Integrand size = 24, antiderivative size = 118 \[ \int (c+d x)^3 \csc ^2(a+b x) \sec ^2(a+b x) \, dx=-\frac {2 i (c+d x)^3}{b}-\frac {2 (c+d x)^3 \cot (2 a+2 b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{4 i (a+b x)}\right )}{b^2}-\frac {3 i d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{4 i (a+b x)}\right )}{2 b^3}+\frac {3 d^3 \operatorname {PolyLog}\left (3,e^{4 i (a+b x)}\right )}{8 b^4} \] Output:
-2*I*(d*x+c)^3/b-2*(d*x+c)^3*cot(2*b*x+2*a)/b+3*d*(d*x+c)^2*ln(1-exp(4*I*( b*x+a)))/b^2-3/2*I*d^2*(d*x+c)*polylog(2,exp(4*I*(b*x+a)))/b^3+3/8*d^3*pol ylog(3,exp(4*I*(b*x+a)))/b^4
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(285\) vs. \(2(118)=236\).
Time = 1.38 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.42 \[ \int (c+d x)^3 \csc ^2(a+b x) \sec ^2(a+b x) \, dx=\frac {-\frac {8 i b^3 (c+d x)^3}{-1+e^{4 i a}}+6 b^2 d (c+d x)^2 \log \left (1-e^{-i (a+b x)}\right )+6 b^2 d (c+d x)^2 \log \left (1+e^{-i (a+b x)}\right )+6 b^2 d (c+d x)^2 \log \left (1+e^{-2 i (a+b x)}\right )+12 i b d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )+12 i b d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )+6 i b d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{-2 i (a+b x)}\right )+12 d^3 \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+12 d^3 \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )+3 d^3 \operatorname {PolyLog}\left (3,-e^{-2 i (a+b x)}\right )+4 b^3 (c+d x)^3 \csc (2 a) \csc (2 (a+b x)) \sin (2 b x)}{2 b^4} \] Input:
Integrate[(c + d*x)^3*Csc[a + b*x]^2*Sec[a + b*x]^2,x]
Output:
(((-8*I)*b^3*(c + d*x)^3)/(-1 + E^((4*I)*a)) + 6*b^2*d*(c + d*x)^2*Log[1 - E^((-I)*(a + b*x))] + 6*b^2*d*(c + d*x)^2*Log[1 + E^((-I)*(a + b*x))] + 6 *b^2*d*(c + d*x)^2*Log[1 + E^((-2*I)*(a + b*x))] + (12*I)*b*d^2*(c + d*x)* PolyLog[2, -E^((-I)*(a + b*x))] + (12*I)*b*d^2*(c + d*x)*PolyLog[2, E^((-I )*(a + b*x))] + (6*I)*b*d^2*(c + d*x)*PolyLog[2, -E^((-2*I)*(a + b*x))] + 12*d^3*PolyLog[3, -E^((-I)*(a + b*x))] + 12*d^3*PolyLog[3, E^((-I)*(a + b* x))] + 3*d^3*PolyLog[3, -E^((-2*I)*(a + b*x))] + 4*b^3*(c + d*x)^3*Csc[2*a ]*Csc[2*(a + b*x)]*Sin[2*b*x])/(2*b^4)
Time = 0.77 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.36, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4919, 3042, 4672, 3042, 25, 4202, 2620, 3011, 2720, 7143}
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.
