Optimal. Leaf size=165 \[ \frac {2 i}{9 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}+\frac {4 i}{15 a d (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {32 i \sqrt {e \sec (c+d x)}}{45 a d e^2 \sqrt {a+i a \tan (c+d x)}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{45 a^2 d (e \sec (c+d x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.20, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3583, 3578,
3569} \begin {gather*} -\frac {16 i \sqrt {a+i a \tan (c+d x)}}{45 a^2 d (e \sec (c+d x))^{3/2}}+\frac {32 i \sqrt {e \sec (c+d x)}}{45 a d e^2 \sqrt {a+i a \tan (c+d x)}}+\frac {4 i}{15 a d \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}+\frac {2 i}{9 d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3569
Rule 3578
Rule 3583
Rubi steps
\begin {align*} \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}} \, dx &=\frac {2 i}{9 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}+\frac {2 \int \frac {1}{(e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx}{3 a}\\ &=\frac {2 i}{9 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}+\frac {4 i}{15 a d (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {8 \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx}{15 a^2}\\ &=\frac {2 i}{9 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}+\frac {4 i}{15 a d (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{45 a^2 d (e \sec (c+d x))^{3/2}}+\frac {16 \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{45 a e^2}\\ &=\frac {2 i}{9 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}+\frac {4 i}{15 a d (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {32 i \sqrt {e \sec (c+d x)}}{45 a d e^2 \sqrt {a+i a \tan (c+d x)}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{45 a^2 d (e \sec (c+d x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.57, size = 100, normalized size = 0.61 \begin {gather*} -\frac {\sec ^3(c+d x) (-81 \cos (c+d x)+5 \cos (3 (c+d x))-54 i \sin (c+d x)+10 i \sin (3 (c+d x)))}{90 a d (e \sec (c+d x))^{3/2} (-i+\tan (c+d x)) \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.89, size = 132, normalized size = 0.80
method | result | size |
default | \(\frac {2 \left (\cos ^{2}\left (d x +c \right )\right ) \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (10 i \left (\cos ^{5}\left (d x +c \right )\right )+10 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+i \left (\cos ^{3}\left (d x +c \right )\right )+6 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+8 i \cos \left (d x +c \right )+16 \sin \left (d x +c \right )\right )}{45 d \,e^{3} a^{2}}\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.55, size = 177, normalized size = 1.07 \begin {gather*} \frac {{\left (5 i \, \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 27 i \, \cos \left (\frac {5}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) - 15 i \, \cos \left (\frac {1}{3} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 135 i \, \cos \left (\frac {1}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 5 \, \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 27 \, \sin \left (\frac {5}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 15 \, \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 135 \, \sin \left (\frac {1}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right )\right )} e^{\left (-\frac {3}{2}\right )}}{180 \, a^{\frac {3}{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 94, normalized size = 0.57 \begin {gather*} \frac {\sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-15 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 120 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 162 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 32 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i\right )} e^{\left (-\frac {9}{2} i \, d x - \frac {9}{2} i \, c - \frac {3}{2}\right )}}{180 \, a^{2} d \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \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*} \int \frac {1}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \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 [B]
time = 4.20, size = 112, normalized size = 0.68 \begin {gather*} \frac {\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,\left (\cos \left (2\,c+2\,d\,x\right )\,12{}\mathrm {i}+\cos \left (4\,c+4\,d\,x\right )\,5{}\mathrm {i}+42\,\sin \left (2\,c+2\,d\,x\right )+5\,\sin \left (4\,c+4\,d\,x\right )+135{}\mathrm {i}\right )}{180\,a\,d\,e^2\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________