Optimal. Leaf size=116 \[ \frac {x}{16 a^4}+\frac {i}{8 d (a+i a \tan (c+d x))^4}+\frac {i}{12 a d (a+i a \tan (c+d x))^3}+\frac {i}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac {i}{16 d \left (a^4+i a^4 \tan (c+d x)\right )} \]
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Rubi [A]
time = 0.05, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3560, 8}
\begin {gather*} \frac {i}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {x}{16 a^4}+\frac {i}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac {i}{12 a d (a+i a \tan (c+d x))^3}+\frac {i}{8 d (a+i a \tan (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3560
Rubi steps
\begin {align*} \int \frac {1}{(a+i a \tan (c+d x))^4} \, dx &=\frac {i}{8 d (a+i a \tan (c+d x))^4}+\frac {\int \frac {1}{(a+i a \tan (c+d x))^3} \, dx}{2 a}\\ &=\frac {i}{8 d (a+i a \tan (c+d x))^4}+\frac {i}{12 a d (a+i a \tan (c+d x))^3}+\frac {\int \frac {1}{(a+i a \tan (c+d x))^2} \, dx}{4 a^2}\\ &=\frac {i}{8 d (a+i a \tan (c+d x))^4}+\frac {i}{12 a d (a+i a \tan (c+d x))^3}+\frac {i}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac {\int \frac {1}{a+i a \tan (c+d x)} \, dx}{8 a^3}\\ &=\frac {i}{8 d (a+i a \tan (c+d x))^4}+\frac {i}{12 a d (a+i a \tan (c+d x))^3}+\frac {i}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac {i}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {\int 1 \, dx}{16 a^4}\\ &=\frac {x}{16 a^4}+\frac {i}{8 d (a+i a \tan (c+d x))^4}+\frac {i}{12 a d (a+i a \tan (c+d x))^3}+\frac {i}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac {i}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 98, normalized size = 0.84 \begin {gather*} \frac {\sec ^4(c+d x) (36 i+64 i \cos (2 (c+d x))+3 (i+8 d x) \cos (4 (c+d x))-32 \sin (2 (c+d x))+3 \sin (4 (c+d x))+24 i d x \sin (4 (c+d x)))}{384 a^4 d (-i+\tan (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 89, normalized size = 0.77
method | result | size |
risch | \(\frac {x}{16 a^{4}}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{4} d}+\frac {3 i {\mathrm e}^{-4 i \left (d x +c \right )}}{32 a^{4} d}+\frac {i {\mathrm e}^{-6 i \left (d x +c \right )}}{24 a^{4} d}+\frac {i {\mathrm e}^{-8 i \left (d x +c \right )}}{128 a^{4} d}\) | \(80\) |
derivativedivides | \(\frac {\frac {i}{8 \left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {i \ln \left (\tan \left (d x +c \right )-i\right )}{32}-\frac {i}{16 \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {1}{12 \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {1}{16 \tan \left (d x +c \right )-16 i}+\frac {i \ln \left (\tan \left (d x +c \right )+i\right )}{32}}{d \,a^{4}}\) | \(89\) |
default | \(\frac {\frac {i}{8 \left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {i \ln \left (\tan \left (d x +c \right )-i\right )}{32}-\frac {i}{16 \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {1}{12 \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {1}{16 \tan \left (d x +c \right )-16 i}+\frac {i \ln \left (\tan \left (d x +c \right )+i\right )}{32}}{d \,a^{4}}\) | \(89\) |
norman | \(\frac {\frac {x}{16 a}+\frac {11 \left (\tan ^{5}\left (d x +c \right )\right )}{48 d a}+\frac {\tan ^{7}\left (d x +c \right )}{16 d a}+\frac {x \left (\tan ^{2}\left (d x +c \right )\right )}{4 a}+\frac {3 x \left (\tan ^{4}\left (d x +c \right )\right )}{8 a}+\frac {x \left (\tan ^{6}\left (d x +c \right )\right )}{4 a}+\frac {x \left (\tan ^{8}\left (d x +c \right )\right )}{16 a}+\frac {i}{3 d a}+\frac {15 \tan \left (d x +c \right )}{16 d a}+\frac {5 \left (\tan ^{3}\left (d x +c \right )\right )}{48 d a}-\frac {2 i \left (\tan ^{2}\left (d x +c \right )\right )}{3 d a}}{a^{3} \left (1+\tan ^{2}\left (d x +c \right )\right )^{4}}\) | \(168\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 65, normalized size = 0.56 \begin {gather*} \frac {{\left (24 \, d x e^{\left (8 i \, d x + 8 i \, c\right )} + 48 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 36 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 16 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{384 \, a^{4} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 189, normalized size = 1.63 \begin {gather*} \begin {cases} \frac {\left (98304 i a^{12} d^{3} e^{18 i c} e^{- 2 i d x} + 73728 i a^{12} d^{3} e^{16 i c} e^{- 4 i d x} + 32768 i a^{12} d^{3} e^{14 i c} e^{- 6 i d x} + 6144 i a^{12} d^{3} e^{12 i c} e^{- 8 i d x}\right ) e^{- 20 i c}}{786432 a^{16} d^{4}} & \text {for}\: a^{16} d^{4} e^{20 i c} \neq 0 \\x \left (\frac {\left (e^{8 i c} + 4 e^{6 i c} + 6 e^{4 i c} + 4 e^{2 i c} + 1\right ) e^{- 8 i c}}{16 a^{4}} - \frac {1}{16 a^{4}}\right ) & \text {otherwise} \end {cases} + \frac {x}{16 a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 92, normalized size = 0.79 \begin {gather*} -\frac {-\frac {12 i \, \log \left (-i \, \tan \left (d x + c\right ) + 1\right )}{a^{4}} + \frac {12 i \, \log \left (-i \, \tan \left (d x + c\right ) - 1\right )}{a^{4}} + \frac {-25 i \, \tan \left (d x + c\right )^{4} - 124 \, \tan \left (d x + c\right )^{3} + 246 i \, \tan \left (d x + c\right )^{2} + 252 \, \tan \left (d x + c\right ) - 153 i}{a^{4} {\left (\tan \left (d x + c\right ) - i\right )}^{4}}}{384 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.50, size = 60, normalized size = 0.52 \begin {gather*} \frac {x}{16\,a^4}-\frac {-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{16}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}}{4}+\frac {19\,\mathrm {tan}\left (c+d\,x\right )}{48}-\frac {1}{3}{}\mathrm {i}}{a^4\,d\,{\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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