Optimal. Leaf size=121 \[ \frac {2 i \sqrt {e \sec (c+d x)}}{9 d (a+i a \tan (c+d x))^{5/2}}+\frac {8 i \sqrt {e \sec (c+d x)}}{45 a d (a+i a \tan (c+d x))^{3/2}}+\frac {16 i \sqrt {e \sec (c+d x)}}{45 a^2 d \sqrt {a+i a \tan (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3583, 3569}
\begin {gather*} \frac {16 i \sqrt {e \sec (c+d x)}}{45 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {8 i \sqrt {e \sec (c+d x)}}{45 a d (a+i a \tan (c+d x))^{3/2}}+\frac {2 i \sqrt {e \sec (c+d x)}}{9 d (a+i a \tan (c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3569
Rule 3583
Rubi steps
\begin {align*} \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^{5/2}} \, dx &=\frac {2 i \sqrt {e \sec (c+d x)}}{9 d (a+i a \tan (c+d x))^{5/2}}+\frac {4 \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^{3/2}} \, dx}{9 a}\\ &=\frac {2 i \sqrt {e \sec (c+d x)}}{9 d (a+i a \tan (c+d x))^{5/2}}+\frac {8 i \sqrt {e \sec (c+d x)}}{45 a d (a+i a \tan (c+d x))^{3/2}}+\frac {8 \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{45 a^2}\\ &=\frac {2 i \sqrt {e \sec (c+d x)}}{9 d (a+i a \tan (c+d x))^{5/2}}+\frac {8 i \sqrt {e \sec (c+d x)}}{45 a d (a+i a \tan (c+d x))^{3/2}}+\frac {16 i \sqrt {e \sec (c+d x)}}{45 a^2 d \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.38, size = 85, normalized size = 0.70 \begin {gather*} -\frac {i \sec ^2(c+d x) \sqrt {e \sec (c+d x)} (9+25 \cos (2 (c+d x))+20 i \sin (2 (c+d x)))}{45 a^2 d (-i+\tan (c+d x))^2 \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.90, size = 128, normalized size = 1.06
method | result | size |
default | \(\frac {2 i \cos \left (d x +c \right ) \sqrt {\frac {e}{\cos \left (d x +c \right )}}\, \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (-20 i \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+20 \left (\cos ^{5}\left (d x +c \right )\right )-3 i \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )-7 \left (\cos ^{3}\left (d x +c \right )\right )-8 i \sin \left (d x +c \right )+4 \cos \left (d x +c \right )\right )}{45 d \,a^{3}}\) | \(128\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.54, size = 129, normalized size = 1.07 \begin {gather*} \frac {{\left (5 i \, \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 18 i \, \cos \left (\frac {5}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 45 i \, \cos \left (\frac {1}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 5 \, \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 18 \, \sin \left (\frac {5}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 45 \, \sin \left (\frac {1}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right )\right )} e^{\frac {1}{2}}}{90 \, a^{\frac {5}{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 88, normalized size = 0.73 \begin {gather*} \frac {\sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (5 i \, e^{\frac {1}{2}} + 45 i \, e^{\left (6 i \, d x + 6 i \, c + \frac {1}{2}\right )} + 63 i \, e^{\left (4 i \, d x + 4 i \, c + \frac {1}{2}\right )} + 23 i \, e^{\left (2 i \, d x + 2 i \, c + \frac {1}{2}\right )}\right )} e^{\left (-\frac {9}{2} i \, d x - \frac {9}{2} i \, c\right )}}{90 \, a^{3} d \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \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*} \int \frac {\sqrt {e \sec {\left (c + d x \right )}}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \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 = 4.22, size = 109, normalized size = 0.90 \begin {gather*} \frac {\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,\left (\cos \left (2\,c+2\,d\,x\right )\,18{}\mathrm {i}+\cos \left (4\,c+4\,d\,x\right )\,5{}\mathrm {i}+18\,\sin \left (2\,c+2\,d\,x\right )+5\,\sin \left (4\,c+4\,d\,x\right )+45{}\mathrm {i}\right )}{90\,a^2\,d\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________