\(\int \frac {\csc ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx\) [66]

Optimal result

Integrand size = 23, antiderivative size = 76 \[ \int \frac {\csc ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx=-\frac {(a-b) \sqrt {b} \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{a^{5/2} f}-\frac {(a-b) \cot (e+f x)}{a^2 f}-\frac {\cot ^3(e+f x)}{3 a f} \]

[Out]

-(a-b)*cot(f*x+e)/a^2/f-1/3*cot(f*x+e)^3/a/f-(a-b)*arctan(b^(1/2)*tan(f*x+e)/a^(1/2))*b^(1/2)/a^(5/2)/f

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3744, 464, 331, 211} \[ \int \frac {\csc ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx=-\frac {\sqrt {b} (a-b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{a^{5/2} f}-\frac {(a-b) \cot (e+f x)}{a^2 f}-\frac {\cot ^3(e+f x)}{3 a f} \]

[In]

Int[Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2),x]

[Out]

-(((a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(5/2)*f)) - ((a - b)*Cot[e + f*x])/(a^2*f) - Cot
[e + f*x]^3/(3*a*f)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 464

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 3744

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff^(m + 1)/f), Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)
^(m/2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1+x^2}{x^4 \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {\cot ^3(e+f x)}{3 a f}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{a f} \\ & = -\frac {(a-b) \cot (e+f x)}{a^2 f}-\frac {\cot ^3(e+f x)}{3 a f}-\frac {((a-b) b) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{a^2 f} \\ & = -\frac {(a-b) \sqrt {b} \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{a^{5/2} f}-\frac {(a-b) \cot (e+f x)}{a^2 f}-\frac {\cot ^3(e+f x)}{3 a f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.96 \[ \int \frac {\csc ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx=\frac {3 \sqrt {b} (-a+b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )-\sqrt {a} \cot (e+f x) \left (2 a-3 b+a \csc ^2(e+f x)\right )}{3 a^{5/2} f} \]

[In]

Integrate[Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2),x]

[Out]

(3*Sqrt[b]*(-a + b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] - Sqrt[a]*Cot[e + f*x]*(2*a - 3*b + a*Csc[e + f*x]^
2))/(3*a^(5/2)*f)

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {-\frac {1}{3 a \tan \left (f x +e \right )^{3}}-\frac {a -b}{a^{2} \tan \left (f x +e \right )}-\frac {b \left (a -b \right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}}{f}\)	\(67\)
default	\(\frac {-\frac {1}{3 a \tan \left (f x +e \right )^{3}}-\frac {a -b}{a^{2} \tan \left (f x +e \right )}-\frac {b \left (a -b \right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}}{f}\)	\(67\)
risch	\(\frac {2 i \left (3 b \,{\mathrm e}^{4 i \left (f x +e \right )}+6 a \,{\mathrm e}^{2 i \left (f x +e \right )}-6 b \,{\mathrm e}^{2 i \left (f x +e \right )}-2 a +3 b \right )}{3 f \,a^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3}}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right )}{2 a^{2} f}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right ) b}{2 a^{3} f}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right )}{2 a^{2} f}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right ) b}{2 a^{3} f}\)	\(259\)

[In]

int(csc(f*x+e)^4/(a+b*tan(f*x+e)^2),x,method=_RETURNVERBOSE)

[Out]

1/f*(-1/3/a/tan(f*x+e)^3-(a-b)/a^2/tan(f*x+e)-b*(a-b)/a^2/(a*b)^(1/2)*arctan(b*tan(f*x+e)/(a*b)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (66) = 132\).

