Optimal. Leaf size=22 \[ \frac {(a+b \tan (c+d x))^4}{4 b d} \]
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Rubi [A]
time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3587, 32}
\begin {gather*} \frac {(a+b \tan (c+d x))^4}{4 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 3587
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+b \tan (c+d x))^3 \, dx &=\frac {\text {Subst}\left (\int (a+x)^3 \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {(a+b \tan (c+d x))^4}{4 b d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(79\) vs. \(2(22)=44\).
time = 0.65, size = 79, normalized size = 3.59 \begin {gather*} \frac {\sec ^4(c+d x) \left (\left (6 a^2 b-2 b^3\right ) \cos (2 (c+d x))+a \left (6 a b+2 \left (a^2+b^2\right ) \sin (2 (c+d x))+\left (a^2-b^2\right ) \sin (4 (c+d x))\right )\right )}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs.
\(2(20)=40\).
time = 0.22, size = 72, normalized size = 3.27
method | result | size |
derivativedivides | \(\frac {\frac {b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4 \cos \left (d x +c \right )^{4}}+\frac {b^{2} a \left (\sin ^{3}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{3}}+\frac {3 a^{2} b}{2 \cos \left (d x +c \right )^{2}}+a^{3} \tan \left (d x +c \right )}{d}\) | \(72\) |
default | \(\frac {\frac {b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4 \cos \left (d x +c \right )^{4}}+\frac {b^{2} a \left (\sin ^{3}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{3}}+\frac {3 a^{2} b}{2 \cos \left (d x +c \right )^{2}}+a^{3} \tan \left (d x +c \right )}{d}\) | \(72\) |
risch | \(-\frac {2 \left (-i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+3 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-3 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-3 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+3 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-3 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-3 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-i a^{3}+i a \,b^{2}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}\) | \(197\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 20, normalized size = 0.91 \begin {gather*} \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{4}}{4 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 78 vs.
\(2 (20) = 40\).
time = 0.37, size = 78, normalized size = 3.55 \begin {gather*} \frac {b^{3} + 2 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (a b^{2} \cos \left (d x + c\right ) + {\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{4}} \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 (a + b \tan {\left (c + d x \right )}\right )^{3} \sec ^{2}{\left (c + d x \right )}\, dx \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. 57 vs.
\(2 (20) = 40\).
time = 0.87, size = 57, normalized size = 2.59 \begin {gather*} \frac {b^{3} \tan \left (d x + c\right )^{4} + 4 \, a b^{2} \tan \left (d x + c\right )^{3} + 6 \, a^{2} b \tan \left (d x + c\right )^{2} + 4 \, a^{3} \tan \left (d x + c\right )}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.57, size = 55, normalized size = 2.50 \begin {gather*} \frac {a^3\,\mathrm {tan}\left (c+d\,x\right )+\frac {3\,a^2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3+\frac {b^3\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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