Integrand size = 21, antiderivative size = 238 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 \left (a^4-5 a^2 b^2+4 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2} d}-\frac {b \left (3 a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}+\frac {\left (7 a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))} \]
-b*(3*a^2-4*b^2)*arctanh(cos(d*x+c))/a^5/d+1/3*(7*a^2-12*b^2)*cot(d*x+c)/a ^4/d-(a^2-2*b^2)*cot(d*x+c)*csc(d*x+c)/a^3/b/d+1/3*(3*a^2-4*b^2)*cot(d*x+c )*csc(d*x+c)/a^2/b/d/(a+b*sin(d*x+c))-1/3*cot(d*x+c)*csc(d*x+c)^2/a/d/(a+b *sin(d*x+c))+2*(a^4-5*a^2*b^2+4*b^4)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2- b^2)^(1/2))/a^5/d/(a^2-b^2)^(1/2)
Time = 6.15 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.69 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 \left (a^4-5 a^2 b^2+4 b^4\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2} d}+\frac {\left (4 a^2 \cos \left (\frac {1}{2} (c+d x)\right )-9 b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{6 a^4 d}+\frac {b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{4 a^3 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^2 d}+\frac {\left (-3 a^2 b+4 b^3\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d}+\frac {\left (3 a^2 b-4 b^3\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d}-\frac {b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{4 a^3 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (-4 a^2 \sin \left (\frac {1}{2} (c+d x)\right )+9 b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^4 d}+\frac {a^2 b \cos (c+d x)-b^3 \cos (c+d x)}{a^4 d (a+b \sin (c+d x))}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 a^2 d} \]
(2*(a^4 - 5*a^2*b^2 + 4*b^4)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]*d) + ((4*a^2 *Cos[(c + d*x)/2] - 9*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^4*d) + (b*Csc[(c + d*x)/2]^2)/(4*a^3*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/( 24*a^2*d) + ((-3*a^2*b + 4*b^3)*Log[Cos[(c + d*x)/2]])/(a^5*d) + ((3*a^2*b - 4*b^3)*Log[Sin[(c + d*x)/2]])/(a^5*d) - (b*Sec[(c + d*x)/2]^2)/(4*a^3*d ) + (Sec[(c + d*x)/2]*(-4*a^2*Sin[(c + d*x)/2] + 9*b^2*Sin[(c + d*x)/2]))/ (6*a^4*d) + (a^2*b*Cos[c + d*x] - b^3*Cos[c + d*x])/(a^4*d*(a + b*Sin[c + d*x])) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^2*d)
Time = 1.45 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.12, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 3203, 3042, 3534, 27, 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.
| \(\displaystyle \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^4 (a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3203 |
| \(\displaystyle \frac {\int \frac {\csc ^3(c+d x) \left (-\left (\left (3 a^2-8 b^2\right ) \sin ^2(c+d x)\right )-a b \sin (c+d x)+6 \left (a^2-2 b^2\right )\right )}{a+b \sin (c+d x)}dx}{3 a^2 b}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-\left (\left (3 a^2-8 b^2\right ) \sin (c+d x)^2\right )-a b \sin (c+d x)+6 \left (a^2-2 b^2\right )}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx}{3 a^2 b}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int -\frac {2 \csc ^2(c+d x) \left (-2 a \sin (c+d x) b^2-3 \left (a^2-2 b^2\right ) \sin ^2(c+d x) b+\left (7 a^2-12 b^2\right ) b\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {3 \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a^2 b}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\csc ^2(c+d x) \left (-2 a \sin (c+d x) b^2-3 \left (a^2-2 b^2\right ) \sin ^2(c+d x) b+\left (7 a^2-12 b^2\right ) b\right )}{a+b \sin (c+d x)}dx}{a}-\frac {3 \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a^2 b}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {-2 a \sin (c+d x) b^2-3 \left (a^2-2 b^2\right ) \sin (c+d x)^2 b+\left (7 a^2-12 b^2\right ) b}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{a}-\frac {3 \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a^2 b}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {3 \csc (c+d x) \left (\left (3 a^2-4 b^2\right ) b^2+a \left (a^2-2 b^2\right ) \sin (c+d x) b\right )}{a+b \sin (c+d x)}dx}{a}-\frac {b \left (7 a^2-12 b^2\right ) \cot (c+d x)}{a d}}{a}-\frac {3 \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a^2 b}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {3 \int \frac {\csc (c+d x) \left (\left (3 a^2-4 b^2\right ) b^2+a \left (a^2-2 