Optimal. Leaf size=126 \[ \frac {a \left (4 a^2-3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a \left (4 a^2-3 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^3(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {b \sec ^3(c+d x) \left (8 \left (6 a^2-b^2\right )+21 a b \tan (c+d x)\right )}{60 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 144, normalized size of antiderivative = 1.14, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3593, 757, 794,
201, 221} \begin {gather*} \frac {b \sec ^3(c+d x) \left (8 \left (6 a^2-b^2\right )+21 a b \tan (c+d x)\right )}{60 d}+\frac {a \left (4 a^2-3 b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {a \left (4 a^2-3 b^2\right ) \sec (c+d x) \sinh ^{-1}(\tan (c+d x))}{8 d \sqrt {\sec ^2(c+d x)}}+\frac {b \sec ^3(c+d x) (a+b \tan (c+d x))^2}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 221
Rule 757
Rule 794
Rule 3593
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+b \tan (c+d x))^3 \, dx &=\frac {\sec (c+d x) \text {Subst}\left (\int (a+x)^3 \sqrt {1+\frac {x^2}{b^2}} \, dx,x,b \tan (c+d x)\right )}{b d \sqrt {\sec ^2(c+d x)}}\\ &=\frac {b \sec ^3(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {(b \sec (c+d x)) \text {Subst}\left (\int (a+x) \left (-2+\frac {5 a^2}{b^2}+\frac {7 a x}{b^2}\right ) \sqrt {1+\frac {x^2}{b^2}} \, dx,x,b \tan (c+d x)\right )}{5 d \sqrt {\sec ^2(c+d x)}}\\ &=\frac {b \sec ^3(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {b \sec ^3(c+d x) \left (8 \left (6 a^2-b^2\right )+21 a b \tan (c+d x)\right )}{60 d}-\frac {\left (a \left (3-\frac {4 a^2}{b^2}\right ) b \sec (c+d x)\right ) \text {Subst}\left (\int \sqrt {1+\frac {x^2}{b^2}} \, dx,x,b \tan (c+d x)\right )}{4 d \sqrt {\sec ^2(c+d x)}}\\ &=\frac {a \left (4 a^2-3 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^3(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {b \sec ^3(c+d x) \left (8 \left (6 a^2-b^2\right )+21 a b \tan (c+d x)\right )}{60 d}-\frac {\left (a \left (3-\frac {4 a^2}{b^2}\right ) b \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{8 d \sqrt {\sec ^2(c+d x)}}\\ &=\frac {a \left (4 a^2-3 b^2\right ) \sinh ^{-1}(\tan (c+d x)) \sec (c+d x)}{8 d \sqrt {\sec ^2(c+d x)}}+\frac {a \left (4 a^2-3 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^3(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {b \sec ^3(c+d x) \left (8 \left (6 a^2-b^2\right )+21 a b \tan (c+d x)\right )}{60 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(464\) vs. \(2(126)=252\).
time = 1.38, size = 464, normalized size = 3.68 \begin {gather*} \frac {\sec ^5(c+d x) \left (960 a^2 b+64 b^3+320 \left (3 a^2 b-b^3\right ) \cos (2 (c+d x))-300 a^3 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+225 a b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-60 a^3 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+45 a b^2 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-150 a \left (4 a^2-3 b^2\right ) \cos (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+300 a^3 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-225 a b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+60 a^3 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-45 a b^2 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+240 a^3 \sin (2 (c+d x))+540 a b^2 \sin (2 (c+d x))+120 a^3 \sin (4 (c+d x))-90 a b^2 \sin (4 (c+d x))\right )}{1920 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 198, normalized size = 1.57
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {a^{2} b}{\cos \left (d x +c \right )^{3}}+3 b^{2} a \left (\frac {\sin ^{3}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{15}\right )}{d}\) | \(198\) |
default | \(\frac {a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {a^{2} b}{\cos \left (d x +c \right )^{3}}+3 b^{2} a \left (\frac {\sin ^{3}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{15}\right )}{d}\) | \(198\) |
risch | \(-\frac {60 i a^{3} {\mathrm e}^{9 i \left (d x +c \right )}-45 i a \,b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+120 i a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+270 i a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-480 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+160 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-960 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-64 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-120 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-270 i a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-480 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+160 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-60 i a^{3} {\mathrm e}^{i \left (d x +c \right )}+45 i a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{2}}{8 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{2}}{8 d}\) | \(318\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 157, normalized size = 1.25 \begin {gather*} \frac {45 \, a b^{2} {\left (\frac {2 \, {\left (\sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {240 \, a^{2} b}{\cos \left (d x + c\right )^{3}} - \frac {16 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} b^{3}}{\cos \left (d x + c\right )^{5}}}{240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 147, normalized size = 1.17 \begin {gather*} \frac {15 \, {\left (4 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (4 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 48 \, b^{3} + 80 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 30 \, {\left (6 \, a b^{2} \cos \left (d x + c\right ) + {\left (4 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \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 ^{3}{\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. 333 vs.
\(2 (115) = 230\).
time = 0.84, size = 333, normalized size = 2.64 \begin {gather*} \frac {15 \, {\left (4 \, a^{3} - 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (4 \, a^{3} - 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 45 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 360 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 270 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 720 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 480 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 80 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 270 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 80 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 45 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 120 \, a^{2} b + 16 \, b^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{120 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.32, size = 293, normalized size = 2.33 \begin {gather*} \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (a^3+\frac {3\,a\,b^2}{4}\right )-2\,a^2\,b-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {9\,a\,b^2}{2}-2\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {9\,a\,b^2}{2}-2\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (4\,a^2\,b-\frac {4\,b^3}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (8\,a^2\,b+\frac {4\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (12\,a^2\,b-4\,b^3\right )+\frac {4\,b^3}{15}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^3+\frac {3\,a\,b^2}{4}\right )-6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a\,b^2}{4}-a^3\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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