Optimal. Leaf size=90 \[ -\frac {2 i a (e \cos (c+d x))^{5/2}}{5 d}+\frac {6 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (e \cos (c+d x))^{5/2}}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a \tan (c+d x) (e \cos (c+d x))^{5/2}}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3515, 3486, 3769, 3771, 2639} \[ -\frac {2 i a (e \cos (c+d x))^{5/2}}{5 d}+\frac {6 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (e \cos (c+d x))^{5/2}}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a \tan (c+d x) (e \cos (c+d x))^{5/2}}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2639
Rule 3486
Rule 3515
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{5/2} (a+i a \tan (c+d x)) \, dx &=\left ((e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}\right ) \int \frac {a+i a \tan (c+d x)}{(e \sec (c+d x))^{5/2}} \, dx\\ &=-\frac {2 i a (e \cos (c+d x))^{5/2}}{5 d}+\left (a (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}\right ) \int \frac {1}{(e \sec (c+d x))^{5/2}} \, dx\\ &=-\frac {2 i a (e \cos (c+d x))^{5/2}}{5 d}+\frac {2 a (e \cos (c+d x))^{5/2} \tan (c+d x)}{5 d}+\frac {\left (3 a (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{5 e^2}\\ &=-\frac {2 i a (e \cos (c+d x))^{5/2}}{5 d}+\frac {2 a (e \cos (c+d x))^{5/2} \tan (c+d x)}{5 d}+\frac {\left (3 a (e \cos (c+d x))^{5/2}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \cos ^{\frac {5}{2}}(c+d x)}\\ &=-\frac {2 i a (e \cos (c+d x))^{5/2}}{5 d}+\frac {6 a (e \cos (c+d x))^{5/2} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a (e \cos (c+d x))^{5/2} \tan (c+d x)}{5 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 12.37, size = 387, normalized size = 4.30 \[ \frac {(\cos (d x)-i \sin (d x)) (a+i a \tan (c+d x)) (e \cos (c+d x))^{5/2} \left (\frac {2 \sqrt {2} (\cot (c)-i) e^{-i d x} \left (e^{2 i d x} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )+3 e^{2 i (c+d x)}-3 \sqrt {1-i e^{i (c+d x)}} \sqrt {e^{i (c+d x)} \left (e^{i (c+d x)}-i\right )} F\left (\left .\sin ^{-1}\left (\sqrt {\sin (c+d x)-i \cos (c+d x)}\right )\right |-1\right )+3 \sqrt {1-i e^{i (c+d x)}} \sqrt {e^{i (c+d x)} \left (e^{i (c+d x)}-i\right )} E\left (\left .\sin ^{-1}\left (\sqrt {\sin (c+d x)-i \cos (c+d x)}\right )\right |-1\right )+3\right )}{5 \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}+\frac {2}{5} \sin (c) \sqrt {\cos (c+d x)} \left ((1-i \cot (c)) \cos (2 d x)+\cot (c) (\sin (2 d x)+5 i)-6 \cot ^2(c)+i \sin (2 d x)-1\right )\right )}{2 d \cos ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ \frac {\sqrt {\frac {1}{2}} {\left (-i \, a e^{2} e^{\left (3 i \, d x + 3 i \, c\right )} + i \, a e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 7 i \, a e^{2} e^{\left (i \, d x + i \, c\right )} - 5 i \, a e^{2}\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + 5 \, {\left (d e^{\left (i \, d x + i \, c\right )} - d\right )} {\rm integral}\left (\frac {\sqrt {\frac {1}{2}} {\left (-6 i \, a e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 12 i \, a e^{2} e^{\left (i \, d x + i \, c\right )} - 6 i \, a e^{2}\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{5 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (3 i \, d x + 3 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \, d e^{\left (i \, d x + i \, c\right )} + d\right )}}, x\right )}{5 \, {\left (d e^{\left (i \, d x + i \, c\right )} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 5.93, size = 205, normalized size = 2.28 \[ \frac {2 a \,e^{3} \left (8 i \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 i \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+6 i \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________