Integrand size = 27, antiderivative size = 198 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 b \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 d}-\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^5 d}-\frac {b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {\left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \] Output:
-2*b*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^5/ d-1/8*(3*a^4-12*a^2*b^2+8*b^4)*arctanh(cos(d*x+c))/a^5/d-1/3*b*(4*a^2-3*b^ 2)*cot(d*x+c)/a^4/d+1/8*(5*a^2-4*b^2)*cot(d*x+c)*csc(d*x+c)/a^3/d+1/3*b*co t(d*x+c)*csc(d*x+c)^2/a^2/d-1/4*cot(d*x+c)*csc(d*x+c)^3/a/d
Leaf count is larger than twice the leaf count of optimal. \(433\) vs. \(2(198)=396\).
Time = 7.15 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.19 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 b \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 d}+\frac {\left (-4 a^2 b \cos \left (\frac {1}{2} (c+d x)\right )+3 b^3 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{6 a^4 d}+\frac {\left (5 a^2-4 b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^3 d}+\frac {b \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^2 d}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 a d}+\frac {\left (-3 a^4+12 a^2 b^2-8 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^5 d}+\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^5 d}+\frac {\left (-5 a^2+4 b^2\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^3 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 a d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (4 a^2 b \sin \left (\frac {1}{2} (c+d x)\right )-3 b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^4 d}-\frac {b \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 a^2 d} \] Input:
Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/(a + b*Sin[c + d*x]),x]
Output:
(-2*b*(a^2 - b^2)^(3/2)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*S in[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*d) + ((-4*a^2*b*Cos[(c + d*x)/2] + 3*b^3*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^4*d) + ((5*a^2 - 4*b^2)*C sc[(c + d*x)/2]^2)/(32*a^3*d) + (b*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(2 4*a^2*d) - Csc[(c + d*x)/2]^4/(64*a*d) + ((-3*a^4 + 12*a^2*b^2 - 8*b^4)*Lo g[Cos[(c + d*x)/2]])/(8*a^5*d) + ((3*a^4 - 12*a^2*b^2 + 8*b^4)*Log[Sin[(c + d*x)/2]])/(8*a^5*d) + ((-5*a^2 + 4*b^2)*Sec[(c + d*x)/2]^2)/(32*a^3*d) + Sec[(c + d*x)/2]^4/(64*a*d) + (Sec[(c + d*x)/2]*(4*a^2*b*Sin[(c + d*x)/2] - 3*b^3*Sin[(c + d*x)/2]))/(6*a^4*d) - (b*Sec[(c + d*x)/2]^2*Tan[(c + d*x )/2])/(24*a^2*d)
Time = 1.51 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.12, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 3372, 3042, 3534, 25, 3042, 3534, 27, 3042, 3480, 3042, 3139, 1083, 217, 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.
| \(\displaystyle \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4}{\sin (c+d x)^5 (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3372 |
| \(\displaystyle -\frac {\int \frac {\csc ^3(c+d x) \left (-4 \left (3 a^2-2 b^2\right ) \sin ^2(c+d x)-a b \sin (c+d x)+3 \left (5 a^2-4 b^2\right )\right )}{a+b \sin (c+d x)}dx}{12 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {-4 \left (3 a^2-2 b^2\right ) \sin (c+d x)^2-a b \sin (c+d x)+3 \left (5 a^2-4 b^2\right )}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx}{12 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {\int -\frac {\csc ^2(c+d x) \left (-3 b \left (5 a^2-4 b^2\right ) \sin ^2(c+d x)+a \left (9 a^2-4 b^2\right ) \sin (c+d x)+8 b \left (4 a^2-3 b^2\right )\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {3 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{12 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int \frac {\csc ^2(c+d x) \left (-3 b \left (5 a^2-4 b^2\right ) \sin ^2(c+d x)+a \left (9 a^2-4 b^2\right ) \sin (c+d x)+8 b \left (4 a^2-3 b^2\right )\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {3 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{12 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {-3 b \left (5 a^2-4 b^2\right ) \sin (c+d x)^2+a \left (9 a^2-4 b^2\right ) \sin (c+d x)+8 b \left (4 a^2-3 b^2\right )}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{2 a}-\frac {3 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{12 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {-\frac {\frac {\int \frac {3 \csc (c+d x) \left (3 a^4-12 b^2 a^2-b \left (5 a^2-4 b^2\right ) \sin (c+d x) a+8 b^4\right )}{a+b \sin (c+d x)}dx}{a}-\frac {8 b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{12 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {3 \int \frac {\csc (c+d x) \left (3 a^4-12 b^2 a^2-b \left (5 a^2-4 b^2\right ) \sin (c+d x) a+8 b^4\right )}{a+b \sin (c+d x)}dx}{a}-\frac {8 b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{12 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {3 \int \frac {3 a^4-12 b^2 a^2-b \left (5 a^2-4 b^2\right ) \sin (c+d x) a+8 b^4}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {8 b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{12 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle -\frac {-\frac {\frac {3 \left (\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {8 b \left (a^2-b^2\right )^2 \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {8 b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{12 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {3 \left (\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {8 b \left (a^2-b^2\right )^2 \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {8 b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{12 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {-\frac {\frac {3 \left (\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {16 b \left (a^2-b^2\right )^2 \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}-\frac {8 b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{12 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {-\frac {\frac {3 \left (\frac {32 b \left (a^2-b^2\right )^2 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}+\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) \int \csc (c+d x)dx}{a}\right )}{a}-\frac {8 b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{12 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {-\frac {\frac {3 \left (\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {16 b \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}\right )}{a}-\frac {8 b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{12 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {-\frac {3 \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {\frac {3 \left (-\frac {16 b \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}-\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{a d}\right )}{a}-\frac {8 b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}}{12 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
Input:
Int[(Cot[c + d*x]^4*Csc[c + d*x])/(a + b*Sin[c + d*x]),x]
Output:
(b*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/ (4*a*d) - (-1/2*((3*((-16*b*(a^2 - b^2)^(3/2)*ArcTan[(2*b + 2*a*Tan[(c + d *x)/2])/(2*Sqrt[a^2 - b^2])])/(a*d) - ((3*a^4 - 12*a^2*b^2 + 8*b^4)*ArcTan h[Cos[c + d*x]])/(a*d)))/a - (8*b*(4*a^2 - 3*b^2)*Cot[c + d*x])/(a*d))/a - (3*(5*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a*d))/(12*a^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[Cos[e + f*x]*(a + b* Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*d*f*(n + 1))), x] + (-Si mp[b*(m + n + 2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x] )^(n + 2)/(a^2*d^2*f*(n + 1)*(n + 2))), x] - Simp[1/(a^2*d^2*(n + 1)*(n + 2 )) Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 2)*Simp[a^2*n*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*m*Sin[e + f*x] - (a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n]) && !m < -1 && LtQ[n, -1] && (LtQ[n, -2] || EqQ[m + n + 4, 0])
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ (b*c - a*d) Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.74 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.47
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}}{4}-\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2}}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{3}+2 a \,b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+10 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{4}}-\frac {2 b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{5} \sqrt {a^{2}-b^{2}}}-\frac {1}{64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-4 a^{2}+4 b^{2}}{32 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (6 a^{4}-24 a^{2} b^{2}+16 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{5}}+\frac {b}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (5 a^{2}-4 b^{2}\right )}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(291\) |
| default | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}}{4}-\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2}}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{3}+2 a \,b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+10 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{4}}-\frac {2 b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{5} \sqrt {a^{2}-b^{2}}}-\frac {1}{64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-4 a^{2}+4 b^{2}}{32 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (6 a^{4}-24 a^{2} b^{2}+16 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{5}}+\frac {b}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (5 a^{2}-4 b^{2}\right )}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(291\) |
| risch | \(\frac {i \left (9 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-12 i a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+15 i a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+15 i a^{3} {\mathrm e}^{i \left (d x +c \right )}-48 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+24 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-12 i a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+96 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-72 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+9 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-80 \,{\mathrm e}^{2 i \left (d x +c \right )} a^{2} b +72 \,{\mathrm e}^{2 i \left (d x +c \right )} b^{3}+32 a^{2} b -24 b^{3}\right )}{12 d \,a^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,a^{3}}-\frac {i \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,a^{5}}-\frac {i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,a^{3}}+\frac {i \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,a^{5}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d a}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{2 d \,a^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{4}}{a^{5} d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d a}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{2 d \,a^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{4}}{a^{5} d}\) | \(590\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(1/16/a^4*(1/4*tan(1/2*d*x+1/2*c)^4*a^3-2/3*b*tan(1/2*d*x+1/2*c)^3*a^2 -2*tan(1/2*d*x+1/2*c)^2*a^3+2*a*b^2*tan(1/2*d*x+1/2*c)^2+10*a^2*b*tan(1/2* d*x+1/2*c)-8*tan(1/2*d*x+1/2*c)*b^3)-2*b*(a^4-2*a^2*b^2+b^4)/a^5/(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))-1/64/a/tan (1/2*d*x+1/2*c)^4-1/32*(-4*a^2+4*b^2)/a^3/tan(1/2*d*x+1/2*c)^2+1/16/a^5*(6 *a^4-24*a^2*b^2+16*b^4)*ln(tan(1/2*d*x+1/2*c))+1/24/a^2*b/tan(1/2*d*x+1/2* c)^3-1/8*b*(5*a^2-4*b^2)/a^4/tan(1/2*d*x+1/2*c))
Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (183) = 366\).
