Optimal. Leaf size=154 \[ \frac {10 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (e \cos (c+d x))^{3/2}}{33 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 i \cos ^2(c+d x) (e \cos (c+d x))^{3/2}}{11 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {2 \sin (c+d x) \cos (c+d x) (e \cos (c+d x))^{3/2}}{11 a^2 d}+\frac {10 \tan (c+d x) (e \cos (c+d x))^{3/2}}{33 a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3515, 3500, 3769, 3771, 2641} \[ \frac {4 i \cos ^2(c+d x) (e \cos (c+d x))^{3/2}}{11 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {10 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (e \cos (c+d x))^{3/2}}{33 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \cos (c+d x) (e \cos (c+d x))^{3/2}}{11 a^2 d}+\frac {10 \tan (c+d x) (e \cos (c+d x))^{3/2}}{33 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2641
Rule 3500
Rule 3515
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2} \, dx &=\left ((e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}\right ) \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2} \, dx\\ &=\frac {4 i \cos ^2(c+d x) (e \cos (c+d x))^{3/2}}{11 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (7 e^2 (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}\right ) \int \frac {1}{(e \sec (c+d x))^{7/2}} \, dx}{11 a^2}\\ &=\frac {2 \cos (c+d x) (e \cos (c+d x))^{3/2} \sin (c+d x)}{11 a^2 d}+\frac {4 i \cos ^2(c+d x) (e \cos (c+d x))^{3/2}}{11 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (5 (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}\right ) \int \frac {1}{(e \sec (c+d x))^{3/2}} \, dx}{11 a^2}\\ &=\frac {2 \cos (c+d x) (e \cos (c+d x))^{3/2} \sin (c+d x)}{11 a^2 d}+\frac {10 (e \cos (c+d x))^{3/2} \tan (c+d x)}{33 a^2 d}+\frac {4 i \cos ^2(c+d x) (e \cos (c+d x))^{3/2}}{11 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (5 (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}\right ) \int \sqrt {e \sec (c+d x)} \, dx}{33 a^2 e^2}\\ &=\frac {2 \cos (c+d x) (e \cos (c+d x))^{3/2} \sin (c+d x)}{11 a^2 d}+\frac {10 (e \cos (c+d x))^{3/2} \tan (c+d x)}{33 a^2 d}+\frac {4 i \cos ^2(c+d x) (e \cos (c+d x))^{3/2}}{11 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (5 (e \cos (c+d x))^{3/2}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{33 a^2 \cos ^{\frac {3}{2}}(c+d x)}\\ &=\frac {10 (e \cos (c+d x))^{3/2} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{33 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \cos (c+d x) (e \cos (c+d x))^{3/2} \sin (c+d x)}{11 a^2 d}+\frac {10 (e \cos (c+d x))^{3/2} \tan (c+d x)}{33 a^2 d}+\frac {4 i \cos ^2(c+d x) (e \cos (c+d x))^{3/2}}{11 d \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.68, size = 131, normalized size = 0.85 \[ \frac {(e \cos (c+d x))^{3/2} \left (\sqrt {\cos (c+d x)} (13 \sin (c+d x)-7 \sin (3 (c+d x))-28 i \cos (c+d x)+4 i \cos (3 (c+d x)))-20 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))\right )}{66 a^2 d \cos ^{\frac {7}{2}}(c+d x) (\tan (c+d x)-i)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ \frac {{\left (132 \, a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} {\rm integral}\left (-\frac {10 i \, \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{33 \, {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}}, x\right ) + \sqrt {\frac {1}{2}} {\left (-11 i \, e e^{\left (6 i \, d x + 6 i \, c\right )} + 41 i \, e e^{\left (4 i \, d x + 4 i \, c\right )} + 15 i \, e e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i \, e\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{132 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 6.10, size = 315, normalized size = 2.05 \[ \frac {2 e^{2} \left (384 i \left (\sin ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-384 \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-1152 i \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+960 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+1440 i \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1008 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-960 i \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+552 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+360 i \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-176 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-72 i \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+28 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+6 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{33 a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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]
________________________________________________________________________________________