Optimal. Leaf size=313 \[ -\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{5/2}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{5/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{5/2}}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{5/2}}+\frac {16 a^2 b}{3 d e^2 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{3 d e (e \cot (c+d x))^{3/2}} \]
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Rubi [A] time = 0.46, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3565, 3628, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{5/2}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{5/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{5/2}}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{5/2}}+\frac {16 a^2 b}{3 d e^2 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{3 d e (e \cot (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 3534
Rule 3565
Rule 3628
Rubi steps
\begin {align*} \int \frac {(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{5/2}} \, dx &=\frac {2 a^2 (a+b \cot (c+d x))}{3 d e (e \cot (c+d x))^{3/2}}-\frac {2 \int \frac {-4 a^2 b e^2+\frac {3}{2} a \left (a^2-3 b^2\right ) e^2 \cot (c+d x)+\frac {1}{2} b \left (a^2-3 b^2\right ) e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2}} \, dx}{3 e^3}\\ &=\frac {16 a^2 b}{3 d e^2 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{3 d e (e \cot (c+d x))^{3/2}}-\frac {2 \int \frac {\frac {3}{2} a \left (a^2-3 b^2\right ) e^3+\frac {3}{2} b \left (3 a^2-b^2\right ) e^3 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{3 e^5}\\ &=\frac {16 a^2 b}{3 d e^2 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{3 d e (e \cot (c+d x))^{3/2}}-\frac {4 \operatorname {Subst}\left (\int \frac {-\frac {3}{2} a \left (a^2-3 b^2\right ) e^4-\frac {3}{2} b \left (3 a^2-b^2\right ) e^3 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{3 d e^5}\\ &=\frac {16 a^2 b}{3 d e^2 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{3 d e (e \cot (c+d x))^{3/2}}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^2}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^2}\\ &=\frac {16 a^2 b}{3 d e^2 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{3 d e (e \cot (c+d x))^{3/2}}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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^{5/2}}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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^{5/2}}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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^2}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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^2}\\ &=\frac {16 a^2 b}{3 d e^2 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{3 d e (e \cot (c+d x))^{3/2}}-\frac {(a+b) \left (a^2-4 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^{5/2}}+\frac {(a+b) \left (a^2-4 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^{5/2}}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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^{5/2}}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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^{5/2}}\\ &=-\frac {(a-b) \left (a^2+4 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^{5/2}}+\frac {(a-b) \left (a^2+4 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^{5/2}}+\frac {16 a^2 b}{3 d e^2 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{3 d e (e \cot (c+d x))^{3/2}}-\frac {(a+b) \left (a^2-4 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^{5/2}}+\frac {(a+b) \left (a^2-4 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^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.38, size = 104, normalized size = 0.33 \[ \frac {-6 b \left (b^2-3 a^2\right ) \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2(c+d x)\right )+2 a \left (a^2-3 b^2\right ) \tan (c+d x) \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\cot ^2(c+d x)\right )+6 b^2 (a \tan (c+d x)+b)}{3 d e^2 \sqrt {e \cot (c+d x)}} \]
Antiderivative was successfully verified.
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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.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \cot \left (d x + c\right ) + a\right )}^{3}}{\left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.46, size = 743, normalized size = 2.37 \[ \frac {\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 ) a^{3}}{4 d \,e^{3}}-\frac {3 \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 ) a \,b^{2}}{4 d \,e^{3}}+\frac {a^{3} \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 )}{2 d \,e^{3}}-\frac {3 \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 ) a \,b^{2}}{2 d \,e^{3}}-\frac {\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 ) a^{3}}{2 d \,e^{3}}+\frac {3 \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 ) a \,b^{2}}{2 d \,e^{3}}+\frac {3 \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} b}{4 d \,e^{2} \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^{3}}{4 d \,e^{2} \left (e^{2}\right )^{\frac {1}{4}}}+\frac {3 \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} b}{2 d \,e^{2} \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^{3}}{2 d \,e^{2} \left (e^{2}\right )^{\frac {1}{4}}}-\frac {3 \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} b}{2 d \,e^{2} \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^{3}}{2 d \,e^{2} \left (e^{2}\right )^{\frac {1}{4}}}+\frac {2 a^{3}}{3 d e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {6 a^{2} b}{d \,e^{2} \sqrt {e \cot \left (d x +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 289, normalized size = 0.92 \[ \frac {e {\left (\frac {3 \, {\left (\frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {\sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} - \frac {\sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}\right )}}{e^{3}} + \frac {8 \, {\left (a^{3} e + \frac {9 \, a^{2} b e}{\tan \left (d x + c\right )}\right )}}{e^{3} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}}}\right )}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.67, size = 1946, normalized size = 6.22 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \cot {\left (c + d x \right )}\right )^{3}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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