Integrand size = 29, antiderivative size = 125 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {7 a^2 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {16 a^2 \cot (c+d x)}{3 d}-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {8 a^2 \cot (c+d x) \csc (c+d x)}{3 d (1-\sin (c+d x))}+\frac {a^4 \cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2} \] Output:
-7/2*a^2*arctanh(cos(d*x+c))/d-16/3*a^2*cot(d*x+c)/d-7/2*a^2*cot(d*x+c)*cs c(d*x+c)/d+8/3*a^2*cot(d*x+c)*csc(d*x+c)/d/(1-sin(d*x+c))+1/3*a^4*cot(d*x+ c)*csc(d*x+c)/d/(a-a*sin(d*x+c))^2
Time = 2.41 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.52 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \left (-24 \cot \left (\frac {1}{2} (c+d x)\right )-3 \csc ^2\left (\frac {1}{2} (c+d x)\right )-84 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+84 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+3 \sec ^2\left (\frac {1}{2} (c+d x)\right )+\frac {8}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {16 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {160 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+24 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{24 d} \] Input:
Integrate[Csc[c + d*x]^3*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]
Output:
(a^2*(-24*Cot[(c + d*x)/2] - 3*Csc[(c + d*x)/2]^2 - 84*Log[Cos[(c + d*x)/2 ]] + 84*Log[Sin[(c + d*x)/2]] + 3*Sec[(c + d*x)/2]^2 + 8/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (16*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (160*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2] ) + 24*Tan[(c + d*x)/2]))/(24*d)
Time = 0.95 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {3042, 3348, 3042, 3245, 3042, 3457, 3042, 3227, 3042, 4254, 24, 4255, 3042, 4257}
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.
| \(\displaystyle \int \csc ^3(c+d x) \sec ^4(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^2}{\sin (c+d x)^3 \cos (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3348 |
| \(\displaystyle a^4 \int \frac {\csc ^3(c+d x)}{(a-a \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 \int \frac {1}{\sin (c+d x)^3 (a-a \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle a^4 \left (\frac {\int \frac {\csc ^3(c+d x) (3 \sin (c+d x) a+5 a)}{a-a \sin (c+d x)}dx}{3 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 \left (\frac {\int \frac {3 \sin (c+d x) a+5 a}{\sin (c+d x)^3 (a-a \sin (c+d x))}dx}{3 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle a^4 \left (\frac {\frac {\int \csc ^3(c+d x) \left (16 \sin (c+d x) a^2+21 a^2\right )dx}{a^2}+\frac {8 \cot (c+d x) \csc (c+d x)}{d (1-\sin (c+d x))}}{3 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 \left (\frac {\frac {\int \frac {16 \sin (c+d x) a^2+21 a^2}{\sin (c+d x)^3}dx}{a^2}+\frac {8 \cot (c+d x) \csc (c+d x)}{d (1-\sin (c+d x))}}{3 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle a^4 \left (\frac {\frac {21 a^2 \int \csc ^3(c+d x)dx+16 a^2 \int \csc ^2(c+d x)dx}{a^2}+\frac {8 \cot (c+d x) \csc (c+d x)}{d (1-\sin (c+d x))}}{3 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 \left (\frac {\frac {16 a^2 \int \csc (c+d x)^2dx+21 a^2 \int \csc (c+d x)^3dx}{a^2}+\frac {8 \cot (c+d x) \csc (c+d x)}{d (1-\sin (c+d x))}}{3 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle a^4 \left (\frac {\frac {21 a^2 \int \csc (c+d x)^3dx-\frac {16 a^2 \int 1d\cot (c+d x)}{d}}{a^2}+\frac {8 \cot (c+d x) \csc (c+d x)}{d (1-\sin (c+d x))}}{3 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a^4 \left (\frac {\frac {21 a^2 \int \csc (c+d x)^3dx-\frac {16 a^2 \cot (c+d x)}{d}}{a^2}+\frac {8 \cot (c+d x) \csc (c+d x)}{d (1-\sin (c+d x))}}{3 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle a^4 \left (\frac {\frac {21 a^2 \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {16 a^2 \cot (c+d x)}{d}}{a^2}+\frac {8 \cot (c+d x) \csc (c+d x)}{d (1-\sin (c+d x))}}{3 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 \left (\frac {\frac {21 a^2 \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {16 a^2 \cot (c+d x)}{d}}{a^2}+\frac {8 \cot (c+d x) \csc (c+d x)}{d (1-\sin (c+d x))}}{3 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle a^4 \left (\frac {\frac {21 a^2 \left (-\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {16 a^2 \cot (c+d x)}{d}}{a^2}+\frac {8 \cot (c+d x) \csc (c+d x)}{d (1-\sin (c+d x))}}{3 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}\right )\) |
Input:
Int[Csc[c + d*x]^3*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]
Output:
a^4*((((-16*a^2*Cot[c + d*x])/d + 21*a^2*(-1/2*ArcTanh[Cos[c + d*x]]/d - ( Cot[c + d*x]*Csc[c + d*x])/(2*d)))/a^2 + (8*Cot[c + d*x]*Csc[c + d*x])/(d* (1 - Sin[c + d*x])))/(3*a^2) + (Cot[c + d*x]*Csc[c + d*x])/(3*d*(a - a*Sin [c + d*x])^2))
