Optimal. Leaf size=142 \[ -\frac {8 a^3 (i A+B)}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {8 a^3 (i A+2 B)}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac {2 a^3 (i A+5 B)}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}+\frac {2 a^3 B}{c^3 f \sqrt {c-i c \tan (e+f x)}} \]
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Rubi [A]
time = 0.14, antiderivative size = 142, 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 = {3669, 78}
\begin {gather*} -\frac {2 a^3 (5 B+i A)}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}+\frac {8 a^3 (2 B+i A)}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac {8 a^3 (B+i A)}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {2 a^3 B}{c^3 f \sqrt {c-i c \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 3669
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))^{7/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {(a+i a x)^2 (A+B x)}{(c-i c x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \text {Subst}\left (\int \left (\frac {4 a^2 (A-i B)}{(c-i c x)^{9/2}}-\frac {4 a^2 (A-2 i B)}{c (c-i c x)^{7/2}}+\frac {a^2 (A-5 i B)}{c^2 (c-i c x)^{5/2}}+\frac {i a^2 B}{c^3 (c-i c x)^{3/2}}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {8 a^3 (i A+B)}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {8 a^3 (i A+2 B)}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac {2 a^3 (i A+5 B)}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}+\frac {2 a^3 B}{c^3 f \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 6.20, size = 141, normalized size = 0.99 \begin {gather*} \frac {a^3 \cos (e+f x) (i (A+13 i B) \cos (e+f x)+(-23 i A+89 B) \cos (3 (e+f x))+14 (A-2 i B+(A-17 i B) \cos (2 (e+f x))) \sin (e+f x)) (\cos (4 e+7 f x)+i \sin (4 e+7 f x)) \sqrt {c-i c \tan (e+f x)}}{105 c^4 f (\cos (f x)+i \sin (f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 105, normalized size = 0.74
method | result | size |
derivativedivides | \(\frac {2 i a^{3} \left (-\frac {4 c^{3} \left (-i B +A \right )}{7 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}+\frac {4 c^{2} \left (-2 i B +A \right )}{5 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {c \left (-5 i B +A \right )}{3 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {i B}{\sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f \,c^{3}}\) | \(105\) |
default | \(\frac {2 i a^{3} \left (-\frac {4 c^{3} \left (-i B +A \right )}{7 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}+\frac {4 c^{2} \left (-2 i B +A \right )}{5 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {c \left (-5 i B +A \right )}{3 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {i B}{\sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f \,c^{3}}\) | \(105\) |
risch | \(-\frac {a^{3} \left (15 i A \,{\mathrm e}^{6 i \left (f x +e \right )}+15 B \,{\mathrm e}^{6 i \left (f x +e \right )}+3 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-39 B \,{\mathrm e}^{4 i \left (f x +e \right )}-4 i A \,{\mathrm e}^{2 i \left (f x +e \right )}+52 B \,{\mathrm e}^{2 i \left (f x +e \right )}+8 i A -104 B \right ) \sqrt {2}}{210 c^{3} \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 106, normalized size = 0.75 \begin {gather*} -\frac {2 i \, {\left (105 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{3} B a^{3} + 35 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2} {\left (A - 5 i \, B\right )} a^{3} c - 84 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} {\left (A - 2 i \, B\right )} a^{3} c^{2} + 60 \, {\left (A - i \, B\right )} a^{3} c^{3}\right )}}{105 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} c^{3} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.46, size = 128, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {2} {\left (15 \, {\left (i \, A + B\right )} a^{3} e^{\left (8 i \, f x + 8 i \, e\right )} + 6 \, {\left (3 i \, A - 4 \, B\right )} a^{3} e^{\left (6 i \, f x + 6 i \, e\right )} - {\left (i \, A - 13 \, B\right )} a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, {\left (i \, A - 13 \, B\right )} a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 \, {\left (i \, A - 13 \, B\right )} a^{3}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{210 \, c^{4} f} \end {gather*}
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 \begin {gather*} - i a^{3} \left (\int \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}}\, dx + \int \left (- \frac {3 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 \frac {A \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 {3 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 + \int \frac {B \tan ^{4}{\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 {3 i 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}}\right )\, dx + \int \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}}\, dx + \int \left (- \frac {3 i 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}}\right )\, dx\right ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.73, size = 161, normalized size = 1.13 \begin {gather*} -\sqrt {c-\frac {c\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}}\,\left (\frac {a^3\,\left (A+B\,13{}\mathrm {i}\right )\,4{}\mathrm {i}}{105\,c^4\,f}+\frac {a^3\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (3\,A+B\,4{}\mathrm {i}\right )\,1{}\mathrm {i}}{35\,c^4\,f}+\frac {a^3\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (A+B\,13{}\mathrm {i}\right )\,2{}\mathrm {i}}{105\,c^4\,f}+\frac {a^3\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\left (A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{14\,c^4\,f}-\frac {a^3\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (A+B\,13{}\mathrm {i}\right )\,1{}\mathrm {i}}{210\,c^4\,f}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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