b^2\right ) \sin (c+d x) b\right )}{a+b \sin (c+d x)}dx}{a}-\frac {b \left (7 a^2-12 b^2\right ) \cot (c+d x)}{a d}}{a}-\frac {3 \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a^2 b}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3 \int \frac {\left (3 a^2-4 b^2\right ) b^2+a \left (a^2-2 b^2\right ) \sin (c+d x) b}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {b \left (7 a^2-12 b^2\right ) \cot (c+d x)}{a d}}{a}-\frac {3 \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a^2 b}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (\frac {b^2 \left (3 a^2-4 b^2\right ) \int \csc (c+d x)dx}{a}+\frac {b \left (a^4-5 a^2 b^2+4 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {b \left (7 a^2-12 b^2\right ) \cot (c+d x)}{a d}}{a}-\frac {3 \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a^2 b}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (\frac {b^2 \left (3 a^2-4 b^2\right ) \int \csc (c+d x)dx}{a}+\frac {b \left (a^4-5 a^2 b^2+4 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {b \left (7 a^2-12 b^2\right ) \cot (c+d x)}{a d}}{a}-\frac {3 \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a^2 b}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (\frac {b^2 \left (3 a^2-4 b^2\right ) \int \csc (c+d x)dx}{a}+\frac {2 b \left (a^4-5 a^2 b^2+4 b^4\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}-\frac {b \left (7 a^2-12 b^2\right ) \cot (c+d x)}{a d}}{a}-\frac {3 \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a^2 b}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (\frac {b^2 \left (3 a^2-4 b^2\right ) \int \csc (c+d x)dx}{a}-\frac {4 b \left (a^4-5 a^2 b^2+4 b^4\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}\right )}{a}-\frac {b \left (7 a^2-12 b^2\right ) \cot (c+d x)}{a d}}{a}-\frac {3 \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a^2 b}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (\frac {b^2 \left (3 a^2-4 b^2\right ) \int \csc (c+d x)dx}{a}+\frac {2 b \left (a^4-5 a^2 b^2+4 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}\right )}{a}-\frac {b \left (7 a^2-12 b^2\right ) \cot (c+d x)}{a d}}{a}-\frac {3 \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a^2 b}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}+\frac {-\frac {3 \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}-\frac {-\frac {b \left (7 a^2-12 b^2\right ) \cot (c+d x)}{a d}-\frac {3 \left (\frac {2 b \left (a^4-5 a^2 b^2+4 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}-\frac {b^2 \left (3 a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{a d}\right )}{a}}{a}}{3 a^2 b}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}\) |
(-(((-3*((2*b*(a^4 - 5*a^2*b^2 + 4*b^4)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2] )/(2*Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d) - (b^2*(3*a^2 - 4*b^2)*ArcTa nh[Cos[c + d*x]])/(a*d)))/a - (b*(7*a^2 - 12*b^2)*Cot[c + d*x])/(a*d))/a) - (3*(a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x])/(a*d))/(3*a^2*b) + ((3*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*b*d*(a + b*Sin[c + d*x])) - (Cot [c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x]))
3.12.33.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> Simp[(-Cos[e + f*x])*((a + b*Sin[e + f*x])^(m + 1)/(3*a*f*Sin[ e + f*x]^3)), x] + (-Simp[(3*a^2 + b^2*(m - 2))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(3*a^2*b*f*(m + 1)*Sin[e + f*x]^2)), x] - Simp[1/(3*a^2*b*( m + 1)) Int[((a + b*Sin[e + f*x])^(m + 1)/Sin[e + f*x]^3)*Simp[6*a^2 - b^ 2*(m - 1)*(m - 2) + a*b*(m + 1)*Sin[e + f*x] - (3*a^2 - b^2*m*(m - 2))*Sin[ e + f*x]^2, x], x], x]) /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && L tQ[m, -1] && IntegerQ[2*m]
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 = 0.71 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.21
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{3}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}+12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{8 a^{4}}-\frac {1}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-5 a^{2}+12 b^{2}}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (3 a^{2}-4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}+\frac {\frac {2 \left (b^{2} \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+b a \left (a^{2}-b^{2}\right )\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {2 \left (a^{4}-5 a^{2} b^{2}+4 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{5}}}{d}\) | \(287\) |
