Integrand size = 13, antiderivative size = 34 \[ \int \frac {\sec ^4(x)}{a+a \csc (x)} \, dx=\frac {\sec ^5(x)}{5 a}-\frac {\tan ^3(x)}{3 a}-\frac {\tan ^5(x)}{5 a} \] Output:
1/5*sec(x)^5/a-1/3*tan(x)^3/a-1/5*tan(x)^5/a
Leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(34)=68\).
Time = 0.18 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.50 \[ \int \frac {\sec ^4(x)}{a+a \csc (x)} \, dx=-\frac {-240+54 \cos (x)+32 \cos (2 x)+18 \cos (3 x)+16 \cos (4 x)-96 \sin (x)+18 \sin (2 x)-32 \sin (3 x)+9 \sin (4 x)}{960 a \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^3 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5} \] Input:
Integrate[Sec[x]^4/(a + a*Csc[x]),x]
Output:
-1/960*(-240 + 54*Cos[x] + 32*Cos[2*x] + 18*Cos[3*x] + 16*Cos[4*x] - 96*Si n[x] + 18*Sin[2*x] - 32*Sin[3*x] + 9*Sin[4*x])/(a*(Cos[x/2] - Sin[x/2])^3* (Cos[x/2] + Sin[x/2])^5)
Time = 0.40 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {3042, 4360, 3042, 3318, 3042, 3086, 15, 3087, 244, 2009}
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.
\(\displaystyle \int \frac {\sec ^4(x)}{a \csc (x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\cos (x)^4 (a \csc (x)+a)}dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int \frac {\tan (x) \sec ^3(x)}{a \sin (x)+a}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (x)}{\cos (x)^4 (a \sin (x)+a)}dx\) |
\(\Big \downarrow \) 3318 |
\(\displaystyle \frac {\int \sec ^5(x) \tan (x)dx}{a}-\frac {\int \sec ^4(x) \tan ^2(x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sec (x)^5 \tan (x)dx}{a}-\frac {\int \sec (x)^4 \tan (x)^2dx}{a}\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle \frac {\int \sec ^4(x)d\sec (x)}{a}-\frac {\int \sec (x)^4 \tan (x)^2dx}{a}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\sec ^5(x)}{5 a}-\frac {\int \sec (x)^4 \tan (x)^2dx}{a}\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle \frac {\sec ^5(x)}{5 a}-\frac {\int \tan ^2(x) \left (\tan ^2(x)+1\right )d\tan (x)}{a}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {\sec ^5(x)}{5 a}-\frac {\int \left (\tan ^4(x)+\tan ^2(x)\right )d\tan (x)}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sec ^5(x)}{5 a}-\frac {\frac {\tan ^5(x)}{5}+\frac {\tan ^3(x)}{3}}{a}\) |
Input:
Int[Sec[x]^4/(a + a*Csc[x]),x]
Output:
Sec[x]^5/(5*a) - (Tan[x]^3/3 + Tan[x]^5/5)/a
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 1])
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]
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.74
method | result | size |
risch | \(\frac {4 i \left (6 i {\mathrm e}^{3 i x}+15 \,{\mathrm e}^{4 i x}+2 i {\mathrm e}^{i x}-2 \,{\mathrm e}^{2 i x}-1\right )}{15 \left ({\mathrm e}^{i x}-i\right )^{3} \left ({\mathrm e}^{i x}+i\right )^{5} a}\) | \(59\) |
parallelrisch | \(\frac {-\frac {2}{5}-2 \tan \left (\frac {x}{2}\right )^{6}-\frac {4 \tan \left (\frac {x}{2}\right )^{5}}{3}-\frac {2 \tan \left (\frac {x}{2}\right )^{4}}{3}+\frac {16 \tan \left (\frac {x}{2}\right )^{3}}{15}-\frac {6 \tan \left (\frac {x}{2}\right )^{2}}{5}-\frac {4 \tan \left (\frac {x}{2}\right )}{5}}{a \left (\tan \left (\frac {x}{2}\right )-1\right )^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(70\) |
default | \(\frac {-\frac {1}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {2}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {3}{8 \left (\tan \left (\frac {x}{2}\right )+1\right )}-\frac {1}{6 \left (\tan \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tan \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {3}{8 \left (\tan \left (\frac {x}{2}\right )-1\right )}}{a}\) | \(87\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {x}{2}\right )^{4}}{3 a}-\frac {2}{5 a}-\frac {2 \tan \left (\frac {x}{2}\right )^{6}}{a}-\frac {6 \tan \left (\frac {x}{2}\right )^{2}}{5 a}-\frac {4 \tan \left (\frac {x}{2}\right )}{5 a}-\frac {4 \tan \left (\frac {x}{2}\right )^{5}}{3 a}+\frac {16 \tan \left (\frac {x}{2}\right )^{3}}{15 a}}{\left (\tan \left (\frac {x}{2}\right )-1\right )^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(88\) |
Input:
int(sec(x)^4/(a+a*csc(x)),x,method=_RETURNVERBOSE)
Output:
4/15*I*(6*I*exp(3*I*x)+15*exp(4*I*x)+2*I*exp(I*x)-2*exp(2*I*x)-1)/(exp(I*x )-I)^3/(exp(I*x)+I)^5/a
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.32 \[ \int \frac {\sec ^4(x)}{a+a \csc (x)} \, dx=-\frac {2 \, \cos \left (x\right )^{4} - \cos \left (x\right )^{2} - {\left (2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) - 4}{15 \, {\left (a \cos \left (x\right )^{3} \sin \left (x\right ) + a \cos \left (x\right )^{3}\right )}} \] Input:
integrate(sec(x)^4/(a+a*csc(x)),x, algorithm="fricas")
Output:
-1/15*(2*cos(x)^4 - cos(x)^2 - (2*cos(x)^2 + 1)*sin(x) - 4)/(a*cos(x)^3*si n(x) + a*cos(x)^3)
\[ \int \frac {\sec ^4(x)}{a+a \csc (x)} \, dx=\frac {\int \frac {\sec ^{4}{\left (x \right )}}{\csc {\left (x \right )} + 1}\, dx}{a} \] Input:
integrate(sec(x)**4/(a+a*csc(x)),x)
Output:
Integral(sec(x)**4/(csc(x) + 1), x)/a
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (28) = 56\).
