Optimal. Leaf size=80 \[ -\frac {3 a b^2 \sin (c+d x)}{d}+\frac {(a-b)^3 \log (\sin (c+d x)+1)}{2 d}-\frac {(a+b)^3 \log (1-\sin (c+d x))}{2 d}-\frac {b^3 \sin ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.11, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2668, 702, 633, 31} \[ -\frac {3 a b^2 \sin (c+d x)}{d}+\frac {(a-b)^3 \log (\sin (c+d x)+1)}{2 d}-\frac {(a+b)^3 \log (1-\sin (c+d x))}{2 d}-\frac {b^3 \sin ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 702
Rule 2668
Rubi steps
\begin {align*} \int \sec (c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {(a+x)^3}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b \operatorname {Subst}\left (\int \left (-3 a-x+\frac {a^3+3 a b^2+\left (3 a^2+b^2\right ) x}{b^2-x^2}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {3 a b^2 \sin (c+d x)}{d}-\frac {b^3 \sin ^2(c+d x)}{2 d}+\frac {b \operatorname {Subst}\left (\int \frac {a^3+3 a b^2+\left (3 a^2+b^2\right ) x}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {3 a b^2 \sin (c+d x)}{d}-\frac {b^3 \sin ^2(c+d x)}{2 d}-\frac {(a-b)^3 \operatorname {Subst}\left (\int \frac {1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{2 d}+\frac {(a+b)^3 \operatorname {Subst}\left (\int \frac {1}{b-x} \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=-\frac {(a+b)^3 \log (1-\sin (c+d x))}{2 d}+\frac {(a-b)^3 \log (1+\sin (c+d x))}{2 d}-\frac {3 a b^2 \sin (c+d x)}{d}-\frac {b^3 \sin ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 67, normalized size = 0.84 \[ -\frac {6 a b^2 \sin (c+d x)+(a-b)^3 (-\log (\sin (c+d x)+1))+(a+b)^3 \log (1-\sin (c+d x))+b^3 \sin ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 93, normalized size = 1.16 \[ \frac {b^{3} \cos \left (d x + c\right )^{2} - 6 \, a b^{2} \sin \left (d x + c\right ) + {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 93, normalized size = 1.16 \[ -\frac {b^{3} \sin \left (d x + c\right )^{2} + 6 \, a b^{2} \sin \left (d x + c\right ) - {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 108, normalized size = 1.35 \[ \frac {a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {3 a^{2} b \ln \left (\cos \left (d x +c \right )\right )}{d}-\frac {3 a \,b^{2} \sin \left (d x +c \right )}{d}+\frac {3 a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {b^{3} \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}-\frac {b^{3} \ln \left (\cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 91, normalized size = 1.14 \[ -\frac {b^{3} \sin \left (d x + c\right )^{2} + 6 \, a b^{2} \sin \left (d x + c\right ) - {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.13, size = 65, normalized size = 0.81 \[ -\frac {\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,{\left (a+b\right )}^3}{2}-\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,{\left (a-b\right )}^3}{2}+\frac {b^3\,{\sin \left (c+d\,x\right )}^2}{2}+3\,a\,b^2\,\sin \left (c+d\,x\right )}{d} \]
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 \[ \int \left (a + b \sin {\left (c + d x \right )}\right )^{3} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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