Integrand size = 21, antiderivative size = 73 \[ \int \frac {\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {x}{8 a}-\frac {\cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac {\sin ^3(c+d x)}{3 a d} \]
Time = 0.42 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.14 \[ \int \frac {\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (24 \sin (c+d x)-8 \sin (3 (c+d x))+3 \left (4 c-4 d x+\sin (4 (c+d x))-4 \tan \left (\frac {c}{2}\right )\right )\right )}{48 a d (1+\sec (c+d x))} \]
(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(24*Sin[c + d*x] - 8*Sin[3*(c + d*x)] + 3 *(4*c - 4*d*x + Sin[4*(c + d*x)] - 4*Tan[c/2])))/(48*a*d*(1 + Sec[c + d*x] ))
Time = 0.53 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 4360, 25, 25, 3042, 3318, 3042, 3044, 15, 3048, 3042, 3115, 24}
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.
\(\displaystyle \int \frac {\sin ^4(c+d x)}{a \sec (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^4}{a-a \csc \left (c+d x-\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int -\frac {\sin ^4(c+d x) \cos (c+d x)}{a (-\cos (c+d x))-a}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\frac {\cos (c+d x) \sin ^4(c+d x)}{\cos (c+d x) a+a}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\sin ^4(c+d x) \cos (c+d x)}{a \cos (c+d x)+a}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \cos \left (c+d x+\frac {\pi }{2}\right )^4}{a \sin \left (c+d x+\frac {\pi }{2}\right )+a}dx\) |
\(\Big \downarrow \) 3318 |
\(\displaystyle \frac {\int \cos (c+d x) \sin ^2(c+d x)dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^2(c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \cos (c+d x) \sin (c+d x)^2dx}{a}-\frac {\int \cos (c+d x)^2 \sin (c+d x)^2dx}{a}\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle \frac {\int \sin ^2(c+d x)d\sin (c+d x)}{a d}-\frac {\int \cos (c+d x)^2 \sin (c+d x)^2dx}{a}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\sin ^3(c+d x)}{3 a d}-\frac {\int \cos (c+d x)^2 \sin (c+d x)^2dx}{a}\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {\sin ^3(c+d x)}{3 a d}-\frac {\frac {1}{4} \int \cos ^2(c+d x)dx-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin ^3(c+d x)}{3 a d}-\frac {\frac {1}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\sin ^3(c+d x)}{3 a d}-\frac {\frac {1}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}}{a}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\sin ^3(c+d x)}{3 a d}-\frac {\frac {1}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}}{a}\) |
Sin[c + d*x]^3/(3*a*d) - (-1/4*(Cos[c + d*x]^3*Sin[c + d*x])/d + (x/2 + (C os[c + d*x]*Sin[c + d*x])/(2*d))/4)/a
3.1.67.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 0.60 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.60
method | result | size |
parallelrisch | \(\frac {-12 d x +24 \sin \left (d x +c \right )-8 \sin \left (3 d x +3 c \right )+3 \sin \left (4 d x +4 c \right )}{96 d a}\) | \(44\) |
risch | \(-\frac {x}{8 a}+\frac {\sin \left (d x +c \right )}{4 a d}+\frac {\sin \left (4 d x +4 c \right )}{32 d a}-\frac {\sin \left (3 d x +3 c \right )}{12 d a}\) | \(56\) |
derivativedivides | \(\frac {-\frac {16 \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{64}-\frac {53 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{192}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{192}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(90\) |
default | \(\frac {-\frac {16 \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{64}-\frac {53 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{192}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{192}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(90\) |
norman | \(\frac {-\frac {x}{8 a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 a d}+\frac {53 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 a d}-\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 a}-\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 a}-\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}\) | \(166\) |
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70 \[ \int \frac {\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {3 \, d x - {\left (6 \, \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right )}{24 \, a d} \]
-1/24*(3*d*x - (6*cos(d*x + c)^3 - 8*cos(d*x + c)^2 - 3*cos(d*x + c) + 8)* sin(d*x + c))/(a*d)
\[ \int \frac {\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\sin ^{4}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (65) = 130\).
Time = 0.29 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.68 \[ \int \frac {\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\frac {\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {11 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {53 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a + \frac {4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{12 \, d} \]
1/12*((3*sin(d*x + c)/(cos(d*x + c) + 1) + 11*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 53*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 3*sin(d*x + c)^7/(cos(d *x + c) + 1)^7)/(a + 4*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a *sin(d*x + c)^8/(cos(d*x + c) + 1)^8) - 3*arctan(sin(d*x + c)/(cos(d*x + c ) + 1))/a)/d
Time = 0.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.19 \[ \int \frac {\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {3 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 53 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \]
-1/24*(3*(d*x + c)/a + 2*(3*tan(1/2*d*x + 1/2*c)^7 - 53*tan(1/2*d*x + 1/2* c)^5 - 11*tan(1/2*d*x + 1/2*c)^3 - 3*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*a))/d
Time = 14.00 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.75 \[ \int \frac {\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\sin \left (c+d\,x\right )}{4\,a\,d}-\frac {x}{8\,a}-\frac {\sin \left (3\,c+3\,d\,x\right )}{12\,a\,d}+\frac {\sin \left (4\,c+4\,d\,x\right )}{32\,a\,d} \]