Time = 0.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 4.91 \[ \int \frac {\sec ^4(x)}{a+a \csc (x)} \, dx=\frac {2 \, {\left (\frac {6 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {9 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {8 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {5 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {10 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 3\right )}}{15 \, {\left (a + \frac {2 \, a \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {2 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {6 \, a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {6 \, a \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {2 \, a \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac {2 \, a \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} - \frac {a \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}}\right )}} \] Input:
integrate(sec(x)^4/(a+a*csc(x)),x, algorithm="maxima")
Output:
2/15*(6*sin(x)/(cos(x) + 1) + 9*sin(x)^2/(cos(x) + 1)^2 - 8*sin(x)^3/(cos( x) + 1)^3 + 5*sin(x)^4/(cos(x) + 1)^4 + 10*sin(x)^5/(cos(x) + 1)^5 + 15*si n(x)^6/(cos(x) + 1)^6 + 3)/(a + 2*a*sin(x)/(cos(x) + 1) - 2*a*sin(x)^2/(co s(x) + 1)^2 - 6*a*sin(x)^3/(cos(x) + 1)^3 + 6*a*sin(x)^5/(cos(x) + 1)^5 + 2*a*sin(x)^6/(cos(x) + 1)^6 - 2*a*sin(x)^7/(cos(x) + 1)^7 - a*sin(x)^8/(co s(x) + 1)^8)
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (28) = 56\).
Time = 0.18 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.21 \[ \int \frac {\sec ^4(x)}{a+a \csc (x)} \, dx=-\frac {9 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 12 \, \tan \left (\frac {1}{2} \, x\right ) + 7}{24 \, a {\left (\tan \left (\frac {1}{2} \, x\right ) - 1\right )}^{3}} + \frac {45 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 60 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 70 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 20 \, \tan \left (\frac {1}{2} \, x\right ) + 13}{120 \, a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \] Input:
integrate(sec(x)^4/(a+a*csc(x)),x, algorithm="giac")
Output:
-1/24*(9*tan(1/2*x)^2 - 12*tan(1/2*x) + 7)/(a*(tan(1/2*x) - 1)^3) + 1/120* (45*tan(1/2*x)^4 + 60*tan(1/2*x)^3 + 70*tan(1/2*x)^2 + 20*tan(1/2*x) + 13) /(a*(tan(1/2*x) + 1)^5)
Time = 16.41 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.03 \[ \int \frac {\sec ^4(x)}{a+a \csc (x)} \, dx=-\frac {2\,\left (15\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+9\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+6\,\mathrm {tan}\left (\frac {x}{2}\right )+3\right )}{15\,a\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \] Input:
int(1/(cos(x)^4*(a + a/sin(x))),x)
Output:
-(2*(6*tan(x/2) + 9*tan(x/2)^2 - 8*tan(x/2)^3 + 5*tan(x/2)^4 + 10*tan(x/2) ^5 + 15*tan(x/2)^6 + 3))/(15*a*(tan(x/2) - 1)^3*(tan(x/2) + 1)^5)
Time = 0.16 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.21 \[ \int \frac {\sec ^4(x)}{a+a \csc (x)} \, dx=\frac {3 \cos \left (x \right ) \sin \left (x \right )^{3}+3 \cos \left (x \right ) \sin \left (x \right )^{2}-3 \cos \left (x \right ) \sin \left (x \right )-3 \cos \left (x \right )+2 \sin \left (x \right )^{4}+2 \sin \left (x \right )^{3}-3 \sin \left (x \right )^{2}-3 \sin \left (x \right )-3}{15 \cos \left (x \right ) a \left (\sin \left (x \right )^{3}+\sin \left (x \right )^{2}-\sin \left (x \right )-1\right )} \] Input:
int(sec(x)^4/(a+a*csc(x)),x)
Output:
(3*cos(x)*sin(x)**3 + 3*cos(x)*sin(x)**2 - 3*cos(x)*sin(x) - 3*cos(x) + 2* sin(x)**4 + 2*sin(x)**3 - 3*sin(x)**2 - 3*sin(x) - 3)/(15*cos(x)*a*(sin(x) **3 + sin(x)**2 - sin(x) - 1))