Optimal. Leaf size=116 \[ \frac {a^2 \tan ^3(c+d x)}{3 d}-\frac {a^2 \tan (c+d x)}{d}+a^2 x+\frac {3 a b \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {a b \tan ^3(c+d x) \sec (c+d x)}{2 d}-\frac {3 a b \tan (c+d x) \sec (c+d x)}{4 d}+\frac {b^2 \tan ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.15, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3886, 3473, 8, 2611, 3770, 2607, 30} \[ \frac {a^2 \tan ^3(c+d x)}{3 d}-\frac {a^2 \tan (c+d x)}{d}+a^2 x+\frac {3 a b \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {a b \tan ^3(c+d x) \sec (c+d x)}{2 d}-\frac {3 a b \tan (c+d x) \sec (c+d x)}{4 d}+\frac {b^2 \tan ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2607
Rule 2611
Rule 3473
Rule 3770
Rule 3886
Rubi steps
\begin {align*} \int (a+b \sec (c+d x))^2 \tan ^4(c+d x) \, dx &=\int \left (a^2 \tan ^4(c+d x)+2 a b \sec (c+d x) \tan ^4(c+d x)+b^2 \sec ^2(c+d x) \tan ^4(c+d x)\right ) \, dx\\ &=a^2 \int \tan ^4(c+d x) \, dx+(2 a b) \int \sec (c+d x) \tan ^4(c+d x) \, dx+b^2 \int \sec ^2(c+d x) \tan ^4(c+d x) \, dx\\ &=\frac {a^2 \tan ^3(c+d x)}{3 d}+\frac {a b \sec (c+d x) \tan ^3(c+d x)}{2 d}-a^2 \int \tan ^2(c+d x) \, dx-\frac {1}{2} (3 a b) \int \sec (c+d x) \tan ^2(c+d x) \, dx+\frac {b^2 \operatorname {Subst}\left (\int x^4 \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {a^2 \tan (c+d x)}{d}-\frac {3 a b \sec (c+d x) \tan (c+d x)}{4 d}+\frac {a^2 \tan ^3(c+d x)}{3 d}+\frac {a b \sec (c+d x) \tan ^3(c+d x)}{2 d}+\frac {b^2 \tan ^5(c+d x)}{5 d}+a^2 \int 1 \, dx+\frac {1}{4} (3 a b) \int \sec (c+d x) \, dx\\ &=a^2 x+\frac {3 a b \tanh ^{-1}(\sin (c+d x))}{4 d}-\frac {a^2 \tan (c+d x)}{d}-\frac {3 a b \sec (c+d x) \tan (c+d x)}{4 d}+\frac {a^2 \tan ^3(c+d x)}{3 d}+\frac {a b \sec (c+d x) \tan ^3(c+d x)}{2 d}+\frac {b^2 \tan ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [B] time = 0.97, size = 355, normalized size = 3.06 \[ \frac {\sec ^5(c+d x) \left (-80 a^2 \sin (c+d x)-160 a^2 \sin (3 (c+d x))-80 a^2 \sin (5 (c+d x))+60 a^2 c \cos (5 (c+d x))+60 a^2 d x \cos (5 (c+d x))-60 a b \sin (2 (c+d x))-150 a b \sin (4 (c+d x))-45 a b \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+45 a b \cos (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+150 a \cos (c+d x) \left (4 a (c+d x)-3 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+3 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+75 a \cos (3 (c+d x)) \left (4 a (c+d x)-3 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+3 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+120 b^2 \sin (c+d x)-60 b^2 \sin (3 (c+d x))+12 b^2 \sin (5 (c+d x))\right )}{960 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 151, normalized size = 1.30 \[ \frac {120 \, a^{2} d x \cos \left (d x + c\right )^{5} + 45 \, a b \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 45 \, a b \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (75 \, a b \cos \left (d x + c\right )^{3} + 4 \, {\left (20 \, a^{2} - 3 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 30 \, a b \cos \left (d x + c\right ) - 4 \, {\left (5 \, a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 12 \, b^{2}\right )} \sin \left (d x + c\right )}{120 \, d \cos \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.39, size = 220, normalized size = 1.90 \[ \frac {60 \, {\left (d x + c\right )} a^{2} + 45 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 45 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (60 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 45 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 320 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 210 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 192 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 320 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 210 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 45 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 164, normalized size = 1.41 \[ \frac {a^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {a^{2} \tan \left (d x +c \right )}{d}+a^{2} x +\frac {a^{2} c}{d}+\frac {a b \left (\sin ^{5}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{4}}-\frac {a b \left (\sin ^{5}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{2}}-\frac {a b \left (\sin ^{3}\left (d x +c \right )\right )}{4 d}-\frac {3 a b \sin \left (d x +c \right )}{4 d}+\frac {3 a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{4 d}+\frac {b^{2} \left (\sin ^{5}\left (d x +c \right )\right )}{5 d \cos \left (d x +c \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 118, normalized size = 1.02 \[ \frac {24 \, b^{2} \tan \left (d x + c\right )^{5} + 40 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{2} + 15 \, a b {\left (\frac {2 \, {\left (5 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.56, size = 332, normalized size = 2.86 \[ \frac {\left (2\,a^2-\frac {3\,a\,b}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (7\,a\,b-\frac {32\,a^2}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {52\,a^2}{3}-\frac {32\,b^2}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {32\,a^2}{3}-7\,b\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a^2+\frac {3\,b\,a}{2}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}+\frac {2\,a^2\,\mathrm {atan}\left (\frac {64\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,a^6+36\,a^4\,b^2}+\frac {36\,a^4\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,a^6+36\,a^4\,b^2}\right )}{d}+\frac {3\,a\,b\,\mathrm {atanh}\left (\frac {48\,a^5\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{48\,a^5\,b+27\,a^3\,b^3}+\frac {27\,a^3\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{48\,a^5\,b+27\,a^3\,b^3}\right )}{2\,d} \]
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 \[ \int \left (a + b \sec {\left (c + d x \right )}\right )^{2} \tan ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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