22*tan(c/2 + (d*x)/2)^6 + 14*tan(c/2 + (d*x)/2)^7 + 13*tan(c/2 + (d*x)/2 )^8 - 1)/(d*(2*a*tan(c/2 + (d*x)/2) + 4*a*tan(c/2 + (d*x)/2)^2 - 4*a*tan(c /2 + (d*x)/2)^3 - 12*a*tan(c/2 + (d*x)/2)^4 + 12*a*tan(c/2 + (d*x)/2)^6 + 4*a*tan(c/2 + (d*x)/2)^7 - 4*a*tan(c/2 + (d*x)/2)^8 - 2*a*tan(c/2 + (d*x)/ 2)^9)) - log(tan(c/2 + (d*x)/2))/(a*d) + tan(c/2 + (d*x)/2)/(2*a*d)
Time = 0.19 (sec) , antiderivative size = 266, normalized size of antiderivative = 2.11 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-60 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4}-60 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3}+60 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2}+60 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )+133 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+133 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-133 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-133 \cos \left (d x +c \right ) \sin \left (d x +c \right )+192 \sin \left (d x +c \right )^{5}+132 \sin \left (d x +c \right )^{4}-348 \sin \left (d x +c \right )^{3}-208 \sin \left (d x +c \right )^{2}+152 \sin \left (d x +c \right )+60}{60 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a d \left (\sin \left (d x +c \right )^{3}+\sin \left (d x +c \right )^{2}-\sin \left (d x +c \right )-1\right )} \] Input:
int(csc(d*x+c)^2*sec(d*x+c)^4/(a+a*sin(d*x+c)),x)
Output:
( - 60*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**4 - 60*cos(c + d*x )*log(tan((c + d*x)/2))*sin(c + d*x)**3 + 60*cos(c + d*x)*log(tan((c + d*x )/2))*sin(c + d*x)**2 + 60*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x) + 133*cos(c + d*x)*sin(c + d*x)**4 + 133*cos(c + d*x)*sin(c + d*x)**3 - 1 33*cos(c + d*x)*sin(c + d*x)**2 - 133*cos(c + d*x)*sin(c + d*x) + 192*sin( c + d*x)**5 + 132*sin(c + d*x)**4 - 348*sin(c + d*x)**3 - 208*sin(c + d*x) **2 + 152*sin(c + d*x) + 60)/(60*cos(c + d*x)*sin(c + d*x)*a*d*(sin(c + d* x)**3 + sin(c + d*x)**2 - sin(c + d*x) - 1))