Optimal. Leaf size=124 \[ \frac {2 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d e^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}-\frac {4 i \left (a^3+i a^3 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3578, 3577,
3856, 2719} \begin {gather*} \frac {2 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d e^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {4 i \left (a^3+i a^3 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}}-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2719
Rule 3577
Rule 3578
Rule 3856
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx &=-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}+\frac {a \int \frac {(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{5/2}} \, dx}{3 e^2}\\ &=-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}-\frac {4 i \left (a^3+i a^3 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}}+\frac {a^3 \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{15 e^4}\\ &=-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}-\frac {4 i \left (a^3+i a^3 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}}+\frac {a^3 \int \sqrt {\cos (c+d x)} \, dx}{15 e^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=\frac {2 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d e^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}-\frac {4 i \left (a^3+i a^3 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 2.49, size = 118, normalized size = 0.95 \begin {gather*} -\frac {a^3 e^{-2 i (c+d x)} \left (11+16 e^{2 i (c+d x)}+5 e^{4 i (c+d x)}+4 \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 )\right ) (-i+\tan (c+d x))^3}{90 d e^2 (e \sec (c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 369 vs. \(2 (132 ) = 264\).
time = 0.56, size = 370, normalized size = 2.98
method | result | size |
risch | \(-\frac {i \left (5 \,{\mathrm e}^{4 i \left (d x +c \right )}+11 \,{\mathrm e}^{2 i \left (d x +c \right )}+12\right ) a^{3} \sqrt {2}}{90 d \,e^{4} \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {i \left (-\frac {2 \left (e \,{\mathrm e}^{2 i \left (d x +c \right )}+e \right )}{e \sqrt {{\mathrm e}^{i \left (d x +c \right )} \left (e \,{\mathrm e}^{2 i \left (d x +c \right )}+e \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, \left (-2 i \EllipticE \left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \EllipticF \left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {e \,{\mathrm e}^{3 i \left (d x +c \right )}+e \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) a^{3} \sqrt {2}\, \sqrt {e \,{\mathrm e}^{i \left (d x +c \right )} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}}{15 d \,e^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}\) | \(339\) |
default | \(\frac {2 a^{3} \left (-20 i \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right )-3 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right )-20 \left (\cos ^{6}\left (d x +c \right )\right )+3 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-3 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )+9 i \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+19 \left (\cos ^{4}\left (d x +c \right )\right )-2 \left (\cos ^{2}\left (d x +c \right )\right )+3 \cos \left (d x +c \right )\right )}{45 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{5} \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {9}{2}}}\) | \(370\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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 = 0.13, size = 100, normalized size = 0.81 \begin {gather*} \frac {{\left (12 i \, \sqrt {2} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right ) + \frac {\sqrt {2} {\left (-5 i \, a^{3} e^{\left (5 i \, d x + 5 i \, c\right )} - 16 i \, a^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - 11 i \, a^{3} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-\frac {9}{2}\right )}}{90 \, 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 [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 (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________