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*((c + d*Sin[e + f*x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/( a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[b*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ [a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && !GtQ[n, 0] && (Intege rsQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[a^(2*m) Int[(d* Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ [b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.83 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.95
| method | result | size |
| parallelrisch | \(\frac {\left (28 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-112 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+5 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+190 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {302}{3}\right ) a^{2}}{8 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(119\) |
| risch | \(\frac {a^{2} \left (-63 i {\mathrm e}^{5 i \left (d x +c \right )}+21 \,{\mathrm e}^{6 i \left (d x +c \right )}+126 i {\mathrm e}^{3 i \left (d x +c \right )}-98 \,{\mathrm e}^{4 i \left (d x +c \right )}-75 i {\mathrm e}^{i \left (d x +c \right )}+97 \,{\mathrm e}^{2 i \left (d x +c \right )}-32\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d}+\frac {7 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}-\frac {7 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}\) | \(148\) |
| derivativedivides | \(\frac {a^{2} \left (\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+2 a^{2} \left (\frac {1}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+a^{2} \left (\frac {1}{3 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}-\frac {5}{6 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {5}{2 \cos \left (d x +c \right )}+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(164\) |
| default | \(\frac {a^{2} \left (\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+2 a^{2} \left (\frac {1}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+a^{2} \left (\frac {1}{3 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}-\frac {5}{6 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {5}{2 \cos \left (d x +c \right )}+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(164\) |
| norman | \(\frac {\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{d}+\frac {a^{2}}{8 d}-\frac {10 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}-\frac {19 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}+\frac {28 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}-\frac {19 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d}-\frac {10 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{d}+\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{8 d}-\frac {21 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2 d}-\frac {35 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{12 d}-\frac {63 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 d}-\frac {91 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{12 d}+\frac {175 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{24 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{3} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {7 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(314\) |
Input:
int(csc(d*x+c)^3*sec(d*x+c)^4*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/8*(28*(tan(1/2*d*x+1/2*c)-1)^3*ln(tan(1/2*d*x+1/2*c))+tan(1/2*d*x+1/2*c) ^5+5*tan(1/2*d*x+1/2*c)^4+cot(1/2*d*x+1/2*c)^2-112*tan(1/2*d*x+1/2*c)^2+5* cot(1/2*d*x+1/2*c)+190*tan(1/2*d*x+1/2*c)-302/3)*a^2/d/(tan(1/2*d*x+1/2*c) -1)^3
Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (114) = 228\).
Time = 0.08 (sec) , antiderivative size = 428, normalized size of antiderivative = 3.42 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {64 \, a^{2} \cos \left (d x + c\right )^{4} + 86 \, a^{2} \cos \left (d x + c\right )^{3} - 54 \, a^{2} \cos \left (d x + c\right )^{2} - 80 \, a^{2} \cos \left (d x + c\right ) - 4 \, a^{2} + 21 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 3 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right ) + 2 \, a^{2} + {\left (a^{2} \cos \left (d x + c\right )^{3} + 2 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right ) - 2 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 21 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 3 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right ) + 2 \, a^{2} + {\left (a^{2} \cos \left (d x + c\right )^{3} + 2 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right ) - 2 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (32 \, a^{2} \cos \left (d x + c\right )^{3} - 11 \, a^{2} \cos \left (d x + c\right )^{2} - 38 \, a^{2} \cos \left (d x + c\right ) + 2 \, a^{2}\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - 2 \, d\right )} \sin \left (d x + c\right ) + 2 \, d\right )}} \] Input:
