Optimal. Leaf size=160 \[ -\frac {a^{3/2} (2 i A+B) \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{f}+\frac {a (2 i A+B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {B (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{2 f} \]
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Rubi [A]
time = 0.17, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3669, 81, 52,
65, 223, 209} \begin {gather*} -\frac {a^{3/2} \sqrt {c} (B+2 i A) \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{f}+\frac {a (B+2 i A) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {B (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{2 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 209
Rule 223
Rule 3669
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {\sqrt {a+i a x} (A+B x)}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {B (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {(a (2 A-i B) c) \text {Subst}\left (\int \frac {\sqrt {a+i a x}}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac {a (2 i A+B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {B (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {\left (a^2 (2 A-i B) c\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac {a (2 i A+B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {B (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{2 f}-\frac {(a (2 i A+B) c) \text {Subst}\left (\int \frac {1}{\sqrt {2 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+i a \tan (e+f x)}\right )}{f}\\ &=\frac {a (2 i A+B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {B (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{2 f}-\frac {(a (2 i A+B) c) \text {Subst}\left (\int \frac {1}{1+\frac {c x^2}{a}} \, dx,x,\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c-i c \tan (e+f x)}}\right )}{f}\\ &=-\frac {a^{3/2} (2 i A+B) \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{f}+\frac {a (2 i A+B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {B (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{2 f}\\ \end {align*}
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Mathematica [A]
time = 2.99, size = 220, normalized size = 1.38 \begin {gather*} \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) \left (-\frac {i (2 A-i B) c e^{-2 i (e+f x)} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \text {ArcTan}\left (e^{i (e+f x)}\right )}{\sqrt {\frac {c}{1+e^{2 i (e+f x)}}}}+\frac {\cos (e) \sqrt {\sec (e+f x)} (i+\tan (e)) (2 A-2 i B+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{2 \cos (f x)+2 i \sin (f x)}\right )}{f \sec ^{\frac {5}{2}}(e+f x) (A \cos (e+f x)+B \sin (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 223, normalized size = 1.39
method | result | size |
derivativedivides | \(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, a \left (i B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )-i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c +2 i A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}+2 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c +2 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{2 f \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}\) | \(223\) |
default | \(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, a \left (i B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )-i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c +2 i A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}+2 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c +2 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{2 f \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}\) | \(223\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 819 vs. \(2 (126) = 252\).
time = 0.67, size = 819, normalized size = 5.12 \begin {gather*} \frac {4 \, {\left (4 \, {\left (2 \, A - 3 i \, B\right )} a \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 4 \, {\left (2 \, A - i \, B\right )} a \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 4 \, {\left (2 i \, A + 3 \, B\right )} a \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 4 \, {\left (2 i \, A + B\right )} a \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 2 \, {\left ({\left (2 \, A - i \, B\right )} a \cos \left (4 \, f x + 4 \, e\right ) + 2 \, {\left (2 \, A - i \, B\right )} a \cos \left (2 \, f x + 2 \, e\right ) - {\left (-2 i \, A - B\right )} a \sin \left (4 \, f x + 4 \, e\right ) - 2 \, {\left (-2 i \, A - B\right )} a \sin \left (2 \, f x + 2 \, e\right ) + {\left (2 \, A - i \, B\right )} a\right )} \arctan \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ), \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) - 2 \, {\left ({\left (2 \, A - i \, B\right )} a \cos \left (4 \, f x + 4 \, e\right ) + 2 \, {\left (2 \, A - i \, B\right )} a \cos \left (2 \, f x + 2 \, e\right ) - {\left (-2 i \, A - B\right )} a \sin \left (4 \, f x + 4 \, e\right ) - 2 \, {\left (-2 i \, A - B\right )} a \sin \left (2 \, f x + 2 \, e\right ) + {\left (2 \, A - i \, B\right )} a\right )} \arctan \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ), -\sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + {\left ({\left (-2 i \, A - B\right )} a \cos \left (4 \, f x + 4 \, e\right ) + 2 \, {\left (-2 i \, A - B\right )} a \cos \left (2 \, f x + 2 \, e\right ) + {\left (2 \, A - i \, B\right )} a \sin \left (4 \, f x + 4 \, e\right ) + 2 \, {\left (2 \, A - i \, B\right )} a \sin \left (2 \, f x + 2 \, e\right ) + {\left (-2 i \, A - B\right )} a\right )} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + {\left ({\left (2 i \, A + B\right )} a \cos \left (4 \, f x + 4 \, e\right ) + 2 \, {\left (2 i \, A + B\right )} a \cos \left (2 \, f x + 2 \, e\right ) - {\left (2 \, A - i \, B\right )} a \sin \left (4 \, f x + 4 \, e\right ) - 2 \, {\left (2 \, A - i \, B\right )} a \sin \left (2 \, f x + 2 \, e\right ) + {\left (2 i \, A + B\right )} a\right )} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} - 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right )\right )} \sqrt {a} \sqrt {c}}{-16 \, f {\left (i \, \cos \left (4 \, f x + 4 \, e\right ) + 2 i \, \cos \left (2 \, f x + 2 \, e\right ) - \sin \left (4 \, f x + 4 \, e\right ) - 2 \, \sin \left (2 \, f x + 2 \, e\right ) + i\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 478 vs. \(2 (126) = 252\).
