Integrand size = 25, antiderivative size = 162 \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {\sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{2 f}+\frac {b (3 a+2 b) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 a f}-\frac {(3 a+2 b) \cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 a f}-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a f} \]
1/2*(3*a+2*b)*arctanh(b^(1/2)*tan(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2))*b^(1/2) /f+1/2*b*(3*a+2*b)*(a+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e)/a/f-1/3*(3*a+2*b)*c ot(f*x+e)*(a+b*tan(f*x+e)^2)^(3/2)/a/f-1/3*cot(f*x+e)^3*(a+b*tan(f*x+e)^2) ^(5/2)/a/f
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.49 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.09 \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {\sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)} \left (-4 (a+2 b) \cot (e+f x)-2 a \cot (e+f x) \csc ^2(e+f x)+\frac {3 \sqrt {2} (3 a+2 b) \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right )}{\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}+3 b \tan (e+f x)\right )}{6 \sqrt {2} f} \]
(Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*(-4*(a + 2*b)*Cot [e + f*x] - 2*a*Cot[e + f*x]*Csc[e + f*x]^2 + (3*Sqrt[2]*(3*a + 2*b)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])/Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b] + 3*b*Tan[e + f*x]))/(6*Sqrt[2]*f)
Time = 0.31 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.89, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 4146, 359, 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.
\(\displaystyle \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \tan (e+f x)^2\right )^{3/2}}{\sin (e+f x)^4}dx\) |
\(\Big \downarrow \) 4146 |
\(\displaystyle \frac {\int \cot ^4(e+f x) \left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{3/2}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 359 |
\(\displaystyle \frac {\frac {(3 a+2 b) \int \cot ^2(e+f x) \left (b \tan ^2(e+f x)+a\right )^{3/2}d\tan (e+f x)}{3 a}-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a}}{f}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {\frac {(3 a+2 b) \left (3 b \int \sqrt {b \tan ^2(e+f x)+a}d\tan (e+f x)-\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}\right )}{3 a}-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a}}{f}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {\frac {(3 a+2 b) \left (3 b \left (\frac {1}{2} a \int \frac {1}{\sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)+\frac {1}{2} \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )-\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}\right )}{3 a}-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a}}{f}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {\frac {(3 a+2 b) \left (3 b \left (\frac {1}{2} a \int \frac {1}{1-\frac {b \tan ^2(e+f x)}{b \tan ^2(e+f x)+a}}d\frac {\tan (e+f x)}{\sqrt {b \tan ^2(e+f x)+a}}+\frac {1}{2} \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )-\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}\right )}{3 a}-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a}}{f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {(3 a+2 b) \left (3 b \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{2 \sqrt {b}}+\frac {1}{2} \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )-\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}\right )}{3 a}-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a}}{f}\) |
(-1/3*(Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(5/2))/a + ((3*a + 2*b)*(-(Co t[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2)) + 3*b*((a*ArcTanh[(Sqrt[b]*Tan[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)))/(3*a))/f
3.2.14.3.1 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[((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[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]
Leaf count of result is larger than twice the leaf count of optimal. \(578\) vs. \(2(142)=284\).