| \(\displaystyle \int (c+d x)^3 \csc ^2(a+b x) \sec ^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 4919 |
| \(\displaystyle 4 \int (c+d x)^3 \csc ^2(2 a+2 b x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 4 \int (c+d x)^3 \csc (2 a+2 b x)^2dx\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle 4 \left (\frac {3 d \int (c+d x)^2 \cot (2 a+2 b x)dx}{2 b}-\frac {(c+d x)^3 \cot (2 a+2 b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 4 \left (\frac {3 d \int -(c+d x)^2 \tan \left (2 a+2 b x+\frac {\pi }{2}\right )dx}{2 b}-\frac {(c+d x)^3 \cot (2 a+2 b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 4 \left (-\frac {3 d \int (c+d x)^2 \tan \left (\frac {1}{2} (4 a+\pi )+2 b x\right )dx}{2 b}-\frac {(c+d x)^3 \cot (2 a+2 b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle 4 \left (-\frac {(c+d x)^3 \cot (2 a+2 b x)}{2 b}-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \int \frac {e^{i (4 a+4 b x+\pi )} (c+d x)^2}{1+e^{i (4 a+4 b x+\pi )}}dx\right )}{2 b}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle 4 \left (-\frac {(c+d x)^3 \cot (2 a+2 b x)}{2 b}-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \int (c+d x) \log \left (1+e^{i (4 a+4 b x+\pi )}\right )dx}{2 b}-\frac {i (c+d x)^2 \log \left (1+e^{i (4 a+4 b x+\pi )}\right )}{4 b}\right )\right )}{2 b}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle 4 \left (-\frac {(c+d x)^3 \cot (2 a+2 b x)}{2 b}-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{i (4 a+4 b x+\pi )}\right )}{4 b}-\frac {i d \int \operatorname {PolyLog}\left (2,-e^{i (4 a+4 b x+\pi )}\right )dx}{4 b}\right )}{2 b}-\frac {i (c+d x)^2 \log \left (1+e^{i (4 a+4 b x+\pi )}\right )}{4 b}\right )\right )}{2 b}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle 4 \left (-\frac {(c+d x)^3 \cot (2 a+2 b x)}{2 b}-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{i (4 a+4 b x+\pi )}\right )}{4 b}-\frac {d \int e^{-i (4 a+4 b x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (4 a+4 b x+\pi )}\right )de^{i (4 a+4 b x+\pi )}}{16 b^2}\right )}{2 b}-\frac {i (c+d x)^2 \log \left (1+e^{i (4 a+4 b x+\pi )}\right )}{4 b}\right )\right )}{2 b}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle 4 \left (-\frac {(c+d x)^3 \cot (2 a+2 b x)}{2 b}-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{i (4 a+4 b x+\pi )}\right )}{4 b}-\frac {d \operatorname {PolyLog}\left (3,-e^{i (4 a+4 b x+\pi )}\right )}{16 b^2}\right )}{2 b}-\frac {i (c+d x)^2 \log \left (1+e^{i (4 a+4 b x+\pi )}\right )}{4 b}\right )\right )}{2 b}\right )\) |
Input:
Int[(c + d*x)^3*Csc[a + b*x]^2*Sec[a + b*x]^2,x]
Output:
4*(-1/2*((c + d*x)^3*Cot[2*a + 2*b*x])/b - (3*d*(((I/3)*(c + d*x)^3)/d - ( 2*I)*(((-1/4*I)*(c + d*x)^2*Log[1 + E^(I*(4*a + Pi + 4*b*x))])/b + ((I/2)* d*(((I/4)*(c + d*x)*PolyLog[2, -E^(I*(4*a + Pi + 4*b*x))])/b - (d*PolyLog[ 3, -E^(I*(4*a + Pi + 4*b*x))])/(16*b^2)))/b)))/(2*b))
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^m*(E^(2*I*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n , x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 686 vs. \(2 (106 ) = 212\).
