Optimal. Leaf size=331 \[ -\frac {7 \text {ArcTan}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d e^{5/2}}+\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d e^{5/2}}-\frac {\text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d e^{5/2}}+\frac {7}{6 a^2 d e (e \cot (c+d x))^{3/2}}-\frac {9}{2 a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {1}{2 d e (e \cot (c+d x))^{3/2} \left (a^2+a^2 \cot (c+d x)\right )}-\frac {\log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d e^{5/2}}+\frac {\log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d e^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.70, 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 =
{3650, 3730, 3734, 12, 16, 3557, 335, 303, 1176, 631, 210, 1179, 642, 3715, 65, 211}
\begin {gather*} -\frac {7 \text {ArcTan}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d e^{5/2}}+\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d e^{5/2}}-\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{2 \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 {\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 {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}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 16
Rule 65
Rule 210
Rule 211
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3557
Rule 3650
Rule 3715
Rule 3730
Rule 3734
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 \text {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 \text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 6.36, size = 467, normalized size = 1.41 \begin {gather*} \frac {\cot ^3(c+d x) \csc ^2(c+d x) (\cos (c+d x)+\sin (c+d x))^2 \left (-\frac {2}{3}+\frac {2}{3} \sec ^2(c+d x)-\frac {\sin (c+d x)}{2 (\cos (c+d x)+\sin (c+d x))}-4 \tan (c+d x)\right )}{d (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^2}+\frac {\cot ^{\frac {5}{2}}(c+d x) \csc ^2(c+d x) (\cos (c+d x)+\sin (c+d x))^2 \left (-\frac {16 \text {ArcTan}\left (\sqrt {\cot (c+d x)}\right ) (1+\cot (c+d x)) \csc ^3(c+d x) \sec (c+d x)}{\left (1+\cot ^2(c+d x)\right )^2 (1+\tan (c+d x))}+\frac {\cos (2 (c+d x)) \csc ^3(c+d x) \left (-\log \left (-1+\sqrt {2} \sqrt {\cot (c+d x)}-\cot (c+d x)\right )+\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right ) \sec (c+d x)}{\sqrt {2} (-1+\cot (c+d x)) \left (1+\cot ^2(c+d x)\right ) (1+\tan (c+d x))}+\frac {\left (-\sqrt {2} \left (-\text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )+\text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )\right )+2 \text {ArcTan}\left (\sqrt {\cot (c+d x)}\right )\right ) (1+\cot (c+d x)) \csc ^2(c+d x) \sec ^2(c+d x) \sin (2 (c+d x))}{2 \left (1+\cot ^2(c+d x)\right ) (1+\tan (c+d x))}\right )}{4 d (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.52, size = 227, normalized size = 0.69
method | result | size |
derivativedivides | \(-\frac {2 e^{3} \left (-\frac {1}{3 e^{4} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {2}{e^{5} \sqrt {e \cot \left (d x +c \right )}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 e^{5} \left (e^{2}\right )^{\frac {1}{4}}}+\frac {\frac {\sqrt {e \cot \left (d x +c \right )}}{2 e \cot \left (d x +c \right )+2 e}+\frac {7 \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 \sqrt {e}}}{2 e^{5}}\right )}{d \,a^{2}}\) | \(227\) |
default | \(-\frac {2 e^{3} \left (-\frac {1}{3 e^{4} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {2}{e^{5} \sqrt {e \cot \left (d x +c \right )}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 e^{5} \left (e^{2}\right )^{\frac {1}{4}}}+\frac {\frac {\sqrt {e \cot \left (d x +c \right )}}{2 e \cot \left (d x +c \right )+2 e}+\frac {7 \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 \sqrt {e}}}{2 e^{5}}\right )}{d \,a^{2}}\) | \(227\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 185, normalized size = 0.56 \begin {gather*} -\frac {{\left (\frac {4 \, {\left (\frac {20}{\tan \left (d x + c\right )} + \frac {27}{\tan \left (d x + c\right )^{2}} - 4\right )}}{\frac {a^{2}}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {a^{2}}{\tan \left (d x + c\right )^{\frac {5}{2}}}} + \frac {3 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )}}{a^{2}} + \frac {84 \, \arctan \left (\frac {1}{\sqrt {\tan \left (d x + c\right )}}\right )}{a^{2}}\right )} e^{\left (-\frac {5}{2}\right )}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.23, size = 425, normalized size = 1.28 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________