Integrand size = 23, antiderivative size = 76 \[ \int \frac {\csc ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx=-\frac {a \sqrt {b} \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{(a+b)^{5/2} f}-\frac {a \cot (e+f x)}{(a+b)^2 f}-\frac {\cot ^3(e+f x)}{3 (a+b) f} \] Output:
-a*b^(1/2)*arctan(b^(1/2)*tan(f*x+e)/(a+b)^(1/2))/(a+b)^(5/2)/f-a*cot(f*x+ e)/(a+b)^2/f-1/3*cot(f*x+e)^3/(a+b)/f
Result contains complex when optimal does not.
Time = 1.97 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.97 \[ \int \frac {\csc ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {(a+2 b+a \cos (2 (e+f x))) \sec ^2(e+f x) \left (3 a b \arctan \left (\frac {\sec (f x) (\cos (2 e)-i \sin (2 e)) (-((a+2 b) \sin (f x))+a \sin (2 e+f x))}{2 \sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}}\right ) (\cos (2 e)-i \sin (2 e))+\frac {1}{4} \sqrt {a+b} \csc (e) \csc ^3(e+f x) \sqrt {b (\cos (e)-i \sin (e))^4} (6 a \sin (f x)-3 b \sin (2 e+f x)+(-2 a+b) \sin (2 e+3 f x))\right )}{6 (a+b)^{5/2} f \left (a+b \sec ^2(e+f x)\right ) \sqrt {b (\cos (e)-i \sin (e))^4}} \] Input:
Integrate[Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2),x]
Output:
((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(3*a*b*ArcTan[(Sec[f*x]*(Co s[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]) + (Sqrt[a + b]*Csc[e]*Csc[e + f*x]^3*Sqrt[b*(Cos[e] - I*Sin[e])^4]*(6*a*Sin[f*x] - 3*b *Sin[2*e + f*x] + (-2*a + b)*Sin[2*e + 3*f*x]))/4))/(6*(a + b)^(5/2)*f*(a + b*Sec[e + f*x]^2)*Sqrt[b*(Cos[e] - I*Sin[e])^4])
Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 4620, 359, 264, 218}
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.
\(\displaystyle \int \frac {\csc ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (e+f x)^4 \left (a+b \sec (e+f x)^2\right )}dx\) |
\(\Big \downarrow \) 4620 |
\(\displaystyle \frac {\int \frac {\cot ^4(e+f x) \left (\tan ^2(e+f x)+1\right )}{b \tan ^2(e+f x)+a+b}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 359 |
\(\displaystyle \frac {\frac {a \int \frac {\cot ^2(e+f x)}{b \tan ^2(e+f x)+a+b}d\tan (e+f x)}{a+b}-\frac {\cot ^3(e+f x)}{3 (a+b)}}{f}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {\frac {a \left (-\frac {b \int \frac {1}{b \tan ^2(e+f x)+a+b}d\tan (e+f x)}{a+b}-\frac {\cot (e+f x)}{a+b}\right )}{a+b}-\frac {\cot ^3(e+f x)}{3 (a+b)}}{f}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\frac {a \left (-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}-\frac {\cot (e+f x)}{a+b}\right )}{a+b}-\frac {\cot ^3(e+f x)}{3 (a+b)}}{f}\) |
Input:
Int[Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2),x]
Output:
(-1/3*Cot[e + f*x]^3/(a + b) + (a*(-((Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x] )/Sqrt[a + b]])/(a + b)^(3/2)) - Cot[e + f*x]/(a + b)))/(a + b))/f
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1)/f Subst[Int[x^m*(ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p/(1 + f f^2*x^2)^(m/2 + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]
Time = 0.59 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {-\frac {1}{3 \left (a +b \right ) \tan \left (f x +e \right )^{3}}-\frac {a}{\left (a +b \right )^{2} \tan \left (f x +e \right )}-\frac {a b \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{\left (a +b \right )^{2} \sqrt {\left (a +b \right ) b}}}{f}\) | \(69\) |