Time = 5.33 (sec) , antiderivative size = 579, normalized size of antiderivative = 3.57
method | result | size |
default | \(-\frac {\left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}} \left (11 \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, b^{\frac {3}{2}} \cos \left (f x +e \right )^{2} \cot \left (f x +e \right )+9 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {b}}\right ) a b \cos \left (f x +e \right )^{2} \cot \left (f x +e \right )^{2}+6 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {b}}\right ) b^{2} \cos \left (f x +e \right )^{2} \cot \left (f x +e \right )^{2}-4 \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {b}\, a \cos \left (f x +e \right )^{2} \cot \left (f x +e \right )^{3}-3 \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, b^{\frac {3}{2}} \cot \left (f x +e \right )-9 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {b}}\right ) a b \cos \left (f x +e \right ) \cot \left (f x +e \right )^{2}-6 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {b}}\right ) b^{2} \cos \left (f x +e \right ) \cot \left (f x +e \right )^{2}+6 \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {b}\, a \cot \left (f x +e \right )^{3}\right )}{6 f \sqrt {b}\, \left (a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}\right ) \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\) | \(579\) |
-1/6/f/b^(1/2)*(a+b*tan(f*x+e)^2)^(3/2)/(a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(( a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*(11*((a*cos(f*x+e )^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*b^(3/2)*cos(f*x+e)^2*cot(f*x +e)+9*arctanh(1/b^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2) ^(1/2)*(cot(f*x+e)+csc(f*x+e)))*a*b*cos(f*x+e)^2*cot(f*x+e)^2+6*arctanh(1/ b^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+ e)+csc(f*x+e)))*b^2*cos(f*x+e)^2*cot(f*x+e)^2-4*((a*cos(f*x+e)^2-b*cos(f*x +e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*b^(1/2)*a*cos(f*x+e)^2*cot(f*x+e)^3-3*((a *cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*b^(3/2)*cot(f*x+e) -9*arctanh(1/b^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1 /2)*(cot(f*x+e)+csc(f*x+e)))*a*b*cos(f*x+e)*cot(f*x+e)^2-6*arctanh(1/b^(1/ 2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)+cs c(f*x+e)))*b^2*cos(f*x+e)*cot(f*x+e)^2+6*((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b )/(cos(f*x+e)+1)^2)^(1/2)*b^(1/2)*a*cot(f*x+e)^3)
Time = 0.86 (sec) , antiderivative size = 497, normalized size of antiderivative = 3.07 \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\left [\frac {3 \, {\left ({\left (3 \, a + 2 \, b\right )} \cos \left (f x + e\right )^{3} - {\left (3 \, a + 2 \, b\right )} \cos \left (f x + e\right )\right )} \sqrt {b} \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 (4 \, a + 11 \, b\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a + 7 \, b\right )} \cos \left (f x + e\right )^{2} + 3 \, b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{24 \, {\left (f \cos \left (f x + e\right )^{3} - f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}, -\frac {3 \, {\left ({\left (3 \, a + 2 \, b\right )} \cos \left (f x + e\right )^{3} - {\left (3 \, a + 2 \, b\right )} \cos \left (f x + e\right )\right )} \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 ) \sin \left (f x + e\right ) + 2 \, {\left ({\left (4 \, a + 11 \, b\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a + 7 \, b\right )} \cos \left (f x + e\right )^{2} + 3 \, b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{12 \, {\left (f \cos \left (f x + e\right )^{3} - f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}\right ] \]
[1/24*(3*((3*a + 2*b)*cos(f*x + e)^3 - (3*a + 2*b)*cos(f*x + e))*sqrt(b)*l og(((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)*co s(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*si n(f*x + e) - 4*((4*a + 11*b)*cos(f*x + e)^4 - 2*(3*a + 7*b)*cos(f*x + e)^2 + 3*b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((f*cos(f*x + e )^3 - f*cos(f*x + e))*sin(f*x + e)), -1/12*(3*((3*a + 2*b)*cos(f*x + e)^3 - (3*a + 2*b)*cos(f*x + e))*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)))*sin(f*x + e) + 2*( (4*a + 11*b)*cos(f*x + e)^4 - 2*(3*a + 7*b)*cos(f*x + e)^2 + 3*b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((f*cos(f*x + e)^3 - f*cos(f*x + e))*sin(f*x + e))]
Timed out. \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.08 \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {9 \, a \sqrt {b} \operatorname {arsinh}\left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right ) + 6 \, b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right ) + 9 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} b \tan \left (f x + e\right ) + \frac {6 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} b^{2} \tan \left (f x + e\right )}{a} - \frac {6 \, {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}{\tan \left (f x + e\right )} - \frac {4 \, {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} b}{a \tan \left (f x + e\right )} - \frac {2 \, {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}{a \tan \left (f x + e\right )^{3}}}{6 \, f} \]
1/6*(9*a*sqrt(b)*arcsinh(b*tan(f*x + e)/sqrt(a*b)) + 6*b^(3/2)*arcsinh(b*t an(f*x + e)/sqrt(a*b)) + 9*sqrt(b*tan(f*x + e)^2 + a)*b*tan(f*x + e) + 6*s qrt(b*tan(f*x + e)^2 + a)*b^2*tan(f*x + e)/a - 6*(b*tan(f*x + e)^2 + a)^(3 /2)/tan(f*x + e) - 4*(b*tan(f*x + e)^2 + a)^(3/2)*b/(a*tan(f*x + e)) - 2*( b*tan(f*x + e)^2 + a)^(5/2)/(a*tan(f*x + e)^3))/f
\[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \csc \left (f x + e\right )^{4} \,d x } \]
Timed out. \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\sin \left (e+f\,x\right )}^4} \,d x \]