Optimal. Leaf size=140 \[ -\frac {2 a^3 (5 B+i A)}{c^2 f \sqrt {c-i c \tan (e+f x)}}+\frac {8 a^3 (2 B+i A)}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac {8 a^3 (B+i A)}{5 f (c-i c \tan (e+f x))^{5/2}}-\frac {2 a^3 B \sqrt {c-i c \tan (e+f x)}}{c^3 f} \]
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Rubi [A] time = 0.20, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {3588, 77} \[ -\frac {2 a^3 (5 B+i A)}{c^2 f \sqrt {c-i c \tan (e+f x)}}+\frac {8 a^3 (2 B+i A)}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac {8 a^3 (B+i A)}{5 f (c-i c \tan (e+f x))^{5/2}}-\frac {2 a^3 B \sqrt {c-i c \tan (e+f x)}}{c^3 f} \]
Antiderivative was successfully verified.
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Rule 77
Rule 3588
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {(a+i a x)^2 (A+B x)}{(c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int \left (\frac {4 a^2 (A-i B)}{(c-i c x)^{7/2}}-\frac {4 a^2 (A-2 i B)}{c (c-i c x)^{5/2}}+\frac {a^2 (A-5 i B)}{c^2 (c-i c x)^{3/2}}+\frac {i a^2 B}{c^3 \sqrt {c-i c x}}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {8 a^3 (i A+B)}{5 f (c-i c \tan (e+f x))^{5/2}}+\frac {8 a^3 (i A+2 B)}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac {2 a^3 (i A+5 B)}{c^2 f \sqrt {c-i c \tan (e+f x)}}-\frac {2 a^3 B \sqrt {c-i c \tan (e+f x)}}{c^3 f}\\ \end {align*}
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Mathematica [A] time = 13.05, size = 135, normalized size = 0.96 \[ \frac {a^3 \sqrt {c-i c \tan (e+f x)} (\sin (3 (e+2 f x))-i \cos (3 (e+2 f x))) (3 (A-11 i B) \cos (e+f x)+(11 A-91 i B) \cos (3 (e+f x))-10 i \sin (e+f x) ((A-17 i B) \cos (2 (e+f x))+A-14 i B))}{15 c^3 f (\cos (f x)+i \sin (f x))^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 100, normalized size = 0.71 \[ \frac {\sqrt {2} {\left ({\left (-3 i \, A - 3 \, B\right )} a^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (i \, A + 11 \, B\right )} a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (-4 i \, A - 44 \, B\right )} a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-8 i \, A - 88 \, B\right )} a^{3}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{15 \, c^{3} f} \]
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 \[ \int \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 105, normalized size = 0.75 \[ \frac {2 i a^{3} \left (i B \sqrt {c -i c \tan \left (f x +e \right )}+\frac {4 c^{2} \left (-2 i B +A \right )}{3 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {4 c^{3} \left (-i B +A \right )}{5 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {c \left (-5 i B +A \right )}{\sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f \,c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 107, normalized size = 0.76 \[ -\frac {2 i \, {\left (-\frac {15 i \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} B a^{3}}{c^{2}} + \frac {15 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2} {\left (A - 5 i \, B\right )} a^{3} - 20 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} {\left (A - 2 i \, B\right )} a^{3} c + 12 \, {\left (A - i \, B\right )} a^{3} c^{2}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c}\right )}}{15 \, c f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.28, size = 208, normalized size = 1.49 \[ -\frac {a^3\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,8{}\mathrm {i}+88\,B+A\,\cos \left (2\,e+2\,f\,x\right )\,4{}\mathrm {i}-A\,\cos \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}+A\,\cos \left (6\,e+6\,f\,x\right )\,3{}\mathrm {i}+44\,B\,\cos \left (2\,e+2\,f\,x\right )-11\,B\,\cos \left (4\,e+4\,f\,x\right )+3\,B\,\cos \left (6\,e+6\,f\,x\right )-4\,A\,\sin \left (2\,e+2\,f\,x\right )+A\,\sin \left (4\,e+4\,f\,x\right )-3\,A\,\sin \left (6\,e+6\,f\,x\right )+B\,\sin \left (2\,e+2\,f\,x\right )\,44{}\mathrm {i}-B\,\sin \left (4\,e+4\,f\,x\right )\,11{}\mathrm {i}+B\,\sin \left (6\,e+6\,f\,x\right )\,3{}\mathrm {i}\right )}{15\,c^3\,f} \]
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 \[ - i a^{3} \left (\int \frac {i A}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {3 A \tan {\left (e + f x \right )}}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \frac {A \tan ^{3}{\left (e + f x \right )}}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {3 B \tan ^{2}{\left (e + f x \right )}}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \frac {B \tan ^{4}{\left (e + f x \right )}}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {3 i A \tan ^{2}{\left (e + f x \right )}}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \frac {i B \tan {\left (e + f x \right )}}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {3 i B \tan ^{3}{\left (e + f x \right )}}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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