| default | \(\frac {\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{3}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}+12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{8 a^{4}}-\frac {1}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-5 a^{2}+12 b^{2}}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (3 a^{2}-4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}+\frac {\frac {2 \left (b^{2} \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+b a \left (a^{2}-b^{2}\right )\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {2 \left (a^{4}-5 a^{2} b^{2}+4 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{5}}}{d}\) | \(287\) |
| risch | \(\frac {-14 a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-24 i b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+\frac {50 i a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}}{3}+14 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+8 i b^{3}-4 a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+20 a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-28 a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+24 i b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+2 i a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-14 i a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+12 a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}-8 i b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-\frac {14 i a^{2} b}{3}-\frac {22 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{3}+2 a^{3} {\mathrm e}^{7 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right ) a^{4} d}+\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{3}}-\frac {4 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) b^{2}}{d \,a^{5}}-\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{3}}+\frac {4 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) b^{2}}{d \,a^{5}}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{3} d}+\frac {4 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{5} d}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{3} d}-\frac {4 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{5} d}\) | \(571\) |
1/d*(1/8/a^4*(1/3*tan(1/2*d*x+1/2*c)^3*a^2-2*tan(1/2*d*x+1/2*c)^2*a*b-5*ta n(1/2*d*x+1/2*c)*a^2+12*tan(1/2*d*x+1/2*c)*b^2)-1/24/a^2/tan(1/2*d*x+1/2*c )^3-1/8*(-5*a^2+12*b^2)/a^4/tan(1/2*d*x+1/2*c)+1/4/a^3*b/tan(1/2*d*x+1/2*c )^2+1/a^5*b*(3*a^2-4*b^2)*ln(tan(1/2*d*x+1/2*c))+2/a^5*((b^2*(a^2-b^2)*tan (1/2*d*x+1/2*c)+b*a*(a^2-b^2))/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2 *c)+a)+(a^4-5*a^2*b^2+4*b^4)/(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))))
Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (227) = 454\).
Time = 0.41 (sec) , antiderivative size = 1149, normalized size of antiderivative = 4.83 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
[-1/6*(4*(2*a^4 - 3*a^2*b^2)*cos(d*x + c)^3 + 3*((a^2*b - 4*b^3)*cos(d*x + c)^4 + a^2*b - 4*b^3 - 2*(a^2*b - 4*b^3)*cos(d*x + c)^2 + (a^3 - 4*a*b^2 - (a^3 - 4*a*b^2)*cos(d*x + c)^2)*sin(d*x + c))*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*cos(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*(a^4 - 2*a^2*b^2)*cos(d*x + c) + 3*( (3*a^2*b^2 - 4*b^4)*cos(d*x + c)^4 + 3*a^2*b^2 - 4*b^4 - 2*(3*a^2*b^2 - 4* b^4)*cos(d*x + c)^2 + (3*a^3*b - 4*a*b^3 - (3*a^3*b - 4*a*b^3)*cos(d*x + c )^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 3*((3*a^2*b^2 - 4*b^4)*co s(d*x + c)^4 + 3*a^2*b^2 - 4*b^4 - 2*(3*a^2*b^2 - 4*b^4)*cos(d*x + c)^2 + (3*a^3*b - 4*a*b^3 - (3*a^3*b - 4*a*b^3)*cos(d*x + c)^2)*sin(d*x + c))*log (-1/2*cos(d*x + c) + 1/2) + 2*((7*a^3*b - 12*a*b^3)*cos(d*x + c)^3 - 3*(3* a^3*b - 4*a*b^3)*cos(d*x + c))*sin(d*x + c))/(a^5*b*d*cos(d*x + c)^4 - 2*a ^5*b*d*cos(d*x + c)^2 + a^5*b*d - (a^6*d*cos(d*x + c)^2 - a^6*d)*sin(d*x + c)), -1/6*(4*(2*a^4 - 3*a^2*b^2)*cos(d*x + c)^3 + 6*((a^2*b - 4*b^3)*cos( d*x + c)^4 + a^2*b - 4*b^3 - 2*(a^2*b - 4*b^3)*cos(d*x + c)^2 + (a^3 - 4*a *b^2 - (a^3 - 4*a*b^2)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(a^2 - b^2)*arcta n(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 6*(a^4 - 2*a^2*b ^2)*cos(d*x + c) + 3*((3*a^2*b^2 - 4*b^4)*cos(d*x + c)^4 + 3*a^2*b^2 - 4*b ^4 - 2*(3*a^2*b^2 - 4*b^4)*cos(d*x + c)^2 + (3*a^3*b - 4*a*b^3 - (3*a^3...
