Optimal. Leaf size=62 \[ \frac {2 a B}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac {2 a (B+i A)}{7 f (c-i c \tan (e+f x))^{7/2}} \]
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Rubi [A] time = 0.10, antiderivative size = 62, 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 = {3588, 43} \[ \frac {2 a B}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac {2 a (B+i A)}{7 f (c-i c \tan (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3588
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))^{7/2}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {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 {A-i B}{(c-i c x)^{9/2}}+\frac {i B}{c (c-i c x)^{7/2}}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {2 a (i A+B)}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {2 a B}{5 c f (c-i c \tan (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [A] time = 11.19, size = 100, normalized size = 1.61 \[ \frac {2 a \cos ^3(e+f x) (\cos (f x)-i \sin (f x)) \sqrt {c-i c \tan (e+f x)} (\cos (4 e+5 f x)+i \sin (4 e+5 f x)) ((2 B-5 i A) \cos (e+f x)-7 i B \sin (e+f x))}{35 c^4 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.56, size = 110, normalized size = 1.77 \[ \frac {\sqrt {2} {\left ({\left (-5 i \, A - 5 \, B\right )} a e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (-20 i \, A - 6 \, B\right )} a e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-30 i \, A + 12 \, B\right )} a e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (-20 i \, A + 22 \, B\right )} a e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-5 i \, A + 9 \, B\right )} a\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{280 \, c^{4} f} \]
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 \[ \int \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 53, normalized size = 0.85 \[ \frac {2 i a \left (-\frac {c \left (-i B +A \right )}{7 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}-\frac {i B}{5 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}\right )}{f c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 46, normalized size = 0.74 \[ -\frac {2 i \, {\left (7 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} B a + 5 \, {\left (A - i \, B\right )} a c\right )}}{35 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} c f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.23, size = 157, normalized size = 2.53 \[ -\sqrt {c-\frac {c\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}}\,\left (\frac {a\,\left (5\,A+B\,9{}\mathrm {i}\right )\,1{}\mathrm {i}}{280\,c^4\,f}+\frac {a\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\left (A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{56\,c^4\,f}+\frac {a\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (5\,A+B\,2{}\mathrm {i}\right )\,3{}\mathrm {i}}{140\,c^4\,f}+\frac {a\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (10\,A+B\,11{}\mathrm {i}\right )\,1{}\mathrm {i}}{140\,c^4\,f}+\frac {a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (10\,A-B\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{140\,c^4\,f}\right ) \]
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 \[ i a \left (\int \left (- \frac {i A}{i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )} - 3 c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 3 i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \frac {A \tan {\left (e + f x \right )}}{i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )} - 3 c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 3 i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \frac {B \tan ^{2}{\left (e + f x \right )}}{i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )} - 3 c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 3 i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {i B \tan {\left (e + f x \right )}}{i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )} - 3 c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 3 i c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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