Integrand size = 25, antiderivative size = 100 \[ \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {3 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{2 f}+\frac {3 b \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}-\frac {\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{f} \]
3/2*a*arctanh(b^(1/2)*tan(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2))*b^(1/2)/f+3/2*b *(a+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e)/f-cot(f*x+e)*(a+b*tan(f*x+e)^2)^(3/2) /f
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 3.61 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.20 \[ \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {\csc (e+f x) \sec ^3(e+f x) \left (-6 a^2-a b+3 b^2-4 \left (2 a^2+b^2\right ) \cos (2 (e+f x))-2 a^2 \cos (4 (e+f x))+a b \cos (4 (e+f x))+b^2 \cos (4 (e+f x))+3 \sqrt {2} a b \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right ) \sin ^2(2 (e+f x))\right )}{8 \sqrt {2} f \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)}} \]
(Csc[e + f*x]*Sec[e + f*x]^3*(-6*a^2 - a*b + 3*b^2 - 4*(2*a^2 + b^2)*Cos[2 *(e + f*x)] - 2*a^2*Cos[4*(e + f*x)] + a*b*Cos[4*(e + f*x)] + b^2*Cos[4*(e + f*x)] + 3*Sqrt[2]*a*b*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[2*(e + f*x)]^2))/(8*Sqrt[2]*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])
Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 4146, 247, 211, 224, 219}
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.
\(\displaystyle \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \tan (e+f x)^2\right )^{3/2}}{\sin (e+f x)^2}dx\) |
\(\Big \downarrow \) 4146 |
\(\displaystyle \frac {\int \cot ^2(e+f x) \left (b \tan ^2(e+f x)+a\right )^{3/2}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {3 b \int \sqrt {b \tan ^2(e+f x)+a}d\tan (e+f x)-\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{f}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {3 b \left (\frac {1}{2} a \int \frac {1}{\sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)+\frac {1}{2} \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )-\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{f}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {3 b \left (\frac {1}{2} a \int \frac {1}{1-\frac {b \tan ^2(e+f x)}{b \tan ^2(e+f x)+a}}d\frac {\tan (e+f x)}{\sqrt {b \tan ^2(e+f x)+a}}+\frac {1}{2} \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )-\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3 b \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{2 \sqrt {b}}+\frac {1}{2} \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )-\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{f}\) |
(-(Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2)) + 3*b*((a*ArcTanh[(Sqrt[b]*T an[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*Sqrt[b]) + (Tan[e + f*x]*Sqrt [a + b*Tan[e + f*x]^2])/2))/f
3.2.13.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
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]}, Sim p[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]
Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.21
method | result | size |
derivativedivides | \(-\frac {\left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}{f a \tan \left (f x +e \right )}+\frac {b \tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{f a}+\frac {3 b \sqrt {a +b \tan \left (f x +e \right )^{2}}\, \tan \left (f x +e \right )}{2 f}+\frac {3 \sqrt {b}\, a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 f}\) | \(121\) |
default | \(-\frac {\left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}{f a \tan \left (f x +e \right )}+\frac {b \tan \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{f a}+\frac {3 b \sqrt {a +b \tan \left (f x +e \right )^{2}}\, \tan \left (f x +e \right )}{2 f}+\frac {3 \sqrt {b}\, a \ln \left (\sqrt {b}\, \tan \left (f x +e \right )+\sqrt {a +b \tan \left (f x +e \right )^{2}}\right )}{2 f}\) | \(121\) |
-1/f/a/tan(f*x+e)*(a+b*tan(f*x+e)^2)^(5/2)+1/f*b/a*tan(f*x+e)*(a+b*tan(f*x +e)^2)^(3/2)+3/2*b*(a+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e)/f+3/2/f*b^(1/2)*a*l n(b^(1/2)*tan(f*x+e)+(a+b*tan(f*x+e)^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (86) = 172\).
Time = 0.46 (sec) , antiderivative size = 387, normalized size of antiderivative = 3.87 \[ \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\left [\frac {3 \, a \sqrt {b} \cos \left (f x + e\right ) \log \left (\frac {{\left (a^{2} - 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 8 \, {\left (a b - 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a - 2 \, b\right )} \cos \left (f x + e\right )^{3} + 2 \, b \cos \left (f x + e\right )\right )} \sqrt {b} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right ) + 8 \, b^{2}}{\cos \left (f x + e\right )^{4}}\right ) \sin \left (f x + e\right ) - 4 \, {\left ({\left (2 \, a + b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{8 \, f \cos \left (f x + e\right ) \sin \left (f x + e\right )}, -\frac {3 \, a \sqrt {-b} \arctan \left (\frac {{\left ({\left (a - 2 \, b\right )} \cos \left (f x + e\right )^{3} + 2 \, b \cos \left (f x + e\right )\right )} \sqrt {-b} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{2 \, {\left ({\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}\right )} \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 2 \, {\left ({\left (2 \, a + b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{4 \, f \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ] \]
[1/8*(3*a*sqrt(b)*cos(f*x + e)*log(((a^2 - 8*a*b + 8*b^2)*cos(f*x + e)^4 + 8*(a*b - 2*b^2)*cos(f*x + e)^2 + 4*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f* x + e))*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*sin(f*x + e) - 4*((2*a + b)*cos(f*x + e)^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)*sin (f*x + e)), -1/4*(3*a*sqrt(-b)*arctan(1/2*((a - 2*b)*cos(f*x + e)^3 + 2*b* cos(f*x + e))*sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/( ((a*b - b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)*sin(f*x + e ) + 2*((2*a + b)*cos(f*x + e)^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos (f*x + e)^2))/(f*cos(f*x + e)*sin(f*x + e))]
\[ \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \csc ^{2}{\left (e + f x \right )}\, dx \]
Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.73 \[ \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {3 \, a \sqrt {b} \operatorname {arsinh}\left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right ) + 3 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} b \tan \left (f x + e\right ) - \frac {2 \, {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}{\tan \left (f x + e\right )}}{2 \, f} \]
1/2*(3*a*sqrt(b)*arcsinh(b*tan(f*x + e)/sqrt(a*b)) + 3*sqrt(b*tan(f*x + e) ^2 + a)*b*tan(f*x + e) - 2*(b*tan(f*x + e)^2 + a)^(3/2)/tan(f*x + e))/f
\[ \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \csc \left (f x + e\right )^{2} \,d x } \]
Timed out. \[ \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\sin \left (e+f\,x\right )}^2} \,d x \]