integrate(csc(d*x+c)^3*sec(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
-1/12*(64*a^2*cos(d*x + c)^4 + 86*a^2*cos(d*x + c)^3 - 54*a^2*cos(d*x + c) ^2 - 80*a^2*cos(d*x + c) - 4*a^2 + 21*(a^2*cos(d*x + c)^4 - a^2*cos(d*x + c)^3 - 3*a^2*cos(d*x + c)^2 + a^2*cos(d*x + c) + 2*a^2 + (a^2*cos(d*x + c) ^3 + 2*a^2*cos(d*x + c)^2 - a^2*cos(d*x + c) - 2*a^2)*sin(d*x + c))*log(1/ 2*cos(d*x + c) + 1/2) - 21*(a^2*cos(d*x + c)^4 - a^2*cos(d*x + c)^3 - 3*a^ 2*cos(d*x + c)^2 + a^2*cos(d*x + c) + 2*a^2 + (a^2*cos(d*x + c)^3 + 2*a^2* cos(d*x + c)^2 - a^2*cos(d*x + c) - 2*a^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 2*(32*a^2*cos(d*x + c)^3 - 11*a^2*cos(d*x + c)^2 - 38*a^2*co s(d*x + c) + 2*a^2)*sin(d*x + c))/(d*cos(d*x + c)^4 - d*cos(d*x + c)^3 - 3 *d*cos(d*x + c)^2 + d*cos(d*x + c) + (d*cos(d*x + c)^3 + 2*d*cos(d*x + c)^ 2 - d*cos(d*x + c) - 2*d)*sin(d*x + c) + 2*d)
Timed out. \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)**3*sec(d*x+c)**4*(a+a*sin(d*x+c))**2,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.28 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {8 \, {\left (\tan \left (d x + c\right )^{3} - \frac {3}{\tan \left (d x + c\right )} + 6 \, \tan \left (d x + c\right )\right )} a^{2} + a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3}} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 2 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )}}{\cos \left (d x + c\right )^{3}} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \] Input:
integrate(csc(d*x+c)^3*sec(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
1/12*(8*(tan(d*x + c)^3 - 3/tan(d*x + c) + 6*tan(d*x + c))*a^2 + a^2*(2*(1 5*cos(d*x + c)^4 - 10*cos(d*x + c)^2 - 2)/(cos(d*x + c)^5 - cos(d*x + c)^3 ) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 2*a^2*(2*(3*cos (d*x + c)^2 + 1)/cos(d*x + c)^3 - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)))/d
Time = 0.18 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.20 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 84 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 24 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {3 \, {\left (42 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - \frac {16 \, {\left (12 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11 \, a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{24 \, d} \] Input:
integrate(csc(d*x+c)^3*sec(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
1/24*(3*a^2*tan(1/2*d*x + 1/2*c)^2 + 84*a^2*log(abs(tan(1/2*d*x + 1/2*c))) + 24*a^2*tan(1/2*d*x + 1/2*c) - 3*(42*a^2*tan(1/2*d*x + 1/2*c)^2 + 8*a^2* tan(1/2*d*x + 1/2*c) + a^2)/tan(1/2*d*x + 1/2*c)^2 - 16*(12*a^2*tan(1/2*d* x + 1/2*c)^2 - 21*a^2*tan(1/2*d*x + 1/2*c) + 11*a^2)/(tan(1/2*d*x + 1/2*c) - 1)^3)/d
Time = 31.57 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.46 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {7\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {-36\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {135\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {239\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{6}+\frac {5\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {a^2}{2}}{d\,\left (-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d} \] Input:
int((a + a*sin(c + d*x))^2/(cos(c + d*x)^4*sin(c + d*x)^3),x)
Output:
(a^2*tan(c/2 + (d*x)/2)^2)/(8*d) + (7*a^2*log(tan(c/2 + (d*x)/2)))/(2*d) - ((135*a^2*tan(c/2 + (d*x)/2)^3)/2 - (239*a^2*tan(c/2 + (d*x)/2)^2)/6 - 36 *a^2*tan(c/2 + (d*x)/2)^4 + a^2/2 + (5*a^2*tan(c/2 + (d*x)/2))/2)/(d*(4*ta n(c/2 + (d*x)/2)^2 - 12*tan(c/2 + (d*x)/2)^3 + 12*tan(c/2 + (d*x)/2)^4 - 4 *tan(c/2 + (d*x)/2)^5)) + (a^2*tan(c/2 + (d*x)/2))/d
Time = 0.18 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.82 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (84 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-252 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+252 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-84 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-112 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+234 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-190 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}{24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )} \] Input:
int(csc(d*x+c)^3*sec(d*x+c)^4*(a+a*sin(d*x+c))^2,x)
Output:
(a**2*(84*log(tan((c + d*x)/2))*tan((c + d*x)/2)**5 - 252*log(tan((c + d*x )/2))*tan((c + d*x)/2)**4 + 252*log(tan((c + d*x)/2))*tan((c + d*x)/2)**3 - 84*log(tan((c + d*x)/2))*tan((c + d*x)/2)**2 + 3*tan((c + d*x)/2)**7 + 1 5*tan((c + d*x)/2)**6 - 112*tan((c + d*x)/2)**5 + 234*tan((c + d*x)/2)**3 - 190*tan((c + d*x)/2)**2 + 15*tan((c + d*x)/2) + 3))/(24*tan((c + d*x)/2) **2*d*(tan((c + d*x)/2)**3 - 3*tan((c + d*x)/2)**2 + 3*tan((c + d*x)/2) - 1))