Integrand size = 27, antiderivative size = 86 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {2 a \cot (c+d x)}{d}-\frac {a \cot ^3(c+d x)}{3 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}+\frac {a \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d} \] Output:
-3/2*a*arctanh(cos(d*x+c))/d-2*a*cot(d*x+c)/d-1/3*a*cot(d*x+c)^3/d-1/2*a*c ot(d*x+c)*csc(d*x+c)/d+a*sec(d*x+c)/d+a*tan(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(205\) vs. \(2(86)=172\).
Time = 5.66 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.38 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {5 a \cot (c+d x)}{3 d}-\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d}-\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \sin \left (\frac {1}{2} (c+d x)\right )}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {a \sin \left (\frac {1}{2} (c+d x)\right )}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {a \tan (c+d x)}{d} \] Input:
Integrate[Csc[c + d*x]^4*Sec[c + d*x]^2*(a + a*Sin[c + d*x]),x]
Output:
(-5*a*Cot[c + d*x])/(3*d) - (a*Csc[(c + d*x)/2]^2)/(8*d) - (a*Cot[c + d*x] *Csc[c + d*x]^2)/(3*d) - (3*a*Log[Cos[(c + d*x)/2]])/(2*d) + (3*a*Log[Sin[ (c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d) + (a*Sin[(c + d*x)/2]) /(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (a*Sin[(c + d*x)/2])/(d*(Cos[ (c + d*x)/2] + Sin[(c + d*x)/2])) + (a*Tan[c + d*x])/d
Time = 0.44 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {3042, 3317, 3042, 3100, 244, 2009, 3102, 252, 262, 219}
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.
| \(\displaystyle \int \csc ^4(c+d x) \sec ^2(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a \sin (c+d x)+a}{\sin (c+d x)^4 \cos (c+d x)^2}dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \csc ^4(c+d x) \sec ^2(c+d x)dx+a \int \csc ^3(c+d x) \sec ^2(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \csc (c+d x)^3 \sec (c+d x)^2dx+a \int \csc (c+d x)^4 \sec (c+d x)^2dx\) |
| \(\Big \downarrow \) 3100 |
| \(\displaystyle \frac {a \int \cot ^4(c+d x) \left (\tan ^2(c+d x)+1\right )^2d\tan (c+d x)}{d}+a \int \csc (c+d x)^3 \sec (c+d x)^2dx\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {a \int \left (\cot ^4(c+d x)+2 \cot ^2(c+d x)+1\right )d\tan (c+d x)}{d}+a \int \csc (c+d x)^3 \sec (c+d x)^2dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a \int \csc (c+d x)^3 \sec (c+d x)^2dx+\frac {a \left (\tan (c+d x)-\frac {1}{3} \cot ^3(c+d x)-2 \cot (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3102 |
| \(\displaystyle \frac {a \int \frac {\sec ^4(c+d x)}{\left (1-\sec ^2(c+d x)\right )^2}d\sec (c+d x)}{d}+\frac {a \left (\tan (c+d x)-\frac {1}{3} \cot ^3(c+d x)-2 \cot (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {a \left (\frac {\sec ^3(c+d x)}{2 \left (1-\sec ^2(c+d x)\right )}-\frac {3}{2} \int \frac {\sec ^2(c+d x)}{1-\sec ^2(c+d x)}d\sec (c+d x)\right )}{d}+\frac {a \left (\tan (c+d x)-\frac {1}{3} \cot ^3(c+d x)-2 \cot (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {a \left (\frac {\sec ^3(c+d x)}{2 \left (1-\sec ^2(c+d x)\right )}-\frac {3}{2} \left (\int \frac {1}{1-\sec ^2(c+d x)}d\sec (c+d x)-\sec (c+d x)\right )\right )}{d}+\frac {a \left (\tan (c+d x)-\frac {1}{3} \cot ^3(c+d x)-2 \cot (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {a \left (\frac {\sec ^3(c+d x)}{2 \left (1-\sec ^2(c+d x)\right )}-\frac {3}{2} (\text {arctanh}(\sec (c+d x))-\sec (c+d x))\right )}{d}+\frac {a \left (\tan (c+d x)-\frac {1}{3} \cot ^3(c+d x)-2 \cot (c+d x)\right )}{d}\) |
Input:
Int[Csc[c + d*x]^4*Sec[c + d*x]^2*(a + a*Sin[c + d*x]),x]
Output:
(a*((-3*(ArcTanh[Sec[c + d*x]] - Sec[c + d*x]))/2 + Sec[c + d*x]^3/(2*(1 - Sec[c + d*x]^2))))/d + (a*(-2*Cot[c + d*x] - Cot[c + d*x]^3/3 + Tan[c + d *x]))/d
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
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[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Time = 0.69 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.19
| method | result | size |
| derivativedivides | \(\frac {a \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )}{d}\) | \(102\) |
| default | \(\frac {a \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )}{d}\) | \(102\) |
| parallelrisch | \(\frac {\left (-90+36 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+2 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+18 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+18 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{24 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(117\) |
| risch | \(\frac {-9 i a \,{\mathrm e}^{5 i \left (d x +c \right )}+9 a \,{\mathrm e}^{6 i \left (d x +c \right )}+24 i a \,{\mathrm e}^{3 i \left (d x +c \right )}-24 a \,{\mathrm e}^{4 i \left (d x +c \right )}-7 i a \,{\mathrm e}^{i \left (d x +c \right )}+39 a \,{\mathrm e}^{2 i \left (d x +c \right )}-16 a}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}\) | \(149\) |
| norman | \(\frac {\frac {a}{24 d}+\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {7 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 d}-\frac {35 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{12 d}-\frac {9 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}-\frac {35 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{12 d}+\frac {7 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 d}+\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{8 d}+\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{24 d}-\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}+\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(218\) |
Input:
int(csc(d*x+c)^4*sec(d*x+c)^2*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(-1/2/sin(d*x+c)^2/cos(d*x+c)+3/2/cos(d*x+c)+3/2*ln(csc(d*x+c)-cot( d*x+c)))+a*(-1/3/sin(d*x+c)^3/cos(d*x+c)+4/3/sin(d*x+c)/cos(d*x+c)-8/3*cot (d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (80) = 160\).