Timed out. \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.49 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.50 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {24 \, {\left (3 \, a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} + \frac {48 \, {\left (a^{4} - 5 \, a^{2} b^{2} + 4 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{5}} + \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}} + \frac {48 \, {\left (a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3} b - a b^{3}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} a^{5}} - \frac {132 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 176 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
1/24*(24*(3*a^2*b - 4*b^3)*log(abs(tan(1/2*d*x + 1/2*c)))/a^5 + 48*(a^4 - 5*a^2*b^2 + 4*b^4)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*ta n(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*a^5) + (a^4*tan (1/2*d*x + 1/2*c)^3 - 6*a^3*b*tan(1/2*d*x + 1/2*c)^2 - 15*a^4*tan(1/2*d*x + 1/2*c) + 36*a^2*b^2*tan(1/2*d*x + 1/2*c))/a^6 + 48*(a^2*b^2*tan(1/2*d*x + 1/2*c) - b^4*tan(1/2*d*x + 1/2*c) + a^3*b - a*b^3)/((a*tan(1/2*d*x + 1/2 *c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)*a^5) - (132*a^2*b*tan(1/2*d*x + 1/2* c)^3 - 176*b^3*tan(1/2*d*x + 1/2*c)^3 - 15*a^3*tan(1/2*d*x + 1/2*c)^2 + 36 *a*b^2*tan(1/2*d*x + 1/2*c)^2 - 6*a^2*b*tan(1/2*d*x + 1/2*c) + a^3)/(a^5*t an(1/2*d*x + 1/2*c)^3))/d
Time = 10.67 (sec) , antiderivative size = 973, normalized size of antiderivative = 4.09 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
(tan(c/2 + (d*x)/2)^3*(28*a^2*b - 40*b^3) - tan(c/2 + (d*x)/2)^2*(8*a*b^2 - (14*a^3)/3) - a^3/3 + (4*a^2*b*tan(c/2 + (d*x)/2))/3 + (tan(c/2 + (d*x)/ 2)^4*(5*a^4 - 16*b^4 + 4*a^2*b^2))/a)/(d*(8*a^5*tan(c/2 + (d*x)/2)^3 + 8*a ^5*tan(c/2 + (d*x)/2)^5 + 16*a^4*b*tan(c/2 + (d*x)/2)^4)) + tan(c/2 + (d*x )/2)^3/(24*a^2*d) - (tan(c/2 + (d*x)/2)*((16*a^2 + 32*b^2)/(64*a^4) + 3/(8 *a^2) - (2*b^2)/a^4))/d - (b*tan(c/2 + (d*x)/2)^2)/(4*a^3*d) + (b*log(tan( c/2 + (d*x)/2))*(3*a^2 - 4*b^2))/(a^5*d) + (atan((((b^2 - a^2)^(1/2)*(a^2 - 4*b^2)*((2*(a^9 + 8*a^5*b^4 - 8*a^7*b^2))/a^8 + (2*tan(c/2 + (d*x)/2)*(5 *a^7*b + 16*a^3*b^5 - 20*a^5*b^3))/a^7 + ((2*a^2*b - (2*tan(c/2 + (d*x)/2) *(3*a^10 - 4*a^8*b^2))/a^7)*(b^2 - a^2)^(1/2)*(a^2 - 4*b^2))/a^5)*1i)/a^5 + ((b^2 - a^2)^(1/2)*(a^2 - 4*b^2)*((2*(a^9 + 8*a^5*b^4 - 8*a^7*b^2))/a^8 + (2*tan(c/2 + (d*x)/2)*(5*a^7*b + 16*a^3*b^5 - 20*a^5*b^3))/a^7 - ((2*a^2 *b - (2*tan(c/2 + (d*x)/2)*(3*a^10 - 4*a^8*b^2))/a^7)*(b^2 - a^2)^(1/2)*(a ^2 - 4*b^2))/a^5)*1i)/a^5)/((4*(3*a^6*b - 16*b^7 + 32*a^2*b^5 - 19*a^4*b^3 ))/a^8 + (4*tan(c/2 + (d*x)/2)*(2*a^6 - 16*b^6 + 28*a^2*b^4 - 14*a^4*b^2)) /a^7 - ((b^2 - a^2)^(1/2)*(a^2 - 4*b^2)*((2*(a^9 + 8*a^5*b^4 - 8*a^7*b^2)) /a^8 + (2*tan(c/2 + (d*x)/2)*(5*a^7*b + 16*a^3*b^5 - 20*a^5*b^3))/a^7 + (( 2*a^2*b - (2*tan(c/2 + (d*x)/2)*(3*a^10 - 4*a^8*b^2))/a^7)*(b^2 - a^2)^(1/ 2)*(a^2 - 4*b^2))/a^5))/a^5 + ((b^2 - a^2)^(1/2)*(a^2 - 4*b^2)*((2*(a^9 + 8*a^5*b^4 - 8*a^7*b^2))/a^8 + (2*tan(c/2 + (d*x)/2)*(5*a^7*b + 16*a^3*b...