Optimal. Leaf size=78 \[ \frac {a^3 \sec (c+d x)}{d}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {3 a^2 b \tan (c+d x)}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \tan (c+d x)}{d}-b^3 x \]
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Rubi [A] time = 0.15, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2912, 3767, 8, 2622, 321, 207, 2606, 3473} \[ \frac {3 a^2 b \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x)}{d}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \tan (c+d x)}{d}-b^3 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 207
Rule 321
Rule 2606
Rule 2622
Rule 2912
Rule 3473
Rule 3767
Rubi steps
\begin {align*} \int \csc (c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx &=\int \left (3 a^2 b \sec ^2(c+d x)+a^3 \csc (c+d x) \sec ^2(c+d x)+3 a b^2 \sec (c+d x) \tan (c+d x)+b^3 \tan ^2(c+d x)\right ) \, dx\\ &=a^3 \int \csc (c+d x) \sec ^2(c+d x) \, dx+\left (3 a^2 b\right ) \int \sec ^2(c+d x) \, dx+\left (3 a b^2\right ) \int \sec (c+d x) \tan (c+d x) \, dx+b^3 \int \tan ^2(c+d x) \, dx\\ &=\frac {b^3 \tan (c+d x)}{d}-b^3 \int 1 \, dx+\frac {a^3 \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}-\frac {\left (3 a^2 b\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}+\frac {\left (3 a b^2\right ) \operatorname {Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}\\ &=-b^3 x+\frac {a^3 \sec (c+d x)}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {3 a^2 b \tan (c+d x)}{d}+\frac {b^3 \tan (c+d x)}{d}+\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-b^3 x-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a^3 \sec (c+d x)}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {3 a^2 b \tan (c+d x)}{d}+\frac {b^3 \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 83, normalized size = 1.06 \[ \frac {a^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+a^3 \left (-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+b \left (3 a^2+b^2\right ) \tan (c+d x)+a \left (a^2+3 b^2\right ) \sec (c+d x)-b^3 c-b^3 d x}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 99, normalized size = 1.27 \[ -\frac {2 \, b^{3} d x \cos \left (d x + c\right ) + a^{3} \cos \left (d x + c\right ) \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - a^{3} \cos \left (d x + c\right ) \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, a^{3} - 6 \, a b^{2} - 2 \, {\left (3 \, a^{2} b + b^{3}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 86, normalized size = 1.10 \[ -\frac {{\left (d x + c\right )} b^{3} - a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {2 \, {\left (3 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3} + 3 \, a b^{2}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 100, normalized size = 1.28 \[ \frac {a^{3}}{d \cos \left (d x +c \right )}+\frac {a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d}+\frac {3 a^{2} b \tan \left (d x +c \right )}{d}+\frac {3 a \,b^{2}}{d \cos \left (d x +c \right )}-b^{3} x +\frac {b^{3} \tan \left (d x +c \right )}{d}-\frac {b^{3} c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 86, normalized size = 1.10 \[ -\frac {2 \, {\left (d x + c - \tan \left (d x + c\right )\right )} b^{3} - a^{3} {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 6 \, a^{2} b \tan \left (d x + c\right ) - \frac {6 \, a b^{2}}{\cos \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.93, size = 154, normalized size = 1.97 \[ \frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^2\,b+2\,b^3\right )+6\,a\,b^2+2\,a^3}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}+\frac {2\,b^3\,\mathrm {atan}\left (\frac {4\,b^6}{4\,a^3\,b^3+4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^6}-\frac {4\,a^3\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,a^3\,b^3+4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^6}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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