time = 3.71, size = 478, normalized size = 2.99 \begin {gather*} -\frac {\sqrt {\frac {{\left (4 \, A^{2} - 4 i \, A B - B^{2}\right )} a^{3} c}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (-2 i \, A - B\right )} a e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-2 i \, A - B\right )} a e^{\left (i \, f x + i \, e\right )}\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}} + \sqrt {\frac {{\left (4 \, A^{2} - 4 i \, A B - B^{2}\right )} a^{3} c}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (2 i \, A + B\right )} a e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (2 i \, A + B\right )} a}\right ) - \sqrt {\frac {{\left (4 \, A^{2} - 4 i \, A B - B^{2}\right )} a^{3} c}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (-2 i \, A - B\right )} a e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-2 i \, A - B\right )} a e^{\left (i \, f x + i \, e\right )}\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}} - \sqrt {\frac {{\left (4 \, A^{2} - 4 i \, A B - B^{2}\right )} a^{3} c}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (2 i \, A + B\right )} a e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (2 i \, A + B\right )} a}\right ) + 4 \, {\left ({\left (-2 i \, A - 3 \, B\right )} a e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-2 i \, A - B\right )} a e^{\left (i \, f x + i \, e\right )}\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}}}{4 \, {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \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*} \int \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}} \sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )} \left (A + B \tan {\left (e + f x \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 566 vs. \(2 (126) = 252\).
time = 1.70, size = 566, normalized size = 3.54 \begin {gather*} \frac {-3 i \, B a^{\frac {3}{2}} \sqrt {c} e^{\left (8 i \, f x + 8 i \, e\right )} \log \left (e^{\left (i \, f x + i \, e\right )} + i\right ) - 12 i \, B a^{\frac {3}{2}} \sqrt {c} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (i \, f x + i \, e\right )} + i\right ) - 18 i \, B a^{\frac {3}{2}} \sqrt {c} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (i \, f x + i \, e\right )} + i\right ) - 12 i \, B a^{\frac {3}{2}} \sqrt {c} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (i \, f x + i \, e\right )} + i\right ) + 3 i \, B a^{\frac {3}{2}} \sqrt {c} e^{\left (8 i \, f x + 8 i \, e\right )} \log \left (e^{\left (i \, f x + i \, e\right )} - i\right ) + 12 i \, B a^{\frac {3}{2}} \sqrt {c} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (i \, f x + i \, e\right )} - i\right ) + 18 i \, B a^{\frac {3}{2}} \sqrt {c} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (i \, f x + i \, e\right )} - i\right ) + 12 i \, B a^{\frac {3}{2}} \sqrt {c} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (i \, f x + i \, e\right )} - i\right ) + 10 \, B a^{\frac {3}{2}} \sqrt {c} e^{\left (7 i \, f x + 7 i \, e\right )} + 26 \, B a^{\frac {3}{2}} \sqrt {c} e^{\left (5 i \, f x + 5 i \, e\right )} + 22 \, B a^{\frac {3}{2}} \sqrt {c} e^{\left (3 i \, f x + 3 i \, e\right )} + 6 \, B a^{\frac {3}{2}} \sqrt {c} e^{\left (i \, f x + i \, e\right )} - 3 i \, B a^{\frac {3}{2}} \sqrt {c} \log \left (e^{\left (i \, f x + i \, e\right )} + i\right ) + 3 i \, B a^{\frac {3}{2}} \sqrt {c} \log \left (e^{\left (i \, f x + i \, e\right )} - i\right )}{8 \, {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} - \frac {i \, {\left (8 \, A a^{\frac {3}{2}} \sqrt {c} - i \, B a^{\frac {3}{2}} \sqrt {c}\right )} \arctan \left (e^{\left (i \, f x + i \, e\right )}\right ) - \frac {i \, {\left (8 \, A a^{\frac {3}{2}} \sqrt {c} e^{\left (3 i \, f x + 3 i \, e\right )} - 7 i \, B a^{\frac {3}{2}} \sqrt {c} e^{\left (3 i \, f x + 3 i \, e\right )} + 8 \, A a^{\frac {3}{2}} \sqrt {c} e^{\left (i \, f x + i \, e\right )} - i \, B a^{\frac {3}{2}} \sqrt {c} e^{\left (i \, f x + i \, e\right )}\right )}}{{\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}^{2}}}{4 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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