Optimal. Leaf size=75 \[ -\frac {3}{2} b x \left (2 a^2+b^2\right )+\frac {6 a b^2 \cos (c+d x)}{d}+\frac {\sec (c+d x) (a+b \sin (c+d x))^3}{d}+\frac {3 b^3 \sin (c+d x) \cos (c+d x)}{2 d} \]
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Rubi [A] time = 0.07, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2861, 12, 2644} \[ -\frac {3}{2} b x \left (2 a^2+b^2\right )+\frac {6 a b^2 \cos (c+d x)}{d}+\frac {\sec (c+d x) (a+b \sin (c+d x))^3}{d}+\frac {3 b^3 \sin (c+d x) \cos (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2644
Rule 2861
Rubi steps
\begin {align*} \int \sec (c+d x) (a+b \sin (c+d x))^3 \tan (c+d x) \, dx &=\frac {\sec (c+d x) (a+b \sin (c+d x))^3}{d}-\int 3 b (a+b \sin (c+d x))^2 \, dx\\ &=\frac {\sec (c+d x) (a+b \sin (c+d x))^3}{d}-(3 b) \int (a+b \sin (c+d x))^2 \, dx\\ &=-\frac {3}{2} b \left (2 a^2+b^2\right ) x+\frac {6 a b^2 \cos (c+d x)}{d}+\frac {3 b^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {\sec (c+d x) (a+b \sin (c+d x))^3}{d}\\ \end {align*}
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Mathematica [A] time = 0.55, size = 91, normalized size = 1.21 \[ \frac {\sec (c+d x) \left (8 a^3+12 a b^2 \cos (2 (c+d x))+36 a b^2+b^3 \sin (3 (c+d x))\right )+3 b \left (\left (8 a^2+3 b^2\right ) \tan (c+d x)-4 \left (2 a^2+b^2\right ) (c+d x)\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 90, normalized size = 1.20 \[ \frac {6 \, a b^{2} \cos \left (d x + c\right )^{2} - 3 \, {\left (2 \, a^{2} b + b^{3}\right )} d x \cos \left (d x + c\right ) + 2 \, a^{3} + 6 \, a b^{2} + {\left (b^{3} \cos \left (d x + c\right )^{2} + 6 \, a^{2} b + 2 \, b^{3}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 148, normalized size = 1.97 \[ -\frac {3 \, {\left (2 \, a^{2} b + b^{3}\right )} {\left (d x + c\right )} + \frac {4 \, {\left (3 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3} + 3 \, a b^{2}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + \frac {2 \, {\left (b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 132, normalized size = 1.76 \[ \frac {\frac {a^{3}}{\cos \left (d x +c \right )}+3 a^{2} b \left (\tan \left (d x +c \right )-d x -c \right )+3 a \,b^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )+b^{3} \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 99, normalized size = 1.32 \[ -\frac {6 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{2} b + {\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} b^{3} - 6 \, a b^{2} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} - \frac {2 \, a^{3}}{\cos \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 16.29, size = 219, normalized size = 2.92 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^2\,b+3\,b^3\right )+2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+12\,a\,b^2+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (4\,a^3+12\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (6\,a^2\,b+3\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (12\,a^2\,b+2\,b^3\right )+2\,a^3}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {3\,b\,\mathrm {atan}\left (\frac {3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2+b^2\right )}{6\,a^2\,b+3\,b^3}\right )\,\left (2\,a^2+b^2\right )}{d} \]
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 \[ \int \left (a + b \sin {\left (c + d x \right )}\right )^{3} \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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