default | \(\frac {-\frac {1}{3 \left (a +b \right ) \tan \left (f x +e \right )^{3}}-\frac {a}{\left (a +b \right )^{2} \tan \left (f x +e \right )}-\frac {a b \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{\left (a +b \right )^{2} \sqrt {\left (a +b \right ) b}}}{f}\) | \(69\) |
risch | \(\frac {2 i \left (3 b \,{\mathrm e}^{4 i \left (f x +e \right )}+6 a \,{\mathrm e}^{2 i \left (f x +e \right )}-2 a +b \right )}{3 f \left (a +b \right )^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3}}+\frac {\sqrt {-\left (a +b \right ) b}\, a \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-\left (a +b \right ) b}+a +2 b}{a}\right )}{2 \left (a +b \right )^{3} f}-\frac {\sqrt {-\left (a +b \right ) b}\, a \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-\left (a +b \right ) b}-a -2 b}{a}\right )}{2 \left (a +b \right )^{3} f}\) | \(158\) |
Input:
int(csc(f*x+e)^4/(a+b*sec(f*x+e)^2),x,method=_RETURNVERBOSE)
Output:
1/f*(-1/3/(a+b)/tan(f*x+e)^3-a/(a+b)^2/tan(f*x+e)-a*b/(a+b)^2/((a+b)*b)^(1 /2)*arctan(b*tan(f*x+e)/((a+b)*b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (66) = 132\).
Time = 0.11 (sec) , antiderivative size = 397, normalized size of antiderivative = 5.22 \[ \int \frac {\csc ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\left [-\frac {4 \, {\left (2 \, a - b\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (a \cos \left (f x + e\right )^{2} - a\right )} \sqrt {-\frac {b}{a + b}} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (a b + b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {b}{a + b}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right ) \sin \left (f x + e\right ) - 12 \, a \cos \left (f x + e\right )}{12 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} f \cos \left (f x + e\right )^{2} - {\left (a^{2} + 2 \, a b + b^{2}\right )} f\right )} \sin \left (f x + e\right )}, -\frac {2 \, {\left (2 \, a - b\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (a \cos \left (f x + e\right )^{2} - a\right )} \sqrt {\frac {b}{a + b}} \arctan \left (\frac {{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {b}{a + b}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 6 \, a \cos \left (f x + e\right )}{6 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} f \cos \left (f x + e\right )^{2} - {\left (a^{2} + 2 \, a b + b^{2}\right )} f\right )} \sin \left (f x + e\right )}\right ] \] Input:
integrate(csc(f*x+e)^4/(a+b*sec(f*x+e)^2),x, algorithm="fricas")
Output:
[-1/12*(4*(2*a - b)*cos(f*x + e)^3 - 3*(a*cos(f*x + e)^2 - a)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f *x + e)^2 + b^2))*sin(f*x + e) - 12*a*cos(f*x + e))/(((a^2 + 2*a*b + b^2)* f*cos(f*x + e)^2 - (a^2 + 2*a*b + b^2)*f)*sin(f*x + e)), -1/6*(2*(2*a - b) *cos(f*x + e)^3 - 3*(a*cos(f*x + e)^2 - a)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))* sin(f*x + e) - 6*a*cos(f*x + e))/(((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 - (a^2 + 2*a*b + b^2)*f)*sin(f*x + e))]
\[ \int \frac {\csc ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\int \frac {\csc ^{4}{\left (e + f x \right )}}{a + b \sec ^{2}{\left (e + f x \right )}}\, dx \] Input:
integrate(csc(f*x+e)**4/(a+b*sec(f*x+e)**2),x)
Output:
Integral(csc(e + f*x)**4/(a + b*sec(e + f*x)**2), x)
Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx=-\frac {\frac {3 \, a b \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {{\left (a + b\right )} b}} + \frac {3 \, a \tan \left (f x + e\right )^{2} + a + b}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{3}}}{3 \, f} \] Input:
integrate(csc(f*x+e)^4/(a+b*sec(f*x+e)^2),x, algorithm="maxima")
Output:
-1/3*(3*a*b*arctan(b*tan(f*x + e)/sqrt((a + b)*b))/((a^2 + 2*a*b + b^2)*sq rt((a + b)*b)) + (3*a*tan(f*x + e)^2 + a + b)/((a^2 + 2*a*b + b^2)*tan(f*x + e)^3))/f
Time = 0.15 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.36 \[ \int \frac {\csc ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx=-\frac {\frac {3 \, {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )\right )} a b}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b + b^{2}}} + \frac {3 \, a \tan \left (f x + e\right )^{2} + a + b}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{3}}}{3 \, f} \] Input:
integrate(csc(f*x+e)^4/(a+b*sec(f*x+e)^2),x, algorithm="giac")
Output:
-1/3*(3*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt( a*b + b^2)))*a*b/((a^2 + 2*a*b + b^2)*sqrt(a*b + b^2)) + (3*a*tan(f*x + e) ^2 + a + b)/((a^2 + 2*a*b + b^2)*tan(f*x + e)^3))/f
Time = 12.42 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.05 \[ \int \frac {\csc ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx=-\frac {\frac {1}{3\,\left (a+b\right )}+\frac {a\,{\mathrm {tan}\left (e+f\,x\right )}^2}{{\left (a+b\right )}^2}}{f\,{\mathrm {tan}\left (e+f\,x\right )}^3}-\frac {a\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\mathrm {tan}\left (e+f\,x\right )\,\left (a^2+2\,a\,b+b^2\right )}{{\left (a+b\right )}^{5/2}}\right )}{f\,{\left (a+b\right )}^{5/2}} \] Input:
int(1/(sin(e + f*x)^4*(a + b/cos(e + f*x)^2)),x)
Output:
- (1/(3*(a + b)) + (a*tan(e + f*x)^2)/(a + b)^2)/(f*tan(e + f*x)^3) - (a*b ^(1/2)*atan((b^(1/2)*tan(e + f*x)*(2*a*b + a^2 + b^2))/(a + b)^(5/2)))/(f* (a + b)^(5/2))
Time = 0.16 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.68 \[ \int \frac {\csc ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {-3 \sqrt {b}\, \sqrt {a +b}\, \mathit {atan} \left (\frac {\sqrt {a +b}\, \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\sqrt {a}}{\sqrt {b}}\right ) \sin \left (f x +e \right )^{3} a -3 \sqrt {b}\, \sqrt {a +b}\, \mathit {atan} \left (\frac {\sqrt {a +b}\, \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\sqrt {a}}{\sqrt {b}}\right ) \sin \left (f x +e \right )^{3} a -2 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a^{2}-\cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a b +\cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b^{2}-\cos \left (f x +e \right ) a^{2}-2 \cos \left (f x +e \right ) a b -\cos \left (f x +e \right ) b^{2}}{3 \sin \left (f x +e \right )^{3} f \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )} \] Input:
int(csc(f*x+e)^4/(a+b*sec(f*x+e)^2),x)
Output:
( - 3*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sq rt(b))*sin(e + f*x)**3*a - 3*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) + sqrt(a))/sqrt(b))*sin(e + f*x)**3*a - 2*cos(e + f*x)*sin(e + f *x)**2*a**2 - cos(e + f*x)*sin(e + f*x)**2*a*b + cos(e + f*x)*sin(e + f*x) **2*b**2 - cos(e + f*x)*a**2 - 2*cos(e + f*x)*a*b - cos(e + f*x)*b**2)/(3* sin(e + f*x)**3*f*(a**3 + 3*a**2*b + 3*a*b**2 + b**3))