Optimal. Leaf size=470 \[ \frac {i e^{5/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {i e^{5/2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {i e^{5/2} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{2 \sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}+\frac {i e^{5/2} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{2 \sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{a d (e \cos (c+d x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.30, antiderivative size = 470, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3596, 3582,
3576, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {i e^{5/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {i e^{5/2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {i e^{5/2} \log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right )}{2 \sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}+\frac {i e^{5/2} \log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right )}{2 \sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{a d (e \cos (c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3576
Rule 3582
Rule 3596
Rubi steps
\begin {align*} \int \frac {1}{(e \cos (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}} \, dx &=\frac {\int \frac {(e \sec (c+d x))^{5/2}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{(e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\\ &=-\frac {i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{a d (e \cos (c+d x))^{5/2}}+\frac {e^2 \int \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)} \, dx}{2 a (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\\ &=-\frac {i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{a d (e \cos (c+d x))^{5/2}}-\frac {\left (2 i e^4\right ) \text {Subst}\left (\int \frac {x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\\ &=-\frac {i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{a d (e \cos (c+d x))^{5/2}}+\frac {\left (i e^3\right ) \text {Subst}\left (\int \frac {a-e x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {\left (i e^3\right ) \text {Subst}\left (\int \frac {a+e x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\\ &=-\frac {i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{a d (e \cos (c+d x))^{5/2}}-\frac {\left (i e^2\right ) \text {Subst}\left (\int \frac {1}{\frac {a}{e}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}+x^2} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{2 d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {\left (i e^2\right ) \text {Subst}\left (\int \frac {1}{\frac {a}{e}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}+x^2} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{2 d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {\left (i e^{5/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {e}}+2 x}{-\frac {a}{e}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}-x^2} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{2 \sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {\left (i e^{5/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {e}}-2 x}{-\frac {a}{e}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}-x^2} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{2 \sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\\ &=-\frac {i e^{5/2} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{2 \sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}+\frac {i e^{5/2} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{2 \sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{a d (e \cos (c+d x))^{5/2}}-\frac {\left (i e^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}+\frac {\left (i e^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\\ &=\frac {i e^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {i e^{5/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {i e^{5/2} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{2 \sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}+\frac {i e^{5/2} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{2 \sqrt {2} \sqrt {a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{a d (e \cos (c+d x))^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.93, size = 250, normalized size = 0.53 \begin {gather*} \frac {i e^{i c-\frac {i d x}{2}} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left (-2 e^{\frac {3 i d x}{2}}+\left (-e^{-2 i c}\right )^{3/4} \left (1+e^{2 i (c+d x)}\right ) \text {ArcTan}\left (\frac {e^{\frac {i d x}{2}}}{\sqrt [4]{-e^{-2 i c}}}\right )-\left (-e^{-2 i c}\right )^{3/4} \left (1+e^{2 i (c+d x)}\right ) \tanh ^{-1}\left (\frac {e^{\frac {i d x}{2}}}{\sqrt [4]{-e^{-2 i c}}}\right )\right )}{d \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \sqrt {\cos (c+d x)} (e \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.99, size = 314, normalized size = 0.67
method | result | size |
default | \(-\frac {\left (\cos ^{2}\left (d x +c \right )\right ) \left (-1+\cos \left (d x +c \right )\right )^{3} \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (i \cos \left (d x +c \right ) \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1+\sin \left (d x +c \right )\right )}{2}\right )-i \cos \left (d x +c \right ) \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1-\sin \left (d x +c \right )\right )}{2}\right )+2 i \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}-\cos \left (d x +c \right ) \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1+\sin \left (d x +c \right )\right )}{2}\right )-\cos \left (d x +c \right ) \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1-\sin \left (d x +c \right )\right )}{2}\right )+2 \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )+2 \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\right )}{2 d \sin \left (d x +c \right )^{5} \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )-1\right ) \left (\frac {1}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}} a}\) | \(314\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 2137 vs. \(2 (310) = 620\).
time = 0.70, size = 2137, normalized size = 4.55 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 465, normalized size = 0.99 \begin {gather*} \frac {-4 i \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (\frac {3}{2} i \, d x + \frac {3}{2} i \, c\right )} - {\left (a d e^{\frac {5}{2}} + a d e^{\left (2 i \, d x + 2 i \, c + \frac {5}{2}\right )}\right )} \sqrt {\frac {i \, e^{\left (-5\right )}}{a d^{2}}} \log \left (i \, a d \sqrt {\frac {i \, e^{\left (-5\right )}}{a d^{2}}} e^{\frac {5}{2}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \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 )}\right ) + {\left (a d e^{\frac {5}{2}} + a d e^{\left (2 i \, d x + 2 i \, c + \frac {5}{2}\right )}\right )} \sqrt {\frac {i \, e^{\left (-5\right )}}{a d^{2}}} \log \left (-i \, a d \sqrt {\frac {i \, e^{\left (-5\right )}}{a d^{2}}} e^{\frac {5}{2}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \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 )}\right ) - {\left (a d e^{\frac {5}{2}} + a d e^{\left (2 i \, d x + 2 i \, c + \frac {5}{2}\right )}\right )} \sqrt {-\frac {i \, e^{\left (-5\right )}}{a d^{2}}} \log \left (i \, a d \sqrt {-\frac {i \, e^{\left (-5\right )}}{a d^{2}}} e^{\frac {5}{2}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \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 )}\right ) + {\left (a d e^{\frac {5}{2}} + a d e^{\left (2 i \, d x + 2 i \, c + \frac {5}{2}\right )}\right )} \sqrt {-\frac {i \, e^{\left (-5\right )}}{a d^{2}}} \log \left (-i \, a d \sqrt {-\frac {i \, e^{\left (-5\right )}}{a d^{2}}} e^{\frac {5}{2}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \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 )}\right )}{2 \, {\left (a d e^{\frac {5}{2}} + a d e^{\left (2 i \, d x + 2 i \, c + \frac {5}{2}\right )}\right )}} \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 = -0.00 \begin {gather*} \int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\sqrt {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]
________________________________________________________________________________________