Time = 0.30 (sec) , antiderivative size = 373, normalized size of antiderivative = 4.91 \[ \int \frac {\csc ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx=\left [-\frac {4 \, {\left (2 \, a - 3 \, b\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left ({\left (a - b\right )} \cos \left (f x + e\right )^{2} - a + b\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {{\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left ({\left (a^{2} + a b\right )} \cos \left (f x + e\right )^{3} - a b \cos \left (f x + e\right )\right )} \sqrt {-\frac {b}{a}} \sin \left (f x + e\right ) + b^{2}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}}\right ) \sin \left (f x + e\right ) - 12 \, {\left (a - b\right )} \cos \left (f x + e\right )}{12 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f\right )} \sin \left (f x + e\right )}, -\frac {2 \, {\left (2 \, a - 3 \, b\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left ({\left (a - b\right )} \cos \left (f x + e\right )^{2} - a + b\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {b}{a}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 6 \, {\left (a - b\right )} \cos \left (f x + e\right )}{6 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f\right )} \sin \left (f x + e\right )}\right ] \]

[In]

integrate(csc(f*x+e)^4/(a+b*tan(f*x+e)^2),x, algorithm="fricas")

[Out]

[-1/12*(4*(2*a - 3*b)*cos(f*x + e)^3 + 3*((a - b)*cos(f*x + e)^2 - a + b)*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*
cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 - 4*((a^2 + a*b)*cos(f*x + e)^3 - a*b*cos(f*x + e))*sqrt(-b/a)
*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2))*sin(f*x + e) -
12*(a - b)*cos(f*x + e))/((a^2*f*cos(f*x + e)^2 - a^2*f)*sin(f*x + e)), -1/6*(2*(2*a - 3*b)*cos(f*x + e)^3 -
3*((a - b)*cos(f*x + e)^2 - a + b)*sqrt(b/a)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(b/a)/(b*cos(f*x + e)
*sin(f*x + e)))*sin(f*x + e) - 6*(a - b)*cos(f*x + e))/((a^2*f*cos(f*x + e)^2 - a^2*f)*sin(f*x + e))]

Sympy [F]

\[ \int \frac {\csc ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx=\int \frac {\csc ^{4}{\left (e + f x \right )}}{a + b \tan ^{2}{\left (e + f x \right )}}\, dx \]

[In]

integrate(csc(f*x+e)**4/(a+b*tan(f*x+e)**2),x)

[Out]

Integral(csc(e + f*x)**4/(a + b*tan(e + f*x)**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89 \[ \int \frac {\csc ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx=-\frac {\frac {3 \, {\left (a b - b^{2}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, {\left (a - b\right )} \tan \left (f x + e\right )^{2} + a}{a^{2} \tan \left (f x + e\right )^{3}}}{3 \, f} \]

[In]

integrate(csc(f*x+e)^4/(a+b*tan(f*x+e)^2),x, algorithm="maxima")

[Out]

-1/3*(3*(a*b - b^2)*arctan(b*tan(f*x + e)/sqrt(a*b))/(sqrt(a*b)*a^2) + (3*(a - b)*tan(f*x + e)^2 + a)/(a^2*tan
(f*x + e)^3))/f

Giac [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.21 \[ \int \frac {\csc ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx=-\frac {\frac {3 \, {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )\right )} {\left (a b - b^{2}\right )}}{\sqrt {a b} a^{2}} + \frac {3 \, a \tan \left (f x + e\right )^{2} - 3 \, b \tan \left (f x + e\right )^{2} + a}{a^{2} \tan \left (f x + e\right )^{3}}}{3 \, f} \]

[In]

integrate(csc(f*x+e)^4/(a+b*tan(f*x+e)^2),x, algorithm="giac")

[Out]

-1/3*(3*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*b)))*(a*b - b^2)/(sqrt(a*b)*a^2) +
(3*a*tan(f*x + e)^2 - 3*b*tan(f*x + e)^2 + a)/(a^2*tan(f*x + e)^3))/f

Mupad [B] (verification not implemented)

Time = 10.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.88 \[ \int \frac {\csc ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx=-\frac {\frac {1}{3\,a}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (a-b\right )}{a^2}}{f\,{\mathrm {tan}\left (e+f\,x\right )}^3}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\mathrm {tan}\left (e+f\,x\right )}{\sqrt {a}}\right )\,\left (a-b\right )}{a^{5/2}\,f} \]

[In]

int(1/(sin(e + f*x)^4*(a + b*tan(e + f*x)^2)),x)

[Out]

- (1/(3*a) + (tan(e + f*x)^2*(a - b))/a^2)/(f*tan(e + f*x)^3) - (b^(1/2)*atan((b^(1/2)*tan(e + f*x))/a^(1/2))*
(a - b))/(a^(5/2)*f)