Time = 0.28 (sec) , antiderivative size = 904, normalized size of antiderivative = 4.57 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)/(a+b*sin(d*x+c)),x, algorithm="fricas")
Output:
[-1/48*(6*(5*a^4 - 4*a^2*b^2)*cos(d*x + c)^3 + 24*((a^2*b - b^3)*cos(d*x + c)^4 + a^2*b - b^3 - 2*(a^2*b - b^3)*cos(d*x + c)^2)*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*co s(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)) - 6*(3*a^4 - 4*a^2*b^2)*cos(d*x + c) + 3*((3*a^4 - 12*a^2*b^2 + 8*b^4)*cos(d*x + c)^4 + 3*a^4 - 12*a^2*b^2 + 8*b^4 - 2*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) - 3*((3*a^4 - 12*a^2*b^2 + 8*b^4)*cos(d*x + c)^4 + 3*a^4 - 12*a ^2*b^2 + 8*b^4 - 2*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cos(d*x + c)^2)*log(-1/2*c os(d*x + c) + 1/2) - 16*((4*a^3*b - 3*a*b^3)*cos(d*x + c)^3 - 3*(a^3*b - a *b^3)*cos(d*x + c))*sin(d*x + c))/(a^5*d*cos(d*x + c)^4 - 2*a^5*d*cos(d*x + c)^2 + a^5*d), -1/48*(6*(5*a^4 - 4*a^2*b^2)*cos(d*x + c)^3 - 48*((a^2*b - b^3)*cos(d*x + c)^4 + a^2*b - b^3 - 2*(a^2*b - b^3)*cos(d*x + c)^2)*sqrt (a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 6*(3*a^4 - 4*a^2*b^2)*cos(d*x + c) + 3*((3*a^4 - 12*a^2*b^2 + 8*b^4)*cos( d*x + c)^4 + 3*a^4 - 12*a^2*b^2 + 8*b^4 - 2*(3*a^4 - 12*a^2*b^2 + 8*b^4)*c os(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) - 3*((3*a^4 - 12*a^2*b^2 + 8*b^ 4)*cos(d*x + c)^4 + 3*a^4 - 12*a^2*b^2 + 8*b^4 - 2*(3*a^4 - 12*a^2*b^2 + 8 *b^4)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2) - 16*((4*a^3*b - 3*a*b^ 3)*cos(d*x + c)^3 - 3*(a^3*b - a*b^3)*cos(d*x + c))*sin(d*x + c))/(a^5*...
\[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\cot ^{4}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)/(a+b*sin(d*x+c)),x)
Output:
Integral(cot(c + d*x)**4*csc(c + d*x)/(a + b*sin(c + d*x)), x)
Exception generated. \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)/(a+b*sin(d*x+c)),x, algorithm="maxima")
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (183) = 366\).
Time = 0.24 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.89 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} + \frac {24 \, {\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} - \frac {384 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{5}} - \frac {150 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 600 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 400 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 96 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{4}}{a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)/(a+b*sin(d*x+c)),x, algorithm="giac")
Output:
1/192*((3*a^3*tan(1/2*d*x + 1/2*c)^4 - 8*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 24 *a^3*tan(1/2*d*x + 1/2*c)^2 + 24*a*b^2*tan(1/2*d*x + 1/2*c)^2 + 120*a^2*b* tan(1/2*d*x + 1/2*c) - 96*b^3*tan(1/2*d*x + 1/2*c))/a^4 + 24*(3*a^4 - 12*a ^2*b^2 + 8*b^4)*log(abs(tan(1/2*d*x + 1/2*c)))/a^5 - 384*(a^4*b - 2*a^2*b^ 3 + b^5)*(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)))/(sqrt(a^2 - b^2)*a^5) - (150*a^4*tan(1/2*d *x + 1/2*c)^4 - 600*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 400*b^4*tan(1/2*d*x + 1/2*c)^4 + 120*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 96*a*b^3*tan(1/2*d*x + 1/2* c)^3 - 24*a^4*tan(1/2*d*x + 1/2*c)^2 + 24*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 - 8*a^3*b*tan(1/2*d*x + 1/2*c) + 3*a^4)/(a^5*tan(1/2*d*x + 1/2*c)^4))/d
