Optimal. Leaf size=159 \[ \frac {3 a \left (2 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d}+\frac {a \left (2 a^2-b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac {3 a \left (2 a^2-b^2\right ) \tan (c+d x) \sec (c+d x)}{16 d}+\frac {b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d} \]
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Rubi [A] time = 0.14, antiderivative size = 177, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3512, 743, 780, 195, 215} \[ \frac {b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d}+\frac {a \left (2 a^2-b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac {3 a \left (2 a^2-b^2\right ) \tan (c+d x) \sec (c+d x)}{16 d}+\frac {3 a \left (2 a^2-b^2\right ) \sec (c+d x) \sinh ^{-1}(\tan (c+d x))}{16 d \sqrt {\sec ^2(c+d x)}}+\frac {b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d} \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 743
Rule 780
Rule 3512
Rubi steps
\begin {align*} \int \sec ^5(c+d x) (a+b \tan (c+d x))^3 \, dx &=\frac {\sec (c+d x) \operatorname {Subst}\left (\int (a+x)^3 \left (1+\frac {x^2}{b^2}\right )^{3/2} \, dx,x,b \tan (c+d x)\right )}{b d \sqrt {\sec ^2(c+d x)}}\\ &=\frac {b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}+\frac {(b \sec (c+d x)) \operatorname {Subst}\left (\int (a+x) \left (-2+\frac {7 a^2}{b^2}+\frac {9 a x}{b^2}\right ) \left (1+\frac {x^2}{b^2}\right )^{3/2} \, dx,x,b \tan (c+d x)\right )}{7 d \sqrt {\sec ^2(c+d x)}}\\ &=\frac {b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}+\frac {b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d}-\frac {\left (\left (\frac {9 a}{b^2}-\frac {6 a \left (-2+\frac {7 a^2}{b^2}\right )}{b^2}\right ) b^3 \sec (c+d x)\right ) \operatorname {Subst}\left (\int \left (1+\frac {x^2}{b^2}\right )^{3/2} \, dx,x,b \tan (c+d x)\right )}{42 d \sqrt {\sec ^2(c+d x)}}\\ &=\frac {a \left (2 a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}+\frac {b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d}-\frac {\left (\left (\frac {9 a}{b^2}-\frac {6 a \left (-2+\frac {7 a^2}{b^2}\right )}{b^2}\right ) b^3 \sec (c+d x)\right ) \operatorname {Subst}\left (\int \sqrt {1+\frac {x^2}{b^2}} \, dx,x,b \tan (c+d x)\right )}{56 d \sqrt {\sec ^2(c+d x)}}\\ &=\frac {3 a \left (2 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a \left (2 a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}+\frac {b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d}-\frac {\left (\left (\frac {9 a}{b^2}-\frac {6 a \left (-2+\frac {7 a^2}{b^2}\right )}{b^2}\right ) b^3 \sec (c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{112 d \sqrt {\sec ^2(c+d x)}}\\ &=\frac {3 a \left (2 a^2-b^2\right ) \sinh ^{-1}(\tan (c+d x)) \sec (c+d x)}{16 d \sqrt {\sec ^2(c+d x)}}+\frac {3 a \left (2 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a \left (2 a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}+\frac {b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d}\\ \end {align*}
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Mathematica [B] time = 2.19, size = 637, normalized size = 4.01 \[ \frac {\sec ^7(c+d x) \left (4340 a^3 \sin (2 (c+d x))+2800 a^3 \sin (4 (c+d x))+420 a^3 \sin (6 (c+d x))-4410 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 )-1470 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 )-210 a^3 \cos (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+4410 a^3 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+1470 a^3 \cos (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+210 a^3 \cos (7 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+3584 \left (3 a^2 b-b^3\right ) \cos (2 (c+d x))-3675 a \left (2 a^2-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 (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+10752 a^2 b+6790 a b^2 \sin (2 (c+d x))-1400 a b^2 \sin (4 (c+d x))-210 a b^2 \sin (6 (c+d x))+2205 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 )+735 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 )+105 a b^2 \cos (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-2205 a b^2 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )-735 a b^2 \cos (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )-105 a b^2 \cos (7 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+1536 b^3\right )}{35840 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 170, normalized size = 1.07 \[ \frac {105 \, {\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 160 \, b^{3} + 224 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 70 \, {\left (3 \, {\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{5} + 8 \, a b^{2} \cos \left (d x + c\right ) + 2 \, {\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{1120 \, d \cos \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.08, size = 465, normalized size = 2.92 \[ \frac {105 \, {\left (2 \, a^{3} - a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (2 \, a^{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 (350 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 105 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 1680 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1540 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3360 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 1120 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 630 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1085 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 5040 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1120 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 6720 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 630 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1085 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3696 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 448 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1540 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 672 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 224 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 350 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 336 \, a^{2} b + 32 \, b^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{7}}}{560 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.54, size = 328, normalized size = 2.06 \[ \frac {a^{3} \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d}+\frac {3 a^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {3 a^{2} b}{5 d \cos \left (d x +c \right )^{5}}+\frac {b^{2} a \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{6}}+\frac {3 b^{2} a \left (\sin ^{3}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{4}}+\frac {3 b^{2} a \left (\sin ^{3}\left (d x +c \right )\right )}{16 d \cos \left (d x +c \right )^{2}}+\frac {3 a \,b^{2} \sin \left (d x +c \right )}{16 d}-\frac {3 b^{2} a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16 d}+\frac {b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{7 d \cos \left (d x +c \right )^{7}}+\frac {3 b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{35 d \cos \left (d x +c \right )^{5}}+\frac {b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{35 d \cos \left (d x +c \right )^{3}}-\frac {b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{35 d \cos \left (d x +c \right )}-\frac {b^{3} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{35 d}-\frac {2 b^{3} \cos \left (d x +c \right )}{35 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 208, normalized size = 1.31 \[ \frac {35 \, a b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 8 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 70 \, a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {672 \, a^{2} b}{\cos \left (d x + c\right )^{5}} - \frac {32 \, {\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} b^{3}}{\cos \left (d x + c\right )^{7}}}{1120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.34, size = 423, normalized size = 2.66 \[ \frac {3\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,a^2-b^2\right )}{8\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,a^3}{4}+\frac {3\,a\,b^2}{8}\right )+\frac {6\,a^2\,b}{5}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {11\,a\,b^2}{2}-3\,a^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {11\,a\,b^2}{2}-3\,a^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {5\,a^3}{4}+\frac {3\,a\,b^2}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {9\,a^3}{4}+\frac {31\,a\,b^2}{8}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {9\,a^3}{4}+\frac {31\,a\,b^2}{8}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (12\,a^2\,b-4\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {12\,a^2\,b}{5}-\frac {4\,b^3}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (18\,a^2\,b+4\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (24\,a^2\,b-8\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {66\,a^2\,b}{5}+\frac {8\,b^3}{5}\right )-\frac {4\,b^3}{35}+6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right )^{3} \sec ^{5}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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