Optimal. Leaf size=77 \[ (a-b)^3 x+\frac {b \left (3 a^2-3 a b+b^2\right ) \tan (c+d x)}{d}+\frac {(3 a-b) b^2 \tan ^3(c+d x)}{3 d}+\frac {b^3 \tan ^5(c+d x)}{5 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3742, 398, 209}
\begin {gather*} \frac {b \left (3 a^2-3 a b+b^2\right ) \tan (c+d x)}{d}+\frac {b^2 (3 a-b) \tan ^3(c+d x)}{3 d}+x (a-b)^3+\frac {b^3 \tan ^5(c+d x)}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 398
Rule 3742
Rubi steps
\begin {align*} \int \left (a+b \tan ^2(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^3}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (b \left (3 a^2-3 a b+b^2\right )+(3 a-b) b^2 x^2+b^3 x^4+\frac {(a-b)^3}{1+x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {b \left (3 a^2-3 a b+b^2\right ) \tan (c+d x)}{d}+\frac {(3 a-b) b^2 \tan ^3(c+d x)}{3 d}+\frac {b^3 \tan ^5(c+d x)}{5 d}+\frac {(a-b)^3 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=(a-b)^3 x+\frac {b \left (3 a^2-3 a b+b^2\right ) \tan (c+d x)}{d}+\frac {(3 a-b) b^2 \tan ^3(c+d x)}{3 d}+\frac {b^3 \tan ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.62, size = 102, normalized size = 1.32 \begin {gather*} \frac {\tan (c+d x) \left (\frac {15 (a-b)^3 \tanh ^{-1}\left (\sqrt {-\tan ^2(c+d x)}\right )}{\sqrt {-\tan ^2(c+d x)}}+b \left (45 a^2-15 a b \left (3-\tan ^2(c+d x)\right )+b^2 \left (15-5 \tan ^2(c+d x)+3 \tan ^4(c+d x)\right )\right )\right )}{15 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 108, normalized size = 1.40
method | result | size |
norman | \(\left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) x +\frac {b \left (3 a^{2}-3 a b +b^{2}\right ) \tan \left (d x +c \right )}{d}+\frac {b^{3} \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}+\frac {\left (3 a -b \right ) b^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}\) | \(88\) |
derivativedivides | \(\frac {\frac {b^{3} \left (\tan ^{5}\left (d x +c \right )\right )}{5}+a \,b^{2} \left (\tan ^{3}\left (d x +c \right )\right )-\frac {b^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+3 a^{2} b \tan \left (d x +c \right )-3 a \,b^{2} \tan \left (d x +c \right )+b^{3} \tan \left (d x +c \right )+\left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(108\) |
default | \(\frac {\frac {b^{3} \left (\tan ^{5}\left (d x +c \right )\right )}{5}+a \,b^{2} \left (\tan ^{3}\left (d x +c \right )\right )-\frac {b^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+3 a^{2} b \tan \left (d x +c \right )-3 a \,b^{2} \tan \left (d x +c \right )+b^{3} \tan \left (d x +c \right )+\left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(108\) |
risch | \(a^{3} x -3 a^{2} b x +3 a \,b^{2} x -b^{3} x +\frac {2 i b \left (45 a^{2} {\mathrm e}^{8 i \left (d x +c \right )}-90 a b \,{\mathrm e}^{8 i \left (d x +c \right )}+45 b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+180 a^{2} {\mathrm e}^{6 i \left (d x +c \right )}-270 a b \,{\mathrm e}^{6 i \left (d x +c \right )}+90 b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+270 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-330 a b \,{\mathrm e}^{4 i \left (d x +c \right )}+140 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+180 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-210 a b \,{\mathrm e}^{2 i \left (d x +c \right )}+70 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+45 a^{2}-60 a b +23 b^{2}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}\) | \(226\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 104, normalized size = 1.35 \begin {gather*} a^{3} x - \frac {3 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{2} b}{d} + \frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a b^{2}}{d} + \frac {{\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} b^{3}}{15 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.60, size = 90, normalized size = 1.17 \begin {gather*} \frac {3 \, b^{3} \tan \left (d x + c\right )^{5} + 5 \, {\left (3 \, a b^{2} - b^{3}\right )} \tan \left (d x + c\right )^{3} + 15 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} d x + 15 \, {\left (3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \tan \left (d x + c\right )}{15 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 126, normalized size = 1.64 \begin {gather*} \begin {cases} a^{3} x - 3 a^{2} b x + \frac {3 a^{2} b \tan {\left (c + d x \right )}}{d} + 3 a b^{2} x + \frac {a b^{2} \tan ^{3}{\left (c + d x \right )}}{d} - \frac {3 a b^{2} \tan {\left (c + d x \right )}}{d} - b^{3} x + \frac {b^{3} \tan ^{5}{\left (c + d x \right )}}{5 d} - \frac {b^{3} \tan ^{3}{\left (c + d x \right )}}{3 d} + \frac {b^{3} \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan ^{2}{\left (c \right )}\right )^{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1027 vs.