Time = 19.97 (sec) , antiderivative size = 953, normalized size of antiderivative = 4.81 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
int(cot(c + d*x)^4/(sin(c + d*x)*(a + b*sin(c + d*x))),x)
Output:
tan(c/2 + (d*x)/2)^4/(64*a*d) - (tan(c/2 + (d*x)/2)^2*(1/(8*a) - b^2/(8*a^ 3)))/d + (tan(c/2 + (d*x)/2)*(b/(8*a^2) + (2*b*(1/(4*a) - b^2/(4*a^3)))/a) )/d - (tan(c/2 + (d*x)/2)^2*(2*a*b^2 - 2*a^3) + tan(c/2 + (d*x)/2)^3*(10*a ^2*b - 8*b^3) + a^3/4 - (2*a^2*b*tan(c/2 + (d*x)/2))/3)/(16*a^4*d*tan(c/2 + (d*x)/2)^4) + (log(tan(c/2 + (d*x)/2))*((3*a^4)/8 + b^4 - (3*a^2*b^2)/2) )/(a^5*d) - (b*tan(c/2 + (d*x)/2)^3)/(24*a^2*d) - (b*atan(((b*(-(a + b)^3* (a - b)^3)^(1/2)*((tan(c/2 + (d*x)/2)*(3*a^9 - 32*a^3*b^6 + 64*a^5*b^4 - 3 4*a^7*b^2))/(4*a^7) - (11*a^9*b + 16*a^5*b^5 - 28*a^7*b^3)/(4*a^8) + (b*(2 *a^2*b - (tan(c/2 + (d*x)/2)*(24*a^10 - 32*a^8*b^2))/(4*a^7))*(-(a + b)^3* (a - b)^3)^(1/2))/a^5)*1i)/a^5 - (b*(-(a + b)^3*(a - b)^3)^(1/2)*((11*a^9* b + 16*a^5*b^5 - 28*a^7*b^3)/(4*a^8) - (tan(c/2 + (d*x)/2)*(3*a^9 - 32*a^3 *b^6 + 64*a^5*b^4 - 34*a^7*b^2))/(4*a^7) + (b*(2*a^2*b - (tan(c/2 + (d*x)/ 2)*(24*a^10 - 32*a^8*b^2))/(4*a^7))*(-(a + b)^3*(a - b)^3)^(1/2))/a^5)*1i) /a^5)/((3*a^8*b + 8*b^9 - 28*a^2*b^7 + 35*a^4*b^5 - 18*a^6*b^3)/(2*a^8) + (tan(c/2 + (d*x)/2)*(8*b^8 - 26*a^2*b^6 + 28*a^4*b^4 - 10*a^6*b^2))/(2*a^7 ) + (b*(-(a + b)^3*(a - b)^3)^(1/2)*((tan(c/2 + (d*x)/2)*(3*a^9 - 32*a^3*b ^6 + 64*a^5*b^4 - 34*a^7*b^2))/(4*a^7) - (11*a^9*b + 16*a^5*b^5 - 28*a^7*b ^3)/(4*a^8) + (b*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^10 - 32*a^8*b^2))/(4 *a^7))*(-(a + b)^3*(a - b)^3)^(1/2))/a^5))/a^5 + (b*(-(a + b)^3*(a - b)^3) ^(1/2)*((11*a^9*b + 16*a^5*b^5 - 28*a^7*b^3)/(4*a^8) - (tan(c/2 + (d*x)...
Time = 0.19 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.52 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-48 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (d x +c \right )^{4} a^{2} b +48 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (d x +c \right )^{4} b^{3}-32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{3} b +24 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a \,b^{3}+15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{4}-12 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2} b^{2}+8 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3} b -6 \cos \left (d x +c \right ) a^{4}+9 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4} a^{4}-36 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4} a^{2} b^{2}+24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4} b^{4}}{24 \sin \left (d x +c \right )^{4} a^{5} d} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)/(a+b*sin(d*x+c)),x)
Output:
( - 48*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))* sin(c + d*x)**4*a**2*b + 48*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b )/sqrt(a**2 - b**2))*sin(c + d*x)**4*b**3 - 32*cos(c + d*x)*sin(c + d*x)** 3*a**3*b + 24*cos(c + d*x)*sin(c + d*x)**3*a*b**3 + 15*cos(c + d*x)*sin(c + d*x)**2*a**4 - 12*cos(c + d*x)*sin(c + d*x)**2*a**2*b**2 + 8*cos(c + d*x )*sin(c + d*x)*a**3*b - 6*cos(c + d*x)*a**4 + 9*log(tan((c + d*x)/2))*sin( c + d*x)**4*a**4 - 36*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**2*b**2 + 24 *log(tan((c + d*x)/2))*sin(c + d*x)**4*b**4)/(24*sin(c + d*x)**4*a**5*d)