Optimal. Leaf size=186 \[ -\frac {2 i \sqrt {a+i a \tan (e+f x)}}{5 a c^2 f \sqrt {c-i c \tan (e+f x)}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{5 a c f (c-i c \tan (e+f x))^{3/2}}-\frac {3 i \sqrt {a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}+\frac {i}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}} \]
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Rubi [A] time = 0.16, 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 = {3523, 45, 37} \[ -\frac {2 i \sqrt {a+i a \tan (e+f x)}}{5 a c^2 f \sqrt {c-i c \tan (e+f x)}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{5 a c f (c-i c \tan (e+f x))^{3/2}}-\frac {3 i \sqrt {a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}+\frac {i}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 3523
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) \operatorname {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) \operatorname {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 \operatorname {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 \operatorname {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 = 4.21, size = 106, normalized size = 0.57 \[ \frac {\sqrt {c-i c \tan (e+f x)} (\cos (3 (e+f x))+i \sin (3 (e+f x))) (-5 \sin (e+f x)+3 \sin (3 (e+f x))-10 i \cos (e+f x)+2 i \cos (3 (e+f x)))}{20 c^3 f \sqrt {a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 122, normalized size = 0.66 \[ \frac {\sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (-i \, e^{\left (8 i \, f x + 8 i \, e\right )} - 6 i \, e^{\left (6 i \, f x + 6 i \, e\right )} - 20 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 16 i \, e^{\left (3 i \, f x + 3 i \, e\right )} - 10 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 16 i \, e^{\left (i \, f x + i \, e\right )} + 5 i\right )} e^{\left (-i \, f x - i \, e\right )}}{40 \, a c^{3} f} \]
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 \[ \int \frac {1}{\sqrt {i \, a \tan \left (f x + e\right ) + a} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 118, normalized size = 0.63 \[ \frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, \left (4 i \left (\tan ^{4}\left (f x +e \right )\right )+2 \left (\tan ^{5}\left (f x +e \right )\right )+6 i \left (\tan ^{2}\left (f x +e \right )\right )+\tan ^{3}\left (f x +e \right )+2 i-\tan \left (f x +e \right )\right )}{5 f \,c^{3} a \left (\tan \left (f x +e \right )+i\right )^{4} \left (-\tan \left (f x +e \right )+i\right )^{2}} \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ -\frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (\cos \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}-10\,\sin \left (2\,e+2\,f\,x\right )-\sin \left (4\,e+4\,f\,x\right )+15{}\mathrm {i}\right )}{40\,a\,c^2\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
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 \[ \int \frac {1}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )} \left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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