Optimal. Leaf size=152 \[ \frac {-B-i A}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac {2 A \tan (e+f x)}{3 a c f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}+\frac {i A}{3 c f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \]
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Rubi [A] time = 0.25, antiderivative size = 150, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3588, 78, 45, 39} \[ -\frac {B+i A}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac {2 A \tan (e+f x)}{3 a c f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}+\frac {i A}{3 c f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 39
Rule 45
Rule 78
Rule 3588
Rubi steps
\begin {align*} \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {A+B x}{(a+i a x)^{5/2} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {i A+B}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac {(a A) \operatorname {Subst}\left (\int \frac {1}{(a+i a x)^{5/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {i A+B}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac {i A}{3 c f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}+\frac {(2 A) \operatorname {Subst}\left (\int \frac {1}{(a+i a x)^{3/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 f}\\ &=-\frac {i A+B}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac {i A}{3 c f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}+\frac {2 A \tan (e+f x)}{3 a c f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 8.51, size = 120, normalized size = 0.79 \[ \frac {\sqrt {c-i c \tan (e+f x)} (\sin (2 (e+f x))-i \cos (2 (e+f x))) (9 A \tan (e+f x)+A \sin (3 (e+f x)) \sec (e+f x)-2 B \cos (2 (e+f x))-2 B)}{12 a c^2 f (\tan (e+f x)-i) \sqrt {a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.32, size = 148, normalized size = 0.97 \[ \frac {{\left ({\left (-i \, A - B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (-10 i \, A - 4 \, B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + 8 \, B e^{\left (5 i \, f x + 5 i \, e\right )} - 6 \, B e^{\left (4 i \, f x + 4 i \, e\right )} + 8 \, B e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (10 i \, A - 4 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, A - B\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-3 i \, f x - 3 i \, e\right )}}{24 \, a^{2} c^{2} f} \]
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 \[ \int \frac {B \tan \left (f x + e\right ) + A}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 113, normalized size = 0.74 \[ -\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, \left (2 A \left (\tan ^{5}\left (f x +e \right )\right )+5 A \left (\tan ^{3}\left (f x +e \right )\right )-B \left (\tan ^{2}\left (f x +e \right )\right )+3 A \tan \left (f x +e \right )-B \right )}{3 f \,c^{2} a^{2} \left (\tan \left (f x +e \right )+i\right )^{3} \left (-\tan \left (f x +e \right )+i\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 199, normalized size = 1.31 \[ -\frac {{\left ({\left (-9 i \, A + 3 \, B\right )} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, {\left (3 \, A + i \, B\right )} \sin \left (2 \, f x + 2 \, e\right ) + 2 \, B\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + {\left (9 i \, A + 3 \, B\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - {\left (3 \, {\left (3 \, A + i \, B\right )} \cos \left (2 \, f x + 2 \, e\right ) - {\left (-9 i \, A + 3 \, B\right )} \sin \left (2 \, f x + 2 \, e\right ) + 2 \, A\right )} \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 3 \, {\left (3 \, A - i \, B\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )}{24 \, a^{\frac {3}{2}} c^{\frac {3}{2}} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.94, size = 198, normalized size = 1.30 \[ \frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (10\,A\,\sin \left (2\,e+2\,f\,x\right )-3\,B+A\,\cos \left (2\,e+2\,f\,x\right )\,8{}\mathrm {i}+A\,\cos \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}-4\,B\,\cos \left (2\,e+2\,f\,x\right )-B\,\cos \left (4\,e+4\,f\,x\right )-A\,9{}\mathrm {i}+A\,\sin \left (4\,e+4\,f\,x\right )+B\,\sin \left (2\,e+2\,f\,x\right )\,2{}\mathrm {i}+B\,\sin \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}\right )}{24\,a^2\,c\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
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 \[ \int \frac {A + B \tan {\left (e + f x \right )}}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}} \left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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