Time = 0.67 (sec) , antiderivative size = 687, normalized size of antiderivative = 5.82
| method | result | size |
| risch | \(-\frac {12 i d^{2} c \,x^{2}}{b}+\frac {6 d^{2} c \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b^{2}}+\frac {6 d^{2} c \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}-\frac {4 i \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}-\frac {6 d^{2} c a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}+\frac {24 d^{2} c a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {6 d^{2} c \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{3}}-\frac {6 i d^{2} c \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {12 i d^{2} c \,a^{2}}{b^{3}}-\frac {3 i d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right ) x}{b^{3}}+\frac {12 i d^{3} a^{2} x}{b^{3}}+\frac {6 d^{2} c \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) x}{b^{2}}-\frac {3 i d^{2} c \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b^{3}}-\frac {6 i d^{2} c \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {6 i d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{2}}+\frac {3 d^{3} \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) x^{2}}{b^{2}}-\frac {4 i d^{3} x^{3}}{b}+\frac {8 i d^{3} a^{3}}{b^{4}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b^{2}}-\frac {6 i d^{3} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}-\frac {24 i d^{2} c a x}{b^{2}}-\frac {3 d^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a^{2}}{b^{4}}+\frac {3 d^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}+\frac {3 d^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{b^{2}}+\frac {3 d^{3} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{4}}-\frac {12 d^{3} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{2}}-\frac {12 d \,c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {3 d^{3} \operatorname {polylog}\left (3, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{4}}+\frac {6 d^{3} \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {6 d^{3} \operatorname {polylog}\left (3, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}\) | \(687\) |
Input:
int((d*x+c)^3*csc(b*x+a)^2*sec(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
6*d^2/b^3*c*ln(1-exp(I*(b*x+a)))*a-6*I*d^2/b^3*c*polylog(2,exp(I*(b*x+a))) -6*I*d^3/b^3*polylog(2,exp(I*(b*x+a)))*x-4*I*(d^3*x^3+3*c*d^2*x^2+3*c^2*d* x+c^3)/b/(exp(2*I*(b*x+a))+1)/(exp(2*I*(b*x+a))-1)-3*d^3/b^4*ln(1-exp(I*(b *x+a)))*a^2+3*d^3/b^2*ln(1-exp(I*(b*x+a)))*x^2+3*d^3/b^2*ln(exp(I*(b*x+a)) +1)*x^2-6*d^2/b^3*c*a*ln(exp(I*(b*x+a))-1)+24*d^2/b^3*c*a*ln(exp(I*(b*x+a) ))+6*d^2/b^2*c*ln(exp(2*I*(b*x+a))+1)*x+12*I*d^3/b^3*a^2*x-3*I*d^3/b^3*pol ylog(2,-exp(2*I*(b*x+a)))*x-12*I*d^2/b*c*x^2-12*I*d^2/b^3*c*a^2-3*I*d^2/b^ 3*c*polylog(2,-exp(2*I*(b*x+a)))-6*I*d^2/b^3*c*polylog(2,-exp(I*(b*x+a)))- 6*I*d^3/b^3*polylog(2,-exp(I*(b*x+a)))*x-24*I*d^2/b^2*c*a*x+3*d^3/b^4*a^2* ln(exp(I*(b*x+a))-1)-12*d^3/b^4*a^2*ln(exp(I*(b*x+a)))+3*d/b^2*c^2*ln(exp( 2*I*(b*x+a))+1)+3*d/b^2*c^2*ln(exp(I*(b*x+a))-1)-12*d/b^2*c^2*ln(exp(I*(b* x+a)))+3*d/b^2*c^2*ln(exp(I*(b*x+a))+1)+3*d^3/b^2*ln(exp(2*I*(b*x+a))+1)*x ^2-4*I*d^3/b*x^3+8*I*d^3/b^4*a^3+6*d^2/b^2*c*ln(exp(I*(b*x+a))+1)*x+6*d^2/ b^2*c*ln(1-exp(I*(b*x+a)))*x+3/2*d^3*polylog(3,-exp(2*I*(b*x+a)))/b^4+6*d^ 3*polylog(3,-exp(I*(b*x+a)))/b^4+6*d^3*polylog(3,exp(I*(b*x+a)))/b^4
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1635 vs. \(2 (103) = 206\).
