Optimal. Leaf size=331 \[ -\frac {\log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{4 \sqrt {2} a^2 d e^{5/2}}+\frac {\log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{4 \sqrt {2} a^2 d e^{5/2}}-\frac {7 \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d e^{5/2}}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d e^{5/2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{2 \sqrt {2} a^2 d e^{5/2}}-\frac {9}{2 a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) (e \cot (c+d x))^{3/2}}+\frac {7}{6 a^2 d e (e \cot (c+d x))^{3/2}} \]
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Rubi [A] time = 1.08, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 16, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3569, 3649, 3653, 12, 16, 3476, 329, 297, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ -\frac {9}{2 a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {\log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{4 \sqrt {2} a^2 d e^{5/2}}+\frac {\log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{4 \sqrt {2} a^2 d e^{5/2}}-\frac {7 \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d e^{5/2}}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d e^{5/2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{2 \sqrt {2} a^2 d e^{5/2}}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) (e \cot (c+d x))^{3/2}}+\frac {7}{6 a^2 d e (e \cot (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 16
Rule 63
Rule 204
Rule 205
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3476
Rule 3569
Rule 3634
Rule 3649
Rule 3653
Rubi steps
\begin {align*} \int \frac {1}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^2} \, dx &=-\frac {1}{2 d e (e \cot (c+d x))^{3/2} \left (a^2+a^2 \cot (c+d x)\right )}-\frac {\int \frac {-\frac {7 a^2 e}{2}+a^2 e \cot (c+d x)-\frac {5}{2} a^2 e \cot ^2(c+d x)}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))} \, dx}{2 a^3 e}\\ &=\frac {7}{6 a^2 d e (e \cot (c+d x))^{3/2}}-\frac {1}{2 d e (e \cot (c+d x))^{3/2} \left (a^2+a^2 \cot (c+d x)\right )}-\frac {\int \frac {\frac {27 a^3 e^3}{4}+\frac {3}{2} a^3 e^3 \cot (c+d x)+\frac {21}{4} a^3 e^3 \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))} \, dx}{3 a^4 e^4}\\ &=\frac {7}{6 a^2 d e (e \cot (c+d x))^{3/2}}-\frac {9}{2 a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {1}{2 d e (e \cot (c+d x))^{3/2} \left (a^2+a^2 \cot (c+d x)\right )}-\frac {2 \int \frac {-\frac {21}{8} a^4 e^5-\frac {3}{4} a^4 e^5 \cot (c+d x)-\frac {27}{8} a^4 e^5 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{3 a^5 e^7}\\ &=\frac {7}{6 a^2 d e (e \cot (c+d x))^{3/2}}-\frac {9}{2 a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {1}{2 d e (e \cot (c+d x))^{3/2} \left (a^2+a^2 \cot (c+d x)\right )}-\frac {\int -\frac {3 a^5 e^5 \cot (c+d x)}{2 \sqrt {e \cot (c+d x)}} \, dx}{3 a^7 e^7}+\frac {7 \int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{4 a e^2}\\ &=\frac {7}{6 a^2 d e (e \cot (c+d x))^{3/2}}-\frac {9}{2 a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {1}{2 d e (e \cot (c+d x))^{3/2} \left (a^2+a^2 \cot (c+d x)\right )}+\frac {\int \frac {\cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{2 a^2 e^2}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-a x)} \, dx,x,-\cot (c+d x)\right )}{4 a d e^2}\\ &=\frac {7}{6 a^2 d e (e \cot (c+d x))^{3/2}}-\frac {9}{2 a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {1}{2 d e (e \cot (c+d x))^{3/2} \left (a^2+a^2 \cot (c+d x)\right )}+\frac {\int \sqrt {e \cot (c+d x)} \, dx}{2 a^2 e^3}-\frac {7 \operatorname {Subst}\left (\int \frac {1}{a+\frac {a x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 a d e^3}\\ &=-\frac {7 \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d e^{5/2}}+\frac {7}{6 a^2 d e (e \cot (c+d x))^{3/2}}-\frac {9}{2 a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {1}{2 d e (e \cot (c+d x))^{3/2} \left (a^2+a^2 \cot (c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {x}}{e^2+x^2} \, dx,x,e \cot (c+d x)\right )}{2 a^2 d e^2}\\ &=-\frac {7 \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d e^{5/2}}+\frac {7}{6 a^2 d e (e \cot (c+d x))^{3/2}}-\frac {9}{2 a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {1}{2 d e (e \cot (c+d x))^{3/2} \left (a^2+a^2 \cot (c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{a^2 d e^2}\\ &=-\frac {7 \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d e^{5/2}}+\frac {7}{6 a^2 d e (e \cot (c+d x))^{3/2}}-\frac {9}{2 a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {1}{2 d e (e \cot (c+d x))^{3/2} \left (a^2+a^2 \cot (c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 a^2 d e^2}-\frac {\operatorname {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 a^2 d e^2}\\ &=-\frac {7 \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d e^{5/2}}+\frac {7}{6 a^2 d e (e \cot (c+d x))^{3/2}}-\frac {9}{2 a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {1}{2 d e (e \cot (c+d x))^{3/2} \left (a^2+a^2 \cot (c+d x)\right )}-\frac {\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 )}{4 \sqrt {2} a^2 d e^{5/2}}-\frac {\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 )}{4 \sqrt {2} a^2 d e^{5/2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{4 a^2 d e^2}-\frac {\operatorname {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{4 a^2 d e^2}\\ &=-\frac {7 \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d e^{5/2}}+\frac {7}{6 a^2 d e (e \cot (c+d x))^{3/2}}-\frac {9}{2 a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {1}{2 d e (e \cot (c+d x))^{3/2} \left (a^2+a^2 \cot (c+d x)\right )}-\frac {\log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d e^{5/2}}+\frac {\log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d e^{5/2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d e^{5/2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d e^{5/2}}\\ &=-\frac {7 \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d e^{5/2}}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d e^{5/2}}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d e^{5/2}}+\frac {7}{6 a^2 d e (e \cot (c+d x))^{3/2}}-\frac {9}{2 a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {1}{2 d e (e \cot (c+d x))^{3/2} \left (a^2+a^2 \cot (c+d x)\right )}-\frac {\log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d e^{5/2}}+\frac {\log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d e^{5/2}}\\ \end {align*}
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Mathematica [A] time = 6.34, size = 467, normalized size = 1.41 \[ \frac {\cot ^3(c+d x) \csc ^2(c+d x) (\sin (c+d x)+\cos (c+d x))^2 \left (-4 \tan (c+d x)+\frac {2}{3} \sec ^2(c+d x)-\frac {\sin (c+d x)}{2 (\sin (c+d x)+\cos (c+d x))}-\frac {2}{3}\right )}{d (a \cot (c+d x)+a)^2 (e \cot (c+d x))^{5/2}}+\frac {\cot ^{\frac {5}{2}}(c+d x) \csc ^2(c+d x) (\sin (c+d x)+\cos (c+d x))^2 \left (-\frac {16 (\cot (c+d x)+1) \csc ^3(c+d x) \sec (c+d x) \tan ^{-1}\left (\sqrt {\cot (c+d x)}\right )}{(\tan (c+d x)+1) \left (\cot ^2(c+d x)+1\right )^2}+\frac {\sin (2 (c+d x)) (\cot (c+d x)+1) \csc ^2(c+d x) \sec ^2(c+d x) \left (2 \tan ^{-1}\left (\sqrt {\cot (c+d x)}\right )-\sqrt {2} \left (\tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-\tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )\right )\right )}{2 (\tan (c+d x)+1) \left (\cot ^2(c+d x)+1\right )}+\frac {\cos (2 (c+d