Integrand size = 25, antiderivative size = 322 \[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{7/2}}-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{7/2}}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\frac {4 a b}{3 d e^2 (e \cot (c+d x))^{3/2}}-\frac {2 \left (a^2-b^2\right )}{d e^3 \sqrt {e \cot (c+d x)}}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{7/2}}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{7/2}} \]
2/5*a^2/d/e/(e*cot(d*x+c))^(5/2)+4/3*a*b/d/e^2/(e*cot(d*x+c))^(3/2)+1/2*(a ^2-2*a*b-b^2)*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/d/e^(7/2)*2^( 1/2)-1/2*(a^2-2*a*b-b^2)*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/d/ e^(7/2)*2^(1/2)-1/4*(a^2+2*a*b-b^2)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)-2^(1/2)* (e*cot(d*x+c))^(1/2))/d/e^(7/2)*2^(1/2)+1/4*(a^2+2*a*b-b^2)*ln(e^(1/2)+cot (d*x+c)*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))/d/e^(7/2)*2^(1/2)-2*(a^2-b^2 )/d/e^3/(e*cot(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.41 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.26 \[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\frac {2 \left (3 \left (a^2-b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},1,-\frac {1}{4},-\cot ^2(c+d x)\right )+b \left (3 b+10 a \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\cot ^2(c+d x)\right )\right )\right )}{15 d e (e \cot (c+d x))^{5/2}} \]
(2*(3*(a^2 - b^2)*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2] + b*(3 *b + 10*a*Cot[c + d*x]*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2]))) /(15*d*e*(e*Cot[c + d*x])^(5/2))
Time = 0.98 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.93, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.760, Rules used = {3042, 4025, 3042, 4012, 25, 3042, 4012, 3042, 4017, 25, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
\(\displaystyle \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
\(\Big \downarrow \) 4025 |
\(\displaystyle \frac {\int \frac {2 a b e-\left (a^2-b^2\right ) e \cot (c+d x)}{(e \cot (c+d x))^{5/2}}dx}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {2 a b e+\left (a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) e}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \frac {\frac {\int -\frac {\left (a^2-b^2\right ) e^2+2 a b \cot (c+d x) e^2}{(e \cot (c+d x))^{3/2}}dx}{e^2}+\frac {4 a b}{3 d (e \cot (c+d x))^{3/2}}}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {4 a b}{3 d (e \cot (c+d x))^{3/2}}-\frac {\int \frac {\left (a^2-b^2\right ) e^2+2 a b \cot (c+d x) e^2}{(e \cot (c+d x))^{3/2}}dx}{e^2}}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {4 a b}{3 d (e \cot (c+d x))^{3/2}}-\frac {\int \frac {\left (a^2-b^2\right ) e^2-2 a b e^2 \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{e^2}}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \frac {\frac {4 a b}{3 d (e \cot (c+d x))^{3/2}}-\frac {\frac {\int \frac {2 a b e^3-\left (a^2-b^2\right ) e^3 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}dx}{e^2}+\frac {2 e \left (a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}}{e^2}}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {4 a b}{3 d (e \cot (c+d x))^{3/2}}-\frac {\frac {\int \frac {2 a b e^3+\left (a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) e^3}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{e^2}+\frac {2 e \left (a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}}{e^2}}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 4017 |
\(\displaystyle \frac {\frac {4 a b}{3 d (e \cot (c+d x))^{3/2}}-\frac {\frac {2 \int -\frac {e^3 \left (2 a b e-\left (a^2-b^2\right ) e \cot (c+d x)\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 e \left (a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}}{e^2}}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {4 a b}{3 d (e \cot (c+d x))^{3/2}}-\frac {\frac {2 e \left (a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}-\frac {2 \int \frac {e^3 \left (2 a b e-\left (a^2-b^2\right ) e \cot (c+d x)\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d e^2}}{e^2}}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {4 a b}{3 d (e \cot (c+d x))^{3/2}}-\frac {\frac {2 e \left (a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}-\frac {2 e \int \frac {2 a b e-\left (a^2-b^2\right ) e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d}}{e^2}}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 1482 |
\(\displaystyle \frac {\frac {4 a b}{3 d (e \cot (c+d x))^{3/2}}-\frac {\frac {2 e \left (a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}-\frac {2 e \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d}}{e^2}}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {\frac {4 a b}{3 d (e \cot (c+d x))^{3/2}}-\frac {\frac {2 e \left (a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}-\frac {2 e \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )\right )}{d}}{e^2}}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\frac {4 a b}{3 d (e \cot (c+d x))^{3/2}}-\frac {\frac {2 e \left (a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}-\frac {2 e \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}}{e^2}}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\frac {4 a b}{3 d (e \cot (c+d x))^{3/2}}-\frac {\frac {2 e \left (a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}-\frac {2 e \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}}{e^2}}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {\frac {4 a b}{3 d (e \cot (c+d x))^{3/2}}-\frac {\frac {2 e \left (a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}-\frac {2 e \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}}{e^2}}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {4 a b}{3 d (e \cot (c+d x))^{3/2}}-\frac {\frac {2 e \left (a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}-\frac {2 e \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}}{e^2}}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {4 a b}{3 d (e \cot (c+d x))^{3/2}}-\frac {\frac {2 e \left (a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}-\frac {2 e \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}}{e^2}}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\frac {4 a b}{3 d (e \cot (c+d x))^{3/2}}-\frac {\frac {2 e \left (a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}-\frac {2 e \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}}{e^2}}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
(2*a^2)/(5*d*e*(e*Cot[c + d*x])^(5/2)) + ((4*a*b)/(3*d*(e*Cot[c + d*x])^(3 /2)) - ((2*(a^2 - b^2)*e)/(d*Sqrt[e*Cot[c + d*x]]) - (2*e*(-1/2*((a^2 - 2* a*b - b^2)*(-(ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2]* Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sq rt[e]))) + ((a^2 + 2*a*b - b^2)*(-1/2*Log[e + e*Cot[c + d*x] - Sqrt[2]*Sqr t[e]*Sqrt[e*Cot[c + d*x]]]/(Sqrt[2]*Sqrt[e]) + Log[e + e*Cot[c + d*x] + Sq rt[2]*Sqrt[e]*Sqrt[e*Cot[c + d*x]]]/(2*Sqrt[2]*Sqrt[e])))/2))/d)/e^2)/e^2
3.1.61.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Time = 0.04 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {-\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}+\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{e^{2}}-\frac {a^{2}}{5 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {-a^{2}+b^{2}}{e^{2} \sqrt {e \cot \left (d x +c \right )}}-\frac {2 a b}{3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d e}\) | \(347\) |
default | \(-\frac {2 \left (\frac {-\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}+\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{e^{2}}-\frac {a^{2}}{5 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {-a^{2}+b^{2}}{e^{2} \sqrt {e \cot \left (d x +c \right )}}-\frac {2 a b}{3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d e}\) | \(347\) |
parts | \(-\frac {2 a^{2} e \left (-\frac {1}{5 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {1}{e^{4} \sqrt {e \cot \left (d x +c \right )}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{4} \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}-\frac {2 b^{2} \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2} \left (e^{2}\right )^{\frac {1}{4}}}-\frac {1}{e^{2} \sqrt {e \cot \left (d x +c \right )}}\right )}{d e}+\frac {2 a b \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e^{4}}+\frac {2}{3 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d}\) | \(493\) |
-2/d/e*(1/e^2*(-1/4*a/e*b*(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)+(e^2)^(1/4 )*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)-(e^2)^(1/4)*(e*c ot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot (d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1))+1 /8*(a^2-b^2)/(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+ c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1 /2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2 )+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)))-1/5*a^2/(e*co t(d*x+c))^(5/2)-(-a^2+b^2)/e^2/(e*cot(d*x+c))^(1/2)-2/3*a*b/e/(e*cot(d*x+c ))^(3/2))
Leaf count of result is larger than twice the leaf count of optimal. 1436 vs. \(2 (265) = 530\).
Time = 0.32 (sec) , antiderivative size = 1436, normalized size of antiderivative = 4.46 \[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\text {Too large to display} \]
-1/30*(15*(d*e^4*cos(2*d*x + 2*c)^2 + 2*d*e^4*cos(2*d*x + 2*c) + d*e^4)*sq rt((d^2*e^7*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/(d^4* e^14)) + 4*a^3*b - 4*a*b^3)/(d^2*e^7))*log((a^8 - 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) + ((a^2 - b^2)*d^3*e^11*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/( d^4*e^14)) - 2*(a^5*b - 6*a^3*b^3 + a*b^5)*d*e^4)*sqrt((d^2*e^7*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/(d^4*e^14)) + 4*a^3*b - 4*a *b^3)/(d^2*e^7))) - 15*(d*e^4*cos(2*d*x + 2*c)^2 + 2*d*e^4*cos(2*d*x + 2*c ) + d*e^4)*sqrt((d^2*e^7*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/(d^4*e^14)) + 4*a^3*b - 4*a*b^3)/(d^2*e^7))*log((a^8 - 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2 *c)) - ((a^2 - b^2)*d^3*e^11*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2 *b^6 + b^8)/(d^4*e^14)) - 2*(a^5*b - 6*a^3*b^3 + a*b^5)*d*e^4)*sqrt((d^2*e ^7*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/(d^4*e^14)) + 4*a^3*b - 4*a*b^3)/(d^2*e^7))) - 15*(d*e^4*cos(2*d*x + 2*c)^2 + 2*d*e^4*co s(2*d*x + 2*c) + d*e^4)*sqrt(-(d^2*e^7*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^ 4 - 12*a^2*b^6 + b^8)/(d^4*e^14)) - 4*a^3*b + 4*a*b^3)/(d^2*e^7))*log((a^8 - 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8)*sqrt((e*cos(2*d*x + 2*c) + e) /sin(2*d*x + 2*c)) + ((a^2 - b^2)*d^3*e^11*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^ 4*b^4 - 12*a^2*b^6 + b^8)/(d^4*e^14)) + 2*(a^5*b - 6*a^3*b^3 + a*b^5)*d...
