Optimal. Leaf size=111 \[ \frac {a b^2 \sin (c+d x)}{2 d}+\frac {(a+2 b) (a-b)^2 \log (\sin (c+d x)+1)}{4 d}-\frac {(a-2 b) (a+b)^2 \log (1-\sin (c+d x))}{4 d}+\frac {\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{2 d} \]
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Rubi [A] time = 0.13, antiderivative size = 111, normalized size of antiderivative = 1.00, 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 = {2668, 739, 774, 633, 31} \[ \frac {a b^2 \sin (c+d x)}{2 d}+\frac {(a+2 b) (a-b)^2 \log (\sin (c+d x)+1)}{4 d}-\frac {(a-2 b) (a+b)^2 \log (1-\sin (c+d x))}{4 d}+\frac {\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 739
Rule 774
Rule 2668
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {b^3 \operatorname {Subst}\left (\int \frac {(a+x)^3}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{2 d}-\frac {b \operatorname {Subst}\left (\int \frac {(a+x) \left (-a^2+2 b^2+a x\right )}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=\frac {a b^2 \sin (c+d x)}{2 d}+\frac {\sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{2 d}+\frac {b \operatorname {Subst}\left (\int \frac {-a b^2-a \left (-a^2+2 b^2\right )-2 b^2 x}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=\frac {a b^2 \sin (c+d x)}{2 d}+\frac {\sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{2 d}+\frac {\left ((a-2 b) (a+b)^2\right ) \operatorname {Subst}\left (\int \frac {1}{b-x} \, dx,x,b \sin (c+d x)\right )}{4 d}-\frac {\left ((a-b)^2 (a+2 b)\right ) \operatorname {Subst}\left (\int \frac {1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{4 