Optimal. Leaf size=137 \[ \frac {i}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{3 a c f \sqrt {c-i c \tan (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {3604, 47, 37}
\begin {gather*} -\frac {2 i \sqrt {a+i a \tan (e+f x)}}{3 a c f \sqrt {c-i c \tan (e+f x)}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}+\frac {i}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 47
Rule 3604
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {1}{(a+i a x)^{3/2} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}+\frac {(2 c) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 f}\\ &=\frac {i}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{3 a c f \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.35, size = 89, normalized size = 0.65 \begin {gather*} \frac {i (\cos (2 (e+f x))+i \sin (2 (e+f x))) (-3+\cos (2 (e+f x))-2 i \sin (2 (e+f x))) \sqrt {c-i c \tan (e+f x)}}{6 c^2 f \sqrt {a+i a \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.40, size = 109, normalized size = 0.80
method | result | size |
risch | \(-\frac {i \left ({\mathrm e}^{4 i \left (f x +e \right )}+6 \,{\mathrm e}^{2 i \left (f x +e \right )}-3\right )}{12 c \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(88\) |
derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (2 i \left (\tan ^{3}\left (f x +e \right )\right )+2 \left (\tan ^{4}\left (f x +e \right )\right )+2 i \tan \left (f x +e \right )+3 \left (\tan ^{2}\left (f x +e \right )\right )+1\right )}{3 f a \,c^{2} \left (\tan \left (f x +e \right )+i\right )^{3} \left (-\tan \left (f x +e \right )+i\right )^{2}}\) | \(109\) |
default | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (2 i \left (\tan ^{3}\left (f x +e \right )\right )+2 \left (\tan ^{4}\left (f x +e \right )\right )+2 i \tan \left (f x +e \right )+3 \left (\tan ^{2}\left (f x +e \right )\right )+1\right )}{3 f a \,c^{2} \left (\tan \left (f x +e \right )+i\right )^{3} \left (-\tan \left (f x +e \right )+i\right )^{2}}\) | \(109\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.90, size = 119, normalized size = 0.87 \begin {gather*} \frac {\sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (-i \, e^{\left (6 i \, f x + 6 i \, e\right )} - 7 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 4 i \, e^{\left (3 i \, f x + 3 i \, e\right )} - 3 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 4 i \, e^{\left (i \, f x + i \, e\right )} + 3 i\right )} e^{\left (-i \, f x - i \, e\right )}}{12 \, a c^{2} f} \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}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )} \left (- i c \left (\tan {\left (e + f 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 = 0.39, size = 117, normalized size = 0.85 \begin {gather*} \frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (\cos \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}+2\,\sin \left (2\,e+2\,f\,x\right )-3{}\mathrm {i}\right )}{6\,a\,c\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________