Time = 0.09 (sec) , antiderivative size = 308, normalized size of antiderivative = 3.58 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {32 \, a \cos \left (d x + c\right )^{4} + 14 \, a \cos \left (d x + c\right )^{3} - 48 \, a \cos \left (d x + c\right )^{2} - 18 \, a \cos \left (d x + c\right ) + 9 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + {\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 9 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + {\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (16 \, a \cos \left (d x + c\right )^{3} + 9 \, a \cos \left (d x + c\right )^{2} - 15 \, a \cos \left (d x + c\right ) - 6 \, a\right )} \sin \left (d x + c\right ) + 12 \, a}{12 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right ) + d\right )}} \] Input:
integrate(csc(d*x+c)^4*sec(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
-1/12*(32*a*cos(d*x + c)^4 + 14*a*cos(d*x + c)^3 - 48*a*cos(d*x + c)^2 - 1 8*a*cos(d*x + c) + 9*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 + (a*cos(d*x + c)^3 + a*cos(d*x + c)^2 - a*cos(d*x + c) - a)*sin(d*x + c) + a)*log(1/2*c os(d*x + c) + 1/2) - 9*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 + (a*cos(d*x + c)^3 + a*cos(d*x + c)^2 - a*cos(d*x + c) - a)*sin(d*x + c) + a)*log(-1/ 2*cos(d*x + c) + 1/2) - 2*(16*a*cos(d*x + c)^3 + 9*a*cos(d*x + c)^2 - 15*a *cos(d*x + c) - 6*a)*sin(d*x + c) + 12*a)/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + (d*cos(d*x + c)^3 + d*cos(d*x + c)^2 - d*cos(d*x + c) - d)*sin(d* x + c) + d)
Timed out. \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)**4*sec(d*x+c)**2*(a+a*sin(d*x+c)),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.14 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 \, a {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 4 \, a {\left (\frac {6 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{3}} - 3 \, \tan \left (d x + c\right )\right )}}{12 \, d} \] Input:
integrate(csc(d*x+c)^4*sec(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
1/12*(3*a*(2*(3*cos(d*x + c)^2 - 2)/(cos(d*x + c)^3 - cos(d*x + c)) - 3*lo g(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 4*a*((6*tan(d*x + c)^2 + 1)/tan(d*x + c)^3 - 3*tan(d*x + c)))/d
Time = 0.17 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.51 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {48 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1} - \frac {66 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \] Input:
integrate(csc(d*x+c)^4*sec(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/24*(a*tan(1/2*d*x + 1/2*c)^3 + 3*a*tan(1/2*d*x + 1/2*c)^2 + 36*a*log(abs (tan(1/2*d*x + 1/2*c))) + 21*a*tan(1/2*d*x + 1/2*c) - 48*a/(tan(1/2*d*x + 1/2*c) - 1) - (66*a*tan(1/2*d*x + 1/2*c)^3 + 21*a*tan(1/2*d*x + 1/2*c)^2 + 3*a*tan(1/2*d*x + 1/2*c) + a)/tan(1/2*d*x + 1/2*c)^3)/d
Time = 26.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.67 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {7\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {-23\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {a}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}+\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d} \] Input:
int((a + a*sin(c + d*x))/(cos(c + d*x)^2*sin(c + d*x)^4),x)
Output:
(7*a*tan(c/2 + (d*x)/2))/(8*d) - (a/3 + (2*a*tan(c/2 + (d*x)/2))/3 + 6*a*t an(c/2 + (d*x)/2)^2 - 23*a*tan(c/2 + (d*x)/2)^3)/(d*(8*tan(c/2 + (d*x)/2)^ 3 - 8*tan(c/2 + (d*x)/2)^4)) + (a*tan(c/2 + (d*x)/2)^2)/(8*d) + (a*tan(c/2 + (d*x)/2)^3)/(24*d) + (3*a*log(tan(c/2 + (d*x)/2)))/(2*d)
Time = 0.17 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.77 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (36 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-36 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+18 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-90 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+18 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )} \] Input:
int(csc(d*x+c)^4*sec(d*x+c)^2*(a+a*sin(d*x+c)),x)
Output:
(a*(36*log(tan((c + d*x)/2))*tan((c + d*x)/2)**4 - 36*log(tan((c + d*x)/2) )*tan((c + d*x)/2)**3 + tan((c + d*x)/2)**7 + 2*tan((c + d*x)/2)**6 + 18*t an((c + d*x)/2)**5 - 90*tan((c + d*x)/2)**4 + 18*tan((c + d*x)/2)**2 + 2*t an((c + d*x)/2) + 1))/(24*tan((c + d*x)/2)**3*d*(tan((c + d*x)/2) - 1))