x)) \csc ^3(c+d x) \sec (c+d x) \left (\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-\log \left (-\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}-1\right )\right )}{\sqrt {2} (\tan (c+d x)+1) (\cot (c+d x)-1) \left (\cot ^2(c+d x)+1\right )}\right )}{4 d (a \cot (c+d x)+a)^2 (e \cot (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {1}{{\left (a \cot \left (d x + c\right ) + a\right )}^{2} \left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.67, size = 276, normalized size = 0.83 \[ -\frac {\sqrt {e \cot \left (d x +c \right )}}{2 d \,a^{2} e^{2} \left (e \cot \left (d x +c \right )+e \right )}-\frac {7 \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 a^{2} d \,e^{\frac {5}{2}}}-\frac {\sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{8 d \,a^{2} e^{2} \left (e^{2}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{4 d \,a^{2} e^{2} \left (e^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{4 d \,a^{2} e^{2} \left (e^{2}\right )^{\frac {1}{4}}}+\frac {2}{3 a^{2} d e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {4}{a^{2} d \,e^{2} \sqrt {e \cot \left (d x +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 274, normalized size = 0.83 \[ \frac {e {\left (\frac {4 \, {\left (4 \, e^{2} - \frac {20 \, e^{2}}{\tan \left (d x + c\right )} - \frac {27 \, e^{2}}{\tan \left (d x + c\right )^{2}}\right )}}{a^{2} e^{4} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}} + a^{2} e^{3} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {5}{2}}} - \frac {3 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}\right )}}{a^{2} e^{3}} - \frac {84 \, \arctan \left (\frac {\sqrt {\frac {e}{\tan \left (d x + c\right )}}}{\sqrt {e}}\right )}{a^{2} e^{\frac {7}{2}}}\right )}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 425, normalized size = 1.28 \[ -\frac {\mathrm {atan}\left (\frac {2048\,a^{10}\,d^5\,e^{18}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {1}{a^8\,d^4\,e^{10}}\right )}^{1/4}}{2048\,a^8\,d^4\,e^{16}+100352\,a^{12}\,d^6\,e^{21}\,\sqrt {-\frac {1}{a^8\,d^4\,e^{10}}}}+\frac {100352\,a^{14}\,d^7\,e^{23}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {1}{a^8\,d^4\,e^{10}}\right )}^{3/4}}{2048\,a^8\,d^4\,e^{16}+100352\,a^{12}\,d^6\,e^{21}\,\sqrt {-\frac {1}{a^8\,d^4\,e^{10}}}}\right )\,{\left (-\frac {1}{a^8\,d^4\,e^{10}}\right )}^{1/4}}{2}-\mathrm {atan}\left (\frac {a^{10}\,d^5\,e^{18}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {1}{256\,a^8\,d^4\,e^{10}}\right )}^{1/4}\,8192{}\mathrm {i}}{2048\,a^8\,d^4\,e^{16}-1605632\,a^{12}\,d^6\,e^{21}\,\sqrt {-\frac {1}{256\,a^8\,d^4\,e^{10}}}}-\frac {a^{14}\,d^7\,e^{23}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {1}{256\,a^8\,d^4\,e^{10}}\right )}^{3/4}\,6422528{}\mathrm {i}}{2048\,a^8\,d^4\,e^{16}-1605632\,a^{12}\,d^6\,e^{21}\,\sqrt {-\frac {1}{256\,a^8\,d^4\,e^{10}}}}\right )\,{\left (-\frac {1}{256\,a^8\,d^4\,e^{10}}\right )}^{1/4}\,2{}\mathrm {i}-\frac {\frac {9\,{\mathrm {cot}\left (c+d\,x\right )}^2}{2}+\frac {10\,\mathrm {cot}\left (c+d\,x\right )}{3}-\frac {2}{3}}{a^2\,d\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}+a^2\,d\,e\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}-\frac {\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {-e^5}\,1{}\mathrm {i}}{e^3}\right )\,\sqrt {-e^5}\,7{}\mathrm {i}}{2\,a^2\,d\,e^5} \]
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 \[ \frac {\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot ^{2}{\left (c + d x \right )} + 2 \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot {\left (c + d x \right )} + \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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