Optimal. Leaf size=322 \[ -\frac {\left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{7/2}}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{7/2}}+\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{7/2}}-\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{7/2}}-\frac {2 \left (a^2-b^2\right )}{d e^3 \sqrt {e \cot (c+d x)}}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\frac {4 a b}{3 d e^2 (e \cot (c+d x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.43, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3542, 3529, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {2 \left (a^2-b^2\right )}{d e^3 \sqrt {e \cot (c+d x)}}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{7/2}}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{7/2}}+\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{7/2}}-\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{7/2}}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\frac {4 a b}{3 d e^2 (e \cot (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 3529
Rule 3534
Rule 3542
Rubi steps
\begin {align*} \int \frac {(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx &=\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\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}}+\frac {4 a b}{3 d e^2 (e \cot (c+d x))^{3/2}}+\frac {\int \frac {-\left (a^2-b^2\right ) e^2-2 a b e^2 \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx}{e^4}\\ &=\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 {\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^6}\\ &=\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 {2 \operatorname {Subst}\left (\int \frac {2 a b e^4-\left (a^2-b^2\right ) e^3 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^6}\\ &=\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 ) \operatorname {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^3}+\frac {\left (a^2+2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^3}\\ &=\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 ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{7/2}}-\frac {\left (a^2+2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{7/2}}-\frac {\left (a^2-2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 d e^3}-\frac {\left (a^2-2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 d e^3}\\ &=\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}}-\frac {\left (a^2-2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,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 ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,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 ) \tan ^{-1}\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 ) \tan ^{-1}\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}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.36, size = 85, normalized size = 0.26 \[ \frac {2 \left (3 \left (a^2-b^2\right ) \, _2F_1\left (-\frac {5}{4},1;-\frac {1}{4};-\cot ^2(c+d x)\right )+b \left (10 a \cot (c+d x) \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\cot ^2(c+d x)\right )+3 b\right )\right )}{15 d e (e \cot (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \cot \left (d x + c\right ) + a\right )}^{2}}{\left (e \cot \left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.45, size = 600, normalized size = 1.86 \[ \frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \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 e^{4} d}+\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{e^{4} d}-\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{e^{4} d}-\frac {\sqrt {2}\, \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 ) a^{2}}{4 e^{3} d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \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 ) b^{2}}{4 e^{3} d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right ) a^{2}}{2 e^{3} d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right ) b^{2}}{2 e^{3} d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right ) a^{2}}{2 e^{3} d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right ) b^{2}}{2 e^{3} d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {2 a^{2}}{5 d e \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {2 a^{2}}{e^{3} d \sqrt {e \cot \left (d x +c \right )}}+\frac {2 b^{2}}{e^{3} d \sqrt {e \cot \left (d x +c \right )}}+\frac {4 a b}{3 d \,e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.53, size = 286, normalized size = 0.89 \[ -\frac {e {\left (\frac {15 \, {\left (\frac {2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} - \frac {\sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} + \frac {\sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}\right )}}{e^{4}} - \frac {8 \, {\left (3 \, a^{2} e^{2} + \frac {10 \, a b e^{2}}{\tan \left (d x + c\right )} - \frac {15 \, {\left (a^{2} - b^{2}\right )} e^{2}}{\tan \left (d x + c\right )^{2}}\right )}}{e^{4} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {5}{2}}}\right )}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.32, size = 1227, normalized size = 3.81 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \cot {\left (c + d x \right )}\right )^{2}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________