Optimal. Leaf size=186 \[ \frac {i}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}-\frac {3 i \sqrt {a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{5 a c f (c-i c \tan (e+f x))^{3/2}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{5 a c^2 f \sqrt {c-i c \tan (e+f x)}} \]
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Rubi [A]
time = 0.11, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps
used = 5, 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)}}{5 a c^2 f \sqrt {c-i c \tan (e+f x)}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{5 a c f (c-i c \tan (e+f x))^{3/2}}-\frac {3 i \sqrt {a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}+\frac {i}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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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))^{5/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {1}{(a+i a x)^{3/2} (c-i c x)^{7/2}} \, 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))^{5/2}}+\frac {(3 c) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} (c-i c x)^{7/2}} \, 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))^{5/2}}-\frac {3 i \sqrt {a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}+\frac {6 \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{5 f}\\ &=\frac {i}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}-\frac {3 i \sqrt {a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{5 a c 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 )}{5 c f}\\ &=\frac {i}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}-\frac {3 i \sqrt {a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{5 a c f (c-i c \tan (e+f x))^{3/2}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{5 a c^2 f \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 2.09, size = 106, normalized size = 0.57 \begin {gather*} \frac {(\cos (3 (e+f x))+i \sin (3 (e+f x))) (-10 i \cos (e+f x)+2 i \cos (3 (e+f x))-5 \sin (e+f x)+3 \sin (3 (e+f x))) \sqrt {c-i c \tan (e+f x)}}{20 c^3 f \sqrt {a+i a \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 118, normalized size = 0.63
method | result | size |
risch | \(-\frac {i \left ({\mathrm e}^{6 i \left (f x +e \right )}+5 \,{\mathrm e}^{4 i \left (f x +e \right )}+15 \,{\mathrm e}^{2 i \left (f x +e \right )}-5\right )}{40 c^{2} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) f}\) | \(99\) |
derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (4 i \left (\tan ^{4}\left (f x +e \right )\right )+2 \left (\tan ^{5}\left (f x +e \right )\right )+6 i \left (\tan ^{2}\left (f x +e \right )\right )+\tan ^{3}\left (f x +e \right )+2 i-\tan \left (f x +e \right )\right )}{5 f a \,c^{3} \left (\tan \left (f x +e \right )+i\right )^{4} \left (-\tan \left (f x +e \right )+i\right )^{2}}\) | \(118\) |
default | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (4 i \left (\tan ^{4}\left (f x +e \right )\right )+2 \left (\tan ^{5}\left (f x +e \right )\right )+6 i \left (\tan ^{2}\left (f x +e \right )\right )+\tan ^{3}\left (f x +e \right )+2 i-\tan \left (f x +e \right )\right )}{5 f a \,c^{3} \left (\tan \left (f x +e \right )+i\right )^{4} \left (-\tan \left (f x +e \right )+i\right )^{2}}\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 1.18, size = 131, normalized size = 0.70 \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 (8 i \, f x + 8 i \, e\right )} - 6 i \, e^{\left (6 i \, f x + 6 i \, e\right )} - 20 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 16 i \, e^{\left (3 i \, f x + 3 i \, e\right )} - 10 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 16 i \, e^{\left (i \, f x + i \, e\right )} + 5 i\right )} e^{\left (-i \, f x - i \, e\right )}}{40 \, a c^{3} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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 {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [B]
time = 5.36, size = 128, normalized size = 0.69 \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 (4\,e+4\,f\,x\right )\,1{}\mathrm {i}-10\,\sin \left (2\,e+2\,f\,x\right )-\sin \left (4\,e+4\,f\,x\right )+15{}\mathrm {i}\right )}{40\,a\,c^2\,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.
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