Optimal. Leaf size=105 \[ \frac {2 a^2 (3 B+i A)}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac {4 a^2 (B+i A)}{7 f (c-i c \tan (e+f x))^{7/2}}-\frac {2 a^2 B}{3 c^2 f (c-i c \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.19, antiderivative size = 105, 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} \[ \frac {2 a^2 (3 B+i A)}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac {4 a^2 (B+i A)}{7 f (c-i c \tan (e+f x))^{7/2}}-\frac {2 a^2 B}{3 c^2 f (c-i c \tan (e+f x))^{3/2}} \]
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))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {(a+i a x) (A+B x)}{(c-i c x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int \left (\frac {2 a (A-i B)}{(c-i c x)^{9/2}}-\frac {a (A-3 i B)}{c (c-i c x)^{7/2}}-\frac {i a B}{c^2 (c-i c x)^{5/2}}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {4 a^2 (i A+B)}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {2 a^2 (i A+3 B)}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac {2 a^2 B}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 13.08, size = 122, normalized size = 1.16 \[ \frac {a^2 \cos ^2(e+f x) \sqrt {c-i c \tan (e+f x)} (\cos (4 e+6 f x)+i \sin (4 e+6 f x)) (7 (3 A+i B) \sin (2 (e+f x))+(-37 B-9 i A) \cos (2 (e+f x))-9 i A+33 B)}{105 c^4 f (\cos (f x)+i \sin (f x))^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 120, normalized size = 1.14 \[ \frac {\sqrt {2} {\left ({\left (-15 i \, A - 15 \, B\right )} a^{2} e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (-39 i \, A + 3 \, B\right )} a^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-27 i \, A + 29 \, B\right )} a^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (3 i \, A - 11 \, B\right )} a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (6 i \, A - 22 \, B\right )} a^{2}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{420 \, c^{4} f} \]
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 \[ \int \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 80, normalized size = 0.76 \[ -\frac {2 i a^{2} \left (\frac {2 c^{2} \left (-i B +A \right )}{7 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}-\frac {i B}{3 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {c \left (-3 i B +A \right )}{5 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}\right )}{f \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 76, normalized size = 0.72 \[ \frac {2 i \, {\left (35 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2} B a^{2} + 21 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} {\left (A - 3 i \, B\right )} a^{2} c - 30 \, {\left (A - i \, B\right )} a^{2} c^{2}\right )}}{105 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} c^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.81, size = 167, normalized size = 1.59 \[ -\sqrt {c-\frac {c\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}}\,\left (-\frac {a^2\,\left (3\,A+B\,11{}\mathrm {i}\right )\,1{}\mathrm {i}}{210\,c^4\,f}-\frac {a^2\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (3\,A+B\,11{}\mathrm {i}\right )\,1{}\mathrm {i}}{420\,c^4\,f}+\frac {a^2\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (13\,A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{140\,c^4\,f}+\frac {a^2\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (27\,A+B\,29{}\mathrm {i}\right )\,1{}\mathrm {i}}{420\,c^4\,f}+\frac {a^2\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\left (A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{28\,c^4\,f}\right ) \]
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 \[ - a^{2} \left (\int \left (- \frac {A}{i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )} - 3 c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 3 i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \frac {A \tan ^{2}{\left (e + f x \right )}}{i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )} - 3 c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 3 i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {B \tan {\left (e + f x \right )}}{i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )} - 3 c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 3 i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \frac {B \tan ^{3}{\left (e + f x \right )}}{i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )} - 3 c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 3 i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {2 i A \tan {\left (e + f x \right )}}{i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )} - 3 c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 3 i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \left (- \frac {2 i B \tan ^{2}{\left (e + f x \right )}}{i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )} - 3 c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 3 i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{3} \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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