Optimal. Leaf size=85 \[ \frac {4 i a e \sqrt {e \cos (c+d x)} \sec (c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}-\frac {2 i (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 86, normalized size of antiderivative = 1.01, number of steps
used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3596, 3578,
3569} \begin {gather*} \frac {4 i a \sec ^2(c+d x) (e \cos (c+d x))^{3/2}}{3 d \sqrt {a+i a \tan (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2}}{3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3569
Rule 3578
Rule 3596
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)} \, dx &=\left ((e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx\\ &=-\frac {2 i (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}{3 d}+\frac {\left (2 a (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}\right ) \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{3 e^2}\\ &=\frac {4 i a (e \cos (c+d x))^{3/2} \sec ^2(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}-\frac {2 i (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.24, size = 56, normalized size = 0.66 \begin {gather*} \frac {2 e \sqrt {e \cos (c+d x)} (i \cos (c+d x)+2 \sin (c+d x)) \sqrt {a+i a \tan (c+d x)}}{3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.08, size = 70, normalized size = 0.82
method | result | size |
risch | \(-\frac {i e \sqrt {2}\, \sqrt {e \cos \left (d x +c \right )}\, \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left (-2 \cos \left (d x +c \right )+4 i \sin \left (d x +c \right )\right )}{3 d}\) | \(65\) |
default | \(\frac {2 \left (i \cos \left (d x +c \right )+2 \sin \left (d x +c \right )\right ) \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{3 d \cos \left (d x +c \right )}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.61, size = 53, normalized size = 0.62 \begin {gather*} \frac {\sqrt {a} {\left (-i \, \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3 i \, \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} e^{\frac {3}{2}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 67, normalized size = 0.79 \begin {gather*} \frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (3 i \, e^{\frac {3}{2}} - i \, e^{\left (2 i \, d x + 2 i \, c + \frac {3}{2}\right )}\right )} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.70, size = 36, normalized size = 0.42 \begin {gather*} \frac {{\left (-i \, \sqrt {a} e^{\left (\frac {3}{2} i \, d x + \frac {3}{2} i \, c\right )} + 3 i \, \sqrt {a} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right )} e^{\frac {3}{2}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.53, size = 88, normalized size = 1.04 \begin {gather*} \frac {2\,e\,\sqrt {e\,\left (2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}\,\left ({\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,2{}\mathrm {i}+2\,\sin \left (c+d\,x\right )-\mathrm {i}\right )\,\sqrt {\frac {a\,\left (2\,{\cos \left (c+d\,x\right )}^2+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{2\,{\cos \left (c+d\,x\right )}^2}}}{3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________