\(2 (73) = 146\).
time = 0.92, size = 1027, normalized size = 13.34 \begin {gather*} \frac {15 \, a^{3} d x \tan \left (d x\right )^{5} \tan \left (c\right )^{5} - 45 \, a^{2} b d x \tan \left (d x\right )^{5} \tan \left (c\right )^{5} + 45 \, a b^{2} d x \tan \left (d x\right )^{5} \tan \left (c\right )^{5} - 15 \, b^{3} d x \tan \left (d x\right )^{5} \tan \left (c\right )^{5} - 75 \, a^{3} d x \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 225 \, a^{2} b d x \tan \left (d x\right )^{4} \tan \left (c\right )^{4} - 225 \, a b^{2} d x \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 75 \, b^{3} d x \tan \left (d x\right )^{4} \tan \left (c\right )^{4} - 45 \, a^{2} b \tan \left (d x\right )^{5} \tan \left (c\right )^{4} + 45 \, a b^{2} \tan \left (d x\right )^{5} \tan \left (c\right )^{4} - 15 \, b^{3} \tan \left (d x\right )^{5} \tan \left (c\right )^{4} - 45 \, a^{2} b \tan \left (d x\right )^{4} \tan \left (c\right )^{5} + 45 \, a b^{2} \tan \left (d x\right )^{4} \tan \left (c\right )^{5} - 15 \, b^{3} \tan \left (d x\right )^{4} \tan \left (c\right )^{5} + 150 \, a^{3} d x \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 450 \, a^{2} b d x \tan \left (d x\right )^{3} \tan \left (c\right )^{3} + 450 \, a b^{2} d x \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 150 \, b^{3} d x \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 15 \, a b^{2} \tan \left (d x\right )^{5} \tan \left (c\right )^{2} + 5 \, b^{3} \tan \left (d x\right )^{5} \tan \left (c\right )^{2} + 180 \, a^{2} b \tan \left (d x\right )^{4} \tan \left (c\right )^{3} - 225 \, a b^{2} \tan \left (d x\right )^{4} \tan \left (c\right )^{3} + 75 \, b^{3} \tan \left (d x\right )^{4} \tan \left (c\right )^{3} + 180 \, a^{2} b \tan \left (d x\right )^{3} \tan \left (c\right )^{4} - 225 \, a b^{2} \tan \left (d x\right )^{3} \tan \left (c\right )^{4} + 75 \, b^{3} \tan \left (d x\right )^{3} \tan \left (c\right )^{4} - 15 \, a b^{2} \tan \left (d x\right )^{2} \tan \left (c\right )^{5} + 5 \, b^{3} \tan \left (d x\right )^{2} \tan \left (c\right )^{5} - 150 \, a^{3} d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 450 \, a^{2} b d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 450 \, a b^{2} d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 150 \, b^{3} d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 3 \, b^{3} \tan \left (d x\right )^{5} + 30 \, a b^{2} \tan \left (d x\right )^{4} \tan \left (c\right ) - 25 \, b^{3} \tan \left (d x\right )^{4} \tan \left (c\right ) - 270 \, a^{2} b \tan \left (d x\right )^{3} \tan \left (c\right )^{2} + 360 \, a b^{2} \tan \left (d x\right )^{3} \tan \left (c\right )^{2} - 150 \, b^{3} \tan \left (d x\right )^{3} \tan \left (c\right )^{2} - 270 \, a^{2} b \tan \left (d x\right )^{2} \tan \left (c\right )^{3} + 360 \, a b^{2} \tan \left (d x\right )^{2} \tan \left (c\right )^{3} - 150 \, b^{3} \tan \left (d x\right )^{2} \tan \left (c\right )^{3} + 30 \, a b^{2} \tan \left (d x\right ) \tan \left (c\right )^{4} - 25 \, b^{3} \tan \left (d x\right ) \tan \left (c\right )^{4} - 3 \, b^{3} \tan \left (c\right )^{5} + 75 \, a^{3} d x \tan \left (d x\right ) \tan \left (c\right ) - 225 \, a^{2} b d x \tan \left (d x\right ) \tan \left (c\right ) + 225 \, a b^{2} d x \tan \left (d x\right ) \tan \left (c\right ) - 75 \, b^{3} d x \tan \left (d x\right ) \tan \left (c\right ) - 15 \, a b^{2} \tan \left (d x\right )^{3} + 5 \, b^{3} \tan \left (d x\right )^{3} + 180 \, a^{2} b \tan \left (d x\right )^{2} \tan \left (c\right ) - 225 \, a b^{2} \tan \left (d x\right )^{2} \tan \left (c\right ) + 75 \, b^{3} \tan \left (d x\right )^{2} \tan \left (c\right ) + 180 \, a^{2} b \tan \left (d x\right ) \tan \left (c\right )^{2} - 225 \, a b^{2} \tan \left (d x\right ) \tan \left (c\right )^{2} + 75 \, b^{3} \tan \left (d x\right ) \tan \left (c\right )^{2} - 15 \, a b^{2} \tan \left (c\right )^{3} + 5 \, b^{3} \tan \left (c\right )^{3} - 15 \, a^{3} d x + 45 \, a^{2} b d x - 45 \, a b^{2} d x + 15 \, b^{3} d x - 45 \, a^{2} b \tan \left (d x\right ) + 45 \, a b^{2} \tan \left (d x\right ) - 15 \, b^{3} \tan \left (d x\right ) - 45 \, a^{2} b \tan \left (c\right ) + 45 \, a b^{2} \tan \left (c\right ) - 15 \, b^{3} \tan \left (c\right )}{15 \, {\left (d \tan \left (d x\right )^{5} \tan \left (c\right )^{5} - 5 \, d \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 10 \, d \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 10 \, d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 5 \, d \tan \left (d x\right ) \tan \left (c\right ) - d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 11.47, size = 115, normalized size = 1.49 \begin {gather*} \frac {b^3\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5\,d}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (3\,a^2\,b-3\,a\,b^2+b^3\right )}{d}+\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,{\left (a-b\right )}^3}{a^3-3\,a^2\,b+3\,a\,b^2-b^3}\right )\,{\left (a-b\right )}^3}{d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (a\,b^2-\frac {b^3}{3}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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