Time = 0.17 (sec) , antiderivative size = 1635, normalized size of antiderivative = 13.86 \[ \int (c+d x)^3 \csc ^2(a+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^3*csc(b*x+a)^2*sec(b*x+a)^2,x, algorithm="fricas")
Output:
1/2*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 6*b^3*c^2*d*x + 2*b^3*c^3 + 6*d^3*c os(b*x + a)*polylog(3, cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a) + 6*d^3 *cos(b*x + a)*polylog(3, cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) + 6*d ^3*cos(b*x + a)*polylog(3, I*cos(b*x + a) + sin(b*x + a))*sin(b*x + a) + 6 *d^3*cos(b*x + a)*polylog(3, I*cos(b*x + a) - sin(b*x + a))*sin(b*x + a) + 6*d^3*cos(b*x + a)*polylog(3, -I*cos(b*x + a) + sin(b*x + a))*sin(b*x + a ) + 6*d^3*cos(b*x + a)*polylog(3, -I*cos(b*x + a) - sin(b*x + a))*sin(b*x + a) + 6*d^3*cos(b*x + a)*polylog(3, -cos(b*x + a) + I*sin(b*x + a))*sin(b *x + a) + 6*d^3*cos(b*x + a)*polylog(3, -cos(b*x + a) - I*sin(b*x + a))*si n(b*x + a) - 6*(I*b*d^3*x + I*b*c*d^2)*cos(b*x + a)*dilog(cos(b*x + a) + I *sin(b*x + a))*sin(b*x + a) - 6*(-I*b*d^3*x - I*b*c*d^2)*cos(b*x + a)*dilo g(cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) - 6*(-I*b*d^3*x - I*b*c*d^2) *cos(b*x + a)*dilog(I*cos(b*x + a) + sin(b*x + a))*sin(b*x + a) - 6*(I*b*d ^3*x + I*b*c*d^2)*cos(b*x + a)*dilog(I*cos(b*x + a) - sin(b*x + a))*sin(b* x + a) - 6*(I*b*d^3*x + I*b*c*d^2)*cos(b*x + a)*dilog(-I*cos(b*x + a) + si n(b*x + a))*sin(b*x + a) - 6*(-I*b*d^3*x - I*b*c*d^2)*cos(b*x + a)*dilog(- I*cos(b*x + a) - sin(b*x + a))*sin(b*x + a) - 6*(-I*b*d^3*x - I*b*c*d^2)*c os(b*x + a)*dilog(-cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a) - 6*(I*b*d^ 3*x + I*b*c*d^2)*cos(b*x + a)*dilog(-cos(b*x + a) - I*sin(b*x + a))*sin(b* x + a) + 3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d)*cos(b*x + a)*log(c...
Timed out. \[ \int (c+d x)^3 \csc ^2(a+b x) \sec ^2(a+b x) \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**3*csc(b*x+a)**2*sec(b*x+a)**2,x)
Output:
Timed out
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2360 vs. \(2 (103) = 206\).
Time = 0.50 (sec) , antiderivative size = 2360, normalized size of antiderivative = 20.00 \[ \int (c+d x)^3 \csc ^2(a+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^3*csc(b*x+a)^2*sec(b*x+a)^2,x, algorithm="maxima")
Output:
-1/2*(2*c^3*(1/tan(b*x + a) - tan(b*x + a)) - 6*a*c^2*d*(1/tan(b*x + a) - tan(b*x + a))/b + 6*a^2*c*d^2*(1/tan(b*x + a) - tan(b*x + a))/b^2 - 2*a^3* d^3*(1/tan(b*x + a) - tan(b*x + a))/b^3 - 3*((cos(4*b*x + 4*a)^2 + sin(4*b *x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a) ^2 - 2*cos(4*b*x + 4*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b *x + a) + 1) + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4* a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) - 8*(b*x + a)*sin(4*b*x + 4*a))*c^2*d/((cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*b) + 6*((cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*c os(2*b*x + 2*a) + 1) + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4* b*x + 4*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*log(c os(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) - 8*(b*x + a)*sin(4*b *x + 4*a))*a*c*d^2/((cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*b^2) - 3*((cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4* b*x + 4*a) + 1)*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a ) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) + (cos...
