Optimal. Leaf size=209 \[ -\frac {256 i \sqrt {a+i a \tan (c+d x)}}{385 a^2 d e^2 \sqrt {e \sec (c+d x)}}-\frac {96 i \sqrt {a+i a \tan (c+d x)}}{385 a^2 d (e \sec (c+d x))^{5/2}}+\frac {128 i}{385 a d e^2 \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {16 i}{77 a d \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}+\frac {2 i}{11 d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.39, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3502, 3497, 3488} \[ -\frac {256 i \sqrt {a+i a \tan (c+d x)}}{385 a^2 d e^2 \sqrt {e \sec (c+d x)}}-\frac {96 i \sqrt {a+i a \tan (c+d x)}}{385 a^2 d (e \sec (c+d x))^{5/2}}+\frac {128 i}{385 a d e^2 \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {16 i}{77 a d \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}+\frac {2 i}{11 d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3488
Rule 3497
Rule 3502
Rubi steps
\begin {align*} \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2}} \, dx &=\frac {2 i}{11 d (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2}}+\frac {8 \int \frac {1}{(e \sec (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}} \, dx}{11 a}\\ &=\frac {2 i}{11 d (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2}}+\frac {16 i}{77 a d (e \sec (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}+\frac {48 \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{5/2}} \, dx}{77 a^2}\\ &=\frac {2 i}{11 d (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2}}+\frac {16 i}{77 a d (e \sec (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}-\frac {96 i \sqrt {a+i a \tan (c+d x)}}{385 a^2 d (e \sec (c+d x))^{5/2}}+\frac {192 \int \frac {1}{\sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx}{385 a e^2}\\ &=\frac {2 i}{11 d (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2}}+\frac {16 i}{77 a d (e \sec (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}+\frac {128 i}{385 a d e^2 \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {96 i \sqrt {a+i a \tan (c+d x)}}{385 a^2 d (e \sec (c+d x))^{5/2}}+\frac {128 \int \frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}} \, dx}{385 a^2 e^2}\\ &=\frac {2 i}{11 d (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2}}+\frac {16 i}{77 a d (e \sec (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}+\frac {128 i}{385 a d e^2 \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {96 i \sqrt {a+i a \tan (c+d x)}}{385 a^2 d (e \sec (c+d x))^{5/2}}-\frac {256 i \sqrt {a+i a \tan (c+d x)}}{385 a^2 d e^2 \sqrt {e \sec (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.55, size = 100, normalized size = 0.48 \[ -\frac {(e \sec (c+d x))^{3/2} (880 i \sin (2 (c+d x))+56 i \sin (4 (c+d x))+660 \cos (2 (c+d x))+21 \cos (4 (c+d x))-385)}{1540 a d e^4 (\tan (c+d x)-i) \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.66, size = 111, normalized size = 0.53 \[ \frac {\sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-77 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 1617 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 770 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 990 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 255 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 35 i\right )} e^{\left (-\frac {11}{2} i \, d x - \frac {11}{2} i \, c\right )}}{3080 \, a^{2} d e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \sec \left (d x + c\right )\right )^{\frac {5}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.17, size = 142, normalized size = 0.68 \[ \frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (70 i \left (\cos ^{6}\left (d x +c \right )\right )+70 \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )+5 i \left (\cos ^{4}\left (d x +c \right )\right )+40 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+16 i \left (\cos ^{2}\left (d x +c \right )\right )+64 \cos \left (d x +c \right ) \sin \left (d x +c \right )-128 i\right )}{385 d \,e^{5} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.16, size = 226, normalized size = 1.08 \[ \frac {35 i \, \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 220 i \, \cos \left (\frac {7}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) - 77 i \, \cos \left (\frac {5}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) + 770 i \, \cos \left (\frac {3}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) - 1540 i \, \cos \left (\frac {1}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) + 35 \, \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 220 \, \sin \left (\frac {7}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) + 77 \, \sin \left (\frac {5}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) + 770 \, \sin \left (\frac {3}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) + 1540 \, \sin \left (\frac {1}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right )}{3080 \, a^{\frac {3}{2}} d e^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.57, size = 127, normalized size = 0.61 \[ \frac {\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,\left (2310\,\sin \left (c+d\,x\right )+297\,\sin \left (3\,c+3\,d\,x\right )+35\,\sin \left (5\,c+5\,d\,x\right )-\cos \left (c+d\,x\right )\,770{}\mathrm {i}+\cos \left (3\,c+3\,d\,x\right )\,143{}\mathrm {i}+\cos \left (5\,c+5\,d\,x\right )\,35{}\mathrm {i}\right )}{3080\,a\,d\,e^3\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________