Optimal. Leaf size=104 \[ \frac {(4 B+i A) (c-i c \tan (e+f x))^{3/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac {(-B+i A) (c-i c \tan (e+f x))^{3/2}}{5 f (a+i a \tan (e+f x))^{5/2}} \]
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Rubi [A] time = 0.23, antiderivative size = 104, 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} \[ \frac {(4 B+i A) (c-i c \tan (e+f x))^{3/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac {(-B+i A) (c-i c \tan (e+f x))^{3/2}}{5 f (a+i a \tan (e+f x))^{5/2}} \]
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)) (c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {(A+B x) \sqrt {c-i c x}}{(a+i a x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(i A-B) (c-i c \tan (e+f x))^{3/2}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac {((A-4 i B) c) \operatorname {Subst}\left (\int \frac {\sqrt {c-i c x}}{(a+i a x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{5 f}\\ &=\frac {(i A-B) (c-i c \tan (e+f x))^{3/2}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac {(i A+4 B) (c-i c \tan (e+f x))^{3/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 8.01, size = 92, normalized size = 0.88 \[ \frac {c (1-i \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} ((A-4 i B) \tan (e+f x)-4 i A-B)}{15 a^2 f (\tan (e+f x)-i)^2 \sqrt {a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 97, normalized size = 0.93 \[ \frac {{\left ({\left (5 i \, A + 5 \, B\right )} c e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (8 i \, A + 2 \, B\right )} c e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (3 i \, A - 3 \, B\right )} c\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-5 i \, f x - 5 i \, e\right )}}{30 \, a^{3} f} \]
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 \[ \int \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 92, normalized size = 0.88 \[ \frac {i \sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (i A \tan \left (f x +e \right )-i B +4 B \tan \left (f x +e \right )+4 A \right )}{15 f \,a^{3} \left (-\tan \left (f x +e \right )+i\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 150, normalized size = 1.44 \[ \frac {{\left (150 \, {\left (A - i \, B\right )} c \cos \left (4 \, f x + 4 \, e\right ) + 60 \, {\left (4 \, A - i \, B\right )} c \cos \left (2 \, f x + 2 \, e\right ) + {\left (150 i \, A + 150 \, B\right )} c \sin \left (4 \, f x + 4 \, e\right ) + {\left (240 i \, A + 60 \, B\right )} c \sin \left (2 \, f x + 2 \, e\right ) + 90 \, {\left (A + i \, B\right )} c\right )} \sqrt {a} \sqrt {c}}{{\left (-900 i \, a^{3} \cos \left (7 \, f x + 7 \, e\right ) - 900 i \, a^{3} \cos \left (5 \, f x + 5 \, e\right ) + 900 \, a^{3} \sin \left (7 \, f x + 7 \, e\right ) + 900 \, a^{3} \sin \left (5 \, f x + 5 \, e\right )\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.07, size = 240, normalized size = 2.31 \[ \frac {c\,\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,\cos \left (2\,e+2\,f\,x\right )\,5{}\mathrm {i}+A\,\cos \left (4\,e+4\,f\,x\right )\,8{}\mathrm {i}+A\,\cos \left (6\,e+6\,f\,x\right )\,3{}\mathrm {i}+5\,B\,\cos \left (2\,e+2\,f\,x\right )+2\,B\,\cos \left (4\,e+4\,f\,x\right )-3\,B\,\cos \left (6\,e+6\,f\,x\right )+5\,A\,\sin \left (2\,e+2\,f\,x\right )+8\,A\,\sin \left (4\,e+4\,f\,x\right )+3\,A\,\sin \left (6\,e+6\,f\,x\right )-B\,\sin \left (2\,e+2\,f\,x\right )\,5{}\mathrm {i}-B\,\sin \left (4\,e+4\,f\,x\right )\,2{}\mathrm {i}+B\,\sin \left (6\,e+6\,f\,x\right )\,3{}\mathrm {i}\right )}{60\,a^3\,f} \]
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 \[ \int \frac {\left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {3}{2}} \left (A + B \tan {\left (e + f x \right )}\right )}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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