\[ \int (c+d x)^3 \csc ^2(a+b x) \sec ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \csc \left (b x + a\right )^{2} \sec \left (b x + a\right )^{2} \,d x } \] Input:
integrate((d*x+c)^3*csc(b*x+a)^2*sec(b*x+a)^2,x, algorithm="giac")
Output:
integrate((d*x + c)^3*csc(b*x + a)^2*sec(b*x + a)^2, x)
Timed out. \[ \int (c+d x)^3 \csc ^2(a+b x) \sec ^2(a+b x) \, dx=\int \frac {{\left (c+d\,x\right )}^3}{{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^2} \,d x \] Input:
int((c + d*x)^3/(cos(a + b*x)^2*sin(a + b*x)^2),x)
Output:
int((c + d*x)^3/(cos(a + b*x)^2*sin(a + b*x)^2), x)
\[ \int (c+d x)^3 \csc ^2(a+b x) \sec ^2(a+b x) \, dx =\text {Too large to display} \] Input:
int((d*x+c)^3*csc(b*x+a)^2*sec(b*x+a)^2,x)
Output:
(112*cos(a + b*x)*int(x**3/(tan((a + b*x)/2)**6 - 2*tan((a + b*x)/2)**4 + tan((a + b*x)/2)**2),x)*sin(a + b*x)*b**4*d**3 - 224*cos(a + b*x)*int(x**3 /(tan((a + b*x)/2)**4 - 2*tan((a + b*x)/2)**2 + 1),x)*sin(a + b*x)*b**4*d* *3 + 336*cos(a + b*x)*int(x**2/(tan((a + b*x)/2)**6 - 2*tan((a + b*x)/2)** 4 + tan((a + b*x)/2)**2),x)*sin(a + b*x)*b**4*c*d**2 - 552*cos(a + b*x)*in t(x**2/(tan((a + b*x)/2)**5 - 2*tan((a + b*x)/2)**3 + tan((a + b*x)/2)),x) *sin(a + b*x)*b**3*d**3 - 672*cos(a + b*x)*int(x**2/(tan((a + b*x)/2)**4 - 2*tan((a + b*x)/2)**2 + 1),x)*sin(a + b*x)*b**4*c*d**2 - 120*cos(a + b*x) *int((tan((a + b*x)/2)**5*x**2)/(tan((a + b*x)/2)**4 - 2*tan((a + b*x)/2)* *2 + 1),x)*sin(a + b*x)*b**3*d**3 - 240*cos(a + b*x)*int((tan((a + b*x)/2) **5*x)/(tan((a + b*x)/2)**4 - 2*tan((a + b*x)/2)**2 + 1),x)*sin(a + b*x)*b **3*c*d**2 - 336*cos(a + b*x)*int(x/(tan((a + b*x)/2)**6 - 2*tan((a + b*x) /2)**4 + tan((a + b*x)/2)**2),x)*sin(a + b*x)*b**2*d**3 - 1104*cos(a + b*x )*int(x/(tan((a + b*x)/2)**5 - 2*tan((a + b*x)/2)**3 + tan((a + b*x)/2)),x )*sin(a + b*x)*b**3*c*d**2 - 480*cos(a + b*x)*log(tan((a + b*x)/2)**2 + 1) *sin(a + b*x)*b**2*c**2*d - 756*cos(a + b*x)*log(tan((a + b*x)/2)**2 + 1)* sin(a + b*x)*d**3 + 240*cos(a + b*x)*log(tan((a + b*x)/2) - 1)*sin(a + b*x )*b**2*c**2*d + 336*cos(a + b*x)*log(tan((a + b*x)/2) - 1)*sin(a + b*x)*b* c*d**2 + 84*cos(a + b*x)*log(tan((a + b*x)/2) - 1)*sin(a + b*x)*d**3 + 240 *cos(a + b*x)*log(tan((a + b*x)/2) + 1)*sin(a + b*x)*b**2*c**2*d - 336*...