Optimal. Leaf size=215 \[ -\frac {59 \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{8 a^3 d e^{5/2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{2 \sqrt {2} a^3 d e^{5/2}}-\frac {63}{8 a^3 d e^2 \sqrt {e \cot (c+d x)}}-\frac {11}{8 a^3 d e (\cot (c+d x)+1) (e \cot (c+d x))^{3/2}}+\frac {55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 (e \cot (c+d x))^{3/2}} \]
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Rubi [A] time = 1.10, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3569, 3649, 3650, 3653, 3532, 205, 3634, 63} \[ -\frac {63}{8 a^3 d e^2 \sqrt {e \cot (c+d x)}}-\frac {59 \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{8 a^3 d e^{5/2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{2 \sqrt {2} a^3 d e^{5/2}}-\frac {11}{8 a^3 d e (\cot (c+d x)+1) (e \cot (c+d x))^{3/2}}+\frac {55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 (e \cot (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 205
Rule 3532
Rule 3569
Rule 3634
Rule 3649
Rule 3650
Rule 3653
Rubi steps
\begin {align*} \int \frac {1}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3} \, dx &=-\frac {1}{4 a d e (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2}-\frac {\int \frac {-\frac {11 a^2 e}{2}+2 a^2 e \cot (c+d x)-\frac {7}{2} a^2 e \cot ^2(c+d x)}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^2} \, dx}{4 a^3 e}\\ &=-\frac {11}{8 a^3 d e (e \cot (c+d x))^{3/2} (1+\cot (c+d x))}-\frac {1}{4 a d e (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2}+\frac {\int \frac {\frac {55 a^4 e^2}{2}-4 a^4 e^2 \cot (c+d x)+\frac {55}{2} a^4 e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))} \, dx}{8 a^6 e^2}\\ &=\frac {55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac {11}{8 a^3 d e (e \cot (c+d x))^{3/2} (1+\cot (c+d x))}-\frac {1}{4 a d e (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2}+\frac {\int \frac {-\frac {189}{4} a^5 e^4-\frac {165}{4} a^5 e^4 \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))} \, dx}{12 a^7 e^5}\\ &=\frac {55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac {63}{8 a^3 d e^2 \sqrt {e \cot (c+d x)}}-\frac {11}{8 a^3 d e (e \cot (c+d x))^{3/2} (1+\cot (c+d x))}-\frac {1}{4 a d e (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2}+\frac {\int \frac {\frac {189 a^6 e^6}{8}+3 a^6 e^6 \cot (c+d x)+\frac {189}{8} a^6 e^6 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{6 a^8 e^8}\\ &=\frac {55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac {63}{8 a^3 d e^2 \sqrt {e \cot (c+d x)}}-\frac {11}{8 a^3 d e (e \cot (c+d x))^{3/2} (1+\cot (c+d x))}-\frac {1}{4 a d e (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2}+\frac {\int \frac {3 a^7 e^6+3 a^7 e^6 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{12 a^{10} e^8}+\frac {59 \int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{16 a^2 e^2}\\ &=\frac {55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac {63}{8 a^3 d e^2 \sqrt {e \cot (c+d x)}}-\frac {11}{8 a^3 d e (e \cot (c+d x))^{3/2} (1+\cot (c+d x))}-\frac {1}{4 a d e (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2}+\frac {59 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-a x)} \, dx,x,-\cot (c+d x)\right )}{16 a^2 d e^2}-\frac {\left (3 a^4 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{-18 a^{14} e^{12}-e x^2} \, dx,x,\frac {3 a^7 e^6-3 a^7 e^6 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{2 d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{2 \sqrt {2} a^3 d e^{5/2}}+\frac {55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac {63}{8 a^3 d e^2 \sqrt {e \cot (c+d x)}}-\frac {11}{8 a^3 d e (e \cot (c+d x))^{3/2} (1+\cot (c+d x))}-\frac {1}{4 a d e (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2}-\frac {59 \operatorname {Subst}\left (\int \frac {1}{a+\frac {a x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{8 a^2 d e^3}\\ &=-\frac {59 \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{8 a^3 d e^{5/2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{2 \sqrt {2} a^3 d e^{5/2}}+\frac {55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac {63}{8 a^3 d e^2 \sqrt {e \cot (c+d x)}}-\frac {11}{8 a^3 d e (e \cot (c+d x))^{3/2} (1+\cot (c+d x))}-\frac {1}{4 a d e (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2}\\ \end {align*}
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Mathematica [A] time = 3.19, size = 167, normalized size = 0.78 \[ \frac {\cot ^{\frac {5}{2}}(c+d x) \left (4 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-4 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-118 \tan ^{-1}\left (\sqrt {\cot (c+d x)}\right )-\frac {\sqrt {\cot (c+d x)} \sec ^2(c+d x) (678 \cos (2 (c+d x))+679 \cot (c+d x)+77 \cos (3 (c+d x)) \csc (c+d x)+614)}{6 (\cot (c+d x)+1)^2}\right )}{16 a^3 d (e \cot (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 718, normalized size = 3.34 \[ \left [-\frac {6 \, \sqrt {2} {\left ({\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (2 \, d x + 2 \, c\right ) + \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sqrt {-e} \log \left (\sqrt {2} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) + 177 \, {\left ({\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (2 \, d x + 2 \, c\right ) + \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sqrt {-e} \log \left (\frac {e \cos \left (2 \, d x + 2 \, c\right ) - e \sin \left (2 \, d x + 2 \, c\right ) + 2 \, \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) + e}{\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1}\right ) - {\left (339 \, \cos \left (2 \, d x + 2 \, c\right )^{2} - 7 \, {\left (11 \, \cos \left (2 \, d x + 2 \, c\right ) + 43\right )} \sin \left (2 \, d x + 2 \, c\right ) - 32 \, \cos \left (2 \, d x + 2 \, c\right ) - 307\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{48 \, {\left (a^{3} d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + a^{3} d e^{3} + {\left (a^{3} d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + a^{3} d e^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )\right )}}, \frac {12 \, \sqrt {2} {\left ({\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (2 \, d x + 2 \, c\right ) + \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sqrt {e} \arctan \left (-\frac {\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) - 354 \, {\left ({\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (2 \, d x + 2 \, c\right ) + \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sqrt {e} \arctan \left (\frac {\sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{\sqrt {e}}\right ) + {\left (339 \, \cos \left (2 \, d x + 2 \, c\right )^{2} - 7 \, {\left (11 \, \cos \left (2 \, d x + 2 \, c\right ) + 43\right )} \sin \left (2 \, d x + 2 \, c\right ) - 32 \, \cos \left (2 \, d x + 2 \, c\right ) - 307\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{48 \, {\left (a^{3} d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + a^{3} d e^{3} + {\left (a^{3} d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + a^{3} d e^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )\right )}}\right ] \]
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 \[ \int \frac {1}{{\left (a \cot \left (d x + c\right ) + a\right )}^{3} \left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.86, size = 482, normalized size = 2.24 \[ -\frac {15 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{8 d \,a^{3} e^{2} \left (e \cot \left (d x +c \right )+e \right )^{2}}-\frac {17 \sqrt {e \cot \left (d x +c \right )}}{8 d \,a^{3} e \left (e \cot \left (d x +c \right )+e \right )^{2}}-\frac {59 \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{8 a^{3} d \,e^{\frac {5}{2}}}-\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 )}{16 d \,a^{3} 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 )}{8 d \,a^{3} 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 )}{8 d \,a^{3} e^{3}}-\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 )}{16 d \,a^{3} 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 )}{8 d \,a^{3} 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 )}{8 d \,a^{3} e^{2} \left (e^{2}\right )^{\frac {1}{4}}}+\frac {2}{3 a^{3} d e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {6}{a^{3} 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 = 0.78, size = 224, normalized size = 1.04 \[ \frac {e {\left (\frac {16 \, e^{3} - \frac {112 \, e^{3}}{\tan \left (d x + c\right )} - \frac {323 \, e^{3}}{\tan \left (d x + c\right )^{2}} - \frac {189 \, e^{3}}{\tan \left (d x + c\right )^{3}}}{a^{3} e^{5} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}} + 2 \, a^{3} e^{4} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {5}{2}} + a^{3} e^{3} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {7}{2}}} - \frac {6 \, {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}}\right )}}{a^{3} e^{3}} - \frac {177 \, \arctan \left (\frac {\sqrt {\frac {e}{\tan \left (d x + c\right )}}}{\sqrt {e}}\right )}{a^{3} e^{\frac {7}{2}}}\right )}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.38, size = 193, normalized size = 0.90 \[ -\frac {\frac {63\,e\,{\mathrm {cot}\left (c+d\,x\right )}^3}{8}+\frac {323\,e\,{\mathrm {cot}\left (c+d\,x\right )}^2}{24}+\frac {14\,e\,\mathrm {cot}\left (c+d\,x\right )}{3}-\frac {2\,e}{3}}{a^3\,d\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{7/2}+2\,a^3\,d\,e\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}+a^3\,d\,e^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}-\frac {59\,\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{8\,a^3\,d\,e^{5/2}}-\frac {\sqrt {2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}+\frac {\sqrt {2}\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{2\,e^{3/2}}\right )\right )}{8\,a^3\,d\,e^{5/2}} \]
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 \[ \frac {\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot ^{3}{\left (c + d x \right )} + 3 \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot ^{2}{\left (c + d x \right )} + 3 \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot {\left (c + d x \right )} + \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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