Optimal. Leaf size=92 \[ \frac {i A \sqrt {c-i c \tan (e+f x)}}{c f \sqrt {a+i a \tan (e+f x)}}-\frac {B+i A}{f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]
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Rubi [A] time = 0.20, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3588, 78, 37} \[ \frac {i A \sqrt {c-i c \tan (e+f x)}}{c f \sqrt {a+i a \tan (e+f x)}}-\frac {B+i A}{f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rule 3588
Rubi steps
\begin {align*} \int \frac {A+B \tan (e+f x)}{\sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {A+B x}{(a+i a x)^{3/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {i A+B}{f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}+\frac {(a A) \operatorname {Subst}\left (\int \frac {1}{(a+i a x)^{3/2} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {i A+B}{f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}+\frac {i A \sqrt {c-i c \tan (e+f x)}}{c f \sqrt {a+i a \tan (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 4.05, size = 77, normalized size = 0.84 \[ -\frac {\sqrt {c-i c \tan (e+f x)} (\cos (e+f x)+i \sin (e+f x)) (B \cos (e+f x)-A \sin (e+f x))}{c f \sqrt {a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.52, size = 114, normalized size = 1.24 \[ \frac {{\left ({\left (-i \, A - B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, B e^{\left (3 i \, f x + 3 i \, e\right )} - 2 \, B e^{\left (2 i \, f x + 2 i \, e\right )} + 2 \, B e^{\left (i \, f x + i \, e\right )} + i \, A - B\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-i \, f x - i \, e\right )}}{2 \, a c f} \]
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 \[ \int \frac {B \tan \left (f x + e\right ) + A}{\sqrt {i \, a \tan \left (f x + e\right ) + a} \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 99, normalized size = 1.08 \[ \frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, \left (A \left (\tan ^{3}\left (f x +e \right )\right )-B \left (\tan ^{2}\left (f x +e \right )\right )+A \tan \left (f x +e \right )-B \right )}{f c a \left (\tan \left (f x +e \right )+i\right )^{2} \left (-\tan \left (f x +e \right )+i\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.80, size = 124, normalized size = 1.35 \[ -\frac {{\left (2 \, {\left (A - i \, B\right )} \cos \left (4 \, f x + 4 \, e\right ) - 4 i \, B \cos \left (2 \, f x + 2 \, e\right ) + {\left (2 i \, A + 2 \, B\right )} \sin \left (4 \, f x + 4 \, e\right ) + 4 \, B \sin \left (2 \, f x + 2 \, e\right ) - 2 \, A - 2 i \, B\right )} \sqrt {a} \sqrt {c}}{{\left (-4 i \, a c \cos \left (3 \, f x + 3 \, e\right ) - 4 i \, a c \cos \left (f x + e\right ) + 4 \, a c \sin \left (3 \, f x + 3 \, e\right ) + 4 \, a c \sin \left (f x + e\right )\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.80, size = 143, normalized size = 1.55 \[ -\frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,1{}\mathrm {i}+B-A\,\cos \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}+B\,\cos \left (2\,e+2\,f\,x\right )-A\,\sin \left (2\,e+2\,f\,x\right )-B\,\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{2\,a\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
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 \[ \int \frac {A + B \tan {\left (e + f x \right )}}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )} \sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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