Optimal. Leaf size=70 \[ -\frac {2 i e^2 \sqrt {e \sec (c+d x)}}{a d}+\frac {2 e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{a d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3582, 3856,
2720} \begin {gather*} \frac {2 e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{a d}-\frac {2 i e^2 \sqrt {e \sec (c+d x)}}{a d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2720
Rule 3582
Rule 3856
Rubi steps
\begin {align*} \int \frac {(e \sec (c+d x))^{5/2}}{a+i a \tan (c+d x)} \, dx &=-\frac {2 i e^2 \sqrt {e \sec (c+d x)}}{a d}+\frac {e^2 \int \sqrt {e \sec (c+d x)} \, dx}{a}\\ &=-\frac {2 i e^2 \sqrt {e \sec (c+d x)}}{a d}+\frac {\left (e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{a}\\ &=-\frac {2 i e^2 \sqrt {e \sec (c+d x)}}{a d}+\frac {2 e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.34, size = 49, normalized size = 0.70 \begin {gather*} \frac {2 e^2 \left (-i+\sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right ) \sqrt {e \sec (c+d x)}}{a d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.64, size = 174, normalized size = 2.49
method | result | size |
default | \(\frac {2 i \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2} \left (\EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+\EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-1\right ) \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \left (\cos ^{2}\left (d x +c \right )\right )}{a d \sin \left (d x +c \right )^{4}}\) | \(174\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.09, size = 56, normalized size = 0.80 \begin {gather*} -\frac {2 \, {\left (i \, \sqrt {2} e^{\frac {5}{2}} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \frac {i \, \sqrt {2} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c + \frac {5}{2}\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )}}{a d} \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 {i \int \frac {\left (e \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}{\tan {\left (c + d x \right )} - i}\, dx}{a} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________