Optimal. Leaf size=30 \[ \frac {(b+a \cot (c+d x))^3 \tan ^3(c+d x)}{3 b d} \]
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Rubi [A]
time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3167, 37}
\begin {gather*} \frac {\tan ^3(c+d x) (a \cot (c+d x)+b)^3}{3 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 3167
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx &=-\frac {\text {Subst}\left (\int \frac {(b+a x)^2}{x^4} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {(b+a \cot (c+d x))^3 \tan ^3(c+d x)}{3 b d}\\ \end {align*}
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Mathematica [A]
time = 0.74, size = 56, normalized size = 1.87 \begin {gather*} \frac {\sec ^2(c+d x) \left (6 a b+\left (3 a^2+b^2+\left (3 a^2-b^2\right ) \cos (2 (c+d x))\right ) \tan (c+d x)\right )}{6 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 48, normalized size = 1.60
method | result | size |
derivativedivides | \(\frac {a^{2} \tan \left (d x +c \right )+\frac {a b}{\cos \left (d x +c \right )^{2}}+\frac {b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}}{d}\) | \(48\) |
default | \(\frac {a^{2} \tan \left (d x +c \right )+\frac {a b}{\cos \left (d x +c \right )^{2}}+\frac {b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}}{d}\) | \(48\) |
risch | \(-\frac {2 i \left (6 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}-3 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+6 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}-6 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-3 a^{2}+b^{2}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}\) | \(99\) |
norman | \(\frac {\frac {4 a b \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {8 b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {8 b^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {4 \left (3 a^{2}-4 b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {4 a b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a b \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}\) | \(206\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 45, normalized size = 1.50 \begin {gather*} \frac {b^{2} \tan \left (d x + c\right )^{3} + 3 \, a^{2} \tan \left (d x + c\right ) - \frac {3 \, a b}{\sin \left (d x + c\right )^{2} - 1}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.29, size = 55, normalized size = 1.83 \begin {gather*} \frac {3 \, a b \cos \left (d x + c\right ) + {\left ({\left (3 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \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 \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right )^{2} \sec ^{4}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 41, normalized size = 1.37 \begin {gather*} \frac {b^{2} \tan \left (d x + c\right )^{3} + 3 \, a b \tan \left (d x + c\right )^{2} + 3 \, a^{2} \tan \left (d x + c\right )}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.49, size = 68, normalized size = 2.27 \begin {gather*} \frac {\frac {b^2\,\sin \left (c+d\,x\right )}{3}+\frac {{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,\left (3\,a^2-b^2\right )}{3}+a\,b\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^2}{d\,{\cos \left (c+d\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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