\[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\int \frac {\left (a + b \cot {\left (c + d x \right )}\right )^{2}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
Exception generated. \[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\int { \frac {{\left (b \cot \left (d x + c\right ) + a\right )}^{2}}{\left (e \cot \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
Time = 15.09 (sec) , antiderivative size = 1227, normalized size of antiderivative = 3.81 \[ \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\text {Too large to display} \]
2*atanh((32*a^4*d^3*e^11*(e*cot(c + d*x))^(1/2)*((a^3*b)/(d^2*e^7) - (b^4* 1i)/(4*d^2*e^7) - (a*b^3)/(d^2*e^7) - (a^4*1i)/(4*d^2*e^7) + (a^2*b^2*3i)/ (2*d^2*e^7))^(1/2))/(16*a^6*d^2*e^8 - 16*b^6*d^2*e^8 + a*b^5*d^2*e^8*32i + a^5*b*d^2*e^8*32i + 112*a^2*b^4*d^2*e^8 - a^3*b^3*d^2*e^8*192i - 112*a^4* b^2*d^2*e^8) + (32*b^4*d^3*e^11*(e*cot(c + d*x))^(1/2)*((a^3*b)/(d^2*e^7) - (b^4*1i)/(4*d^2*e^7) - (a*b^3)/(d^2*e^7) - (a^4*1i)/(4*d^2*e^7) + (a^2*b ^2*3i)/(2*d^2*e^7))^(1/2))/(16*a^6*d^2*e^8 - 16*b^6*d^2*e^8 + a*b^5*d^2*e^ 8*32i + a^5*b*d^2*e^8*32i + 112*a^2*b^4*d^2*e^8 - a^3*b^3*d^2*e^8*192i - 1 12*a^4*b^2*d^2*e^8) - (192*a^2*b^2*d^3*e^11*(e*cot(c + d*x))^(1/2)*((a^3*b )/(d^2*e^7) - (b^4*1i)/(4*d^2*e^7) - (a*b^3)/(d^2*e^7) - (a^4*1i)/(4*d^2*e ^7) + (a^2*b^2*3i)/(2*d^2*e^7))^(1/2))/(16*a^6*d^2*e^8 - 16*b^6*d^2*e^8 + a*b^5*d^2*e^8*32i + a^5*b*d^2*e^8*32i + 112*a^2*b^4*d^2*e^8 - a^3*b^3*d^2* e^8*192i - 112*a^4*b^2*d^2*e^8))*(-((a^3*b*4i - a*b^3*4i + a^4 + b^4 - 6*a ^2*b^2)*1i)/(4*d^2*e^7))^(1/2) - 2*atanh((32*a^4*d^3*e^11*(e*cot(c + d*x)) ^(1/2)*((a^4*1i)/(4*d^2*e^7) + (b^4*1i)/(4*d^2*e^7) - (a*b^3)/(d^2*e^7) + (a^3*b)/(d^2*e^7) - (a^2*b^2*3i)/(2*d^2*e^7))^(1/2))/(16*b^6*d^2*e^8 - 16* a^6*d^2*e^8 + a*b^5*d^2*e^8*32i + a^5*b*d^2*e^8*32i - 112*a^2*b^4*d^2*e^8 - a^3*b^3*d^2*e^8*192i + 112*a^4*b^2*d^2*e^8) + (32*b^4*d^3*e^11*(e*cot(c + d*x))^(1/2)*((a^4*1i)/(4*d^2*e^7) + (b^4*1i)/(4*d^2*e^7) - (a*b^3)/(d^2* e^7) + (a^3*b)/(d^2*e^7) - (a^2*b^2*3i)/(2*d^2*e^7))^(1/2))/(16*b^6*d^2...