Optimal. Leaf size=194 \[ -\frac {6 i e^{c (a+b x)} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}+\frac {12 i e^{c (a+b x)} \, _2F_1\left (2,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}-\frac {8 i e^{c (a+b x)} \, _2F_1\left (3,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}+\frac {i e^{c (a+b x)}}{b c} \]
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Rubi [A] time = 0.20, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4442, 2194, 2251} \[ -\frac {6 i e^{c (a+b x)} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}+\frac {12 i e^{c (a+b x)} \, _2F_1\left (2,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}-\frac {8 i e^{c (a+b x)} \, _2F_1\left (3,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}+\frac {i e^{c (a+b x)}}{b c} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2251
Rule 4442
Rubi steps
\begin {align*} \int e^{c (a+b x)} \tan ^3(d+e x) \, dx &=-\left (i \int \left (-e^{c (a+b x)}+\frac {8 e^{c (a+b x)}}{\left (1+e^{2 i (d+e x)}\right )^3}-\frac {12 e^{c (a+b x)}}{\left (1+e^{2 i (d+e x)}\right )^2}+\frac {6 e^{c (a+b x)}}{1+e^{2 i (d+e x)}}\right ) \, dx\right )\\ &=i \int e^{c (a+b x)} \, dx-6 i \int \frac {e^{c (a+b x)}}{1+e^{2 i (d+e x)}} \, dx-8 i \int \frac {e^{c (a+b x)}}{\left (1+e^{2 i (d+e x)}\right )^3} \, dx+12 i \int \frac {e^{c (a+b x)}}{\left (1+e^{2 i (d+e x)}\right )^2} \, dx\\ &=\frac {i e^{c (a+b x)}}{b c}-\frac {6 i e^{c (a+b x)} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}+\frac {12 i e^{c (a+b x)} \, _2F_1\left (2,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}-\frac {8 i e^{c (a+b x)} \, _2F_1\left (3,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}\\ \end {align*}
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Mathematica [A] time = 2.14, size = 212, normalized size = 1.09 \[ \frac {1}{2} e^{c (a+b x)} \left (\frac {2 e^{2 i d} \left (b^2 c^2-2 e^2\right ) \left (b c e^{2 i e x} \, _2F_1\left (1,1-\frac {i b c}{2 e};2-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )-(b c+2 i e) \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )\right )}{b c \left (1+e^{2 i d}\right ) e^2 (-2 e+i b c)}-\frac {b c \sec (d) \sin (e x) \sec (d+e x)}{e^2}-\frac {2 \tan (d)}{b c}+\frac {\sec ^2(d+e x)}{e}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\left (b c x + a c\right )} \tan \left (e x + d\right )^{3}, x\right ) \]
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 \[ \int e^{\left ({\left (b x + a\right )} c\right )} \tan \left (e x + d\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{c \left (b x +a \right )} \left (\tan ^{3}\left (e x +d \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {4 \, e \cos \left (2 \, e x + 2 \, d\right )^{2} e^{\left (b c x + a c\right )} - b c e^{\left (b c x + a c\right )} \sin \left (2 \, e x + 2 \, d\right ) + 4 \, e e^{\left (b c x + a c\right )} \sin \left (2 \, e x + 2 \, d\right )^{2} + 2 \, e \cos \left (2 \, e x + 2 \, d\right ) e^{\left (b c x + a c\right )} + {\left (b c e^{\left (b c x + a c\right )} \sin \left (2 \, e x + 2 \, d\right ) + 2 \, e \cos \left (2 \, e x + 2 \, d\right ) e^{\left (b c x + a c\right )}\right )} \cos \left (4 \, e x + 4 \, d\right ) - {\left (b c \cos \left (2 \, e x + 2 \, d\right ) e^{\left (b c x + a c\right )} + b c e^{\left (b c x + a c\right )} - 2 \, e e^{\left (b c x + a c\right )} \sin \left (2 \, e x + 2 \, d\right )\right )} \sin \left (4 \, e x + 4 \, d\right ) + \frac {{\left (b^{2} c^{2} e^{4} e^{\left (a c\right )} - 2 \, e^{6} e^{\left (a c\right )} + {\left (b^{2} c^{2} e^{4} e^{\left (a c\right )} - 2 \, e^{6} e^{\left (a c\right )}\right )} \cos \left (4 \, e x + 4 \, d\right )^{2} + 4 \, {\left (b^{2} c^{2} e^{4} e^{\left (a c\right )} - 2 \, e^{6} e^{\left (a c\right )}\right )} \cos \left (2 \, e x + 2 \, d\right )^{2} + {\left (b^{2} c^{2} e^{4} e^{\left (a c\right )} - 2 \, e^{6} e^{\left (a c\right )}\right )} \sin \left (4 \, e x + 4 \, d\right )^{2} + 4 \, {\left (b^{2} c^{2} e^{4} e^{\left (a c\right )} - 2 \, e^{6} e^{\left (a c\right )}\right )} \sin \left (4 \, e x + 4 \, d\right ) \sin \left (2 \, e x + 2 \, d\right ) + 4 \, {\left (b^{2} c^{2} e^{4} e^{\left (a c\right )} - 2 \, e^{6} e^{\left (a c\right )}\right )} \sin \left (2 \, e x + 2 \, d\right )^{2} + 2 \, {\left (b^{2} c^{2} e^{4} e^{\left (a c\right )} - 2 \, e^{6} e^{\left (a c\right )} + 2 \, {\left (b^{2} c^{2} e^{4} e^{\left (a c\right )} - 2 \, e^{6} e^{\left (a c\right )}\right )} \cos \left (2 \, e x + 2 \, d\right )\right )} \cos \left (4 \, e x + 4 \, d\right ) + 4 \, {\left (b^{2} c^{2} e^{4} e^{\left (a c\right )} - 2 \, e^{6} e^{\left (a c\right )}\right )} \cos \left (2 \, e x + 2 \, d\right )\right )} \int \frac {e^{\left (b c x\right )} \sin \left (2 \, e x + 2 \, d\right )}{\cos \left (2 \, e x + 2 \, d\right )^{2} + \sin \left (2 \, e x + 2 \, d\right )^{2} + 2 \, \cos \left (2 \, e x + 2 \, d\right ) + 1}\,{d x}}{e^{4}}}{e^{2} \cos \left (4 \, e x + 4 \, d\right )^{2} + 4 \, e^{2} \cos \left (2 \, e x + 2 \, d\right )^{2} + e^{2} \sin \left (4 \, e x + 4 \, d\right )^{2} + 4 \, e^{2} \sin \left (4 \, e x + 4 \, d\right ) \sin \left (2 \, e x + 2 \, d\right ) + 4 \, e^{2} \sin \left (2 \, e x + 2 \, d\right )^{2} + 4 \, e^{2} \cos \left (2 \, e x + 2 \, d\right ) + e^{2} + 2 \, {\left (2 \, e^{2} \cos \left (2 \, e x + 2 \, d\right ) + e^{2}\right )} \cos \left (4 \, e x + 4 \, d\right )} \]
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 \[ \int {\mathrm {e}}^{c\,\left (a+b\,x\right )}\,{\mathrm {tan}\left (d+e\,x\right )}^3 \,d x \]
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 \[ e^{a c} \int e^{b c x} \tan ^{3}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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