Optimal. Leaf size=60 \[ -\frac {2 a (i A+B)}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {2 a B}{c f \sqrt {c-i c \tan (e+f x)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {3669, 45}
\begin {gather*} \frac {2 a B}{c f \sqrt {c-i c \tan (e+f x)}}-\frac {2 a (B+i A)}{3 f (c-i c \tan (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3669
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x)) (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {A+B x}{(c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \text {Subst}\left (\int \left (\frac {A-i B}{(c-i c x)^{5/2}}+\frac {i B}{c (c-i c x)^{3/2}}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {2 a (i A+B)}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {2 a B}{c f \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 1.23, size = 98, normalized size = 1.63 \begin {gather*} \frac {2 a \cos (e+f x) (\cos (f x)-i \sin (f x)) ((-i A+2 B) \cos (e+f x)-3 i B \sin (e+f x)) (\cos (2 e+3 f x)+i \sin (2 e+3 f x)) \sqrt {c-i c \tan (e+f x)}}{3 c^2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 53, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {2 i a \left (-\frac {i B}{\sqrt {c -i c \tan \left (f x +e \right )}}-\frac {c \left (-i B +A \right )}{3 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\right )}{f c}\) | \(53\) |
default | \(\frac {2 i a \left (-\frac {i B}{\sqrt {c -i c \tan \left (f x +e \right )}}-\frac {c \left (-i B +A \right )}{3 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\right )}{f c}\) | \(53\) |
risch | \(-\frac {a \left (i A \,{\mathrm e}^{2 i \left (f x +e \right )}+B \,{\mathrm e}^{2 i \left (f x +e \right )}+i A -5 B \right ) \sqrt {2}}{6 c \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 47, normalized size = 0.78 \begin {gather*} -\frac {2 i \, {\left (3 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} B a + {\left (A - i \, B\right )} a c\right )}}{3 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.37, size = 78, normalized size = 1.30 \begin {gather*} \frac {\sqrt {2} {\left ({\left (-i \, A - B\right )} a e^{\left (4 i \, f x + 4 i \, e\right )} - 2 \, {\left (i \, A - 2 \, B\right )} a e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, A + 5 \, B\right )} a\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{6 \, c^{2} 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*} i a \left (\int \left (- \frac {i A}{- i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \frac {A \tan {\left (e + f x \right )}}{- i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \frac {B \tan ^{2}{\left (e + f x \right )}}{- i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {i B \tan {\left (e + f x \right )}}{- i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx\right ) \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 = 10.19, size = 170, normalized size = 2.83 \begin {gather*} -\frac {a\,\sqrt {-\frac {c\,\left (-2\,{\cos \left (e+f\,x\right )}^2+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{2\,{\cos \left (e+f\,x\right )}^2}}\,\left (B\,\left (2\,{\cos \left (2\,e+2\,f\,x\right )}^2-1\right )-4\,B\,\left (2\,{\cos \left (e+f\,x\right )}^2-1\right )-2\,A\,\sin \left (2\,e+2\,f\,x\right )-A\,\sin \left (4\,e+4\,f\,x\right )-5\,B+A\,1{}\mathrm {i}+A\,\left (2\,{\cos \left (e+f\,x\right )}^2-1\right )\,2{}\mathrm {i}-B\,\sin \left (2\,e+2\,f\,x\right )\,4{}\mathrm {i}+B\,\sin \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}+A\,\left (2\,{\cos \left (2\,e+2\,f\,x\right )}^2-1\right )\,1{}\mathrm {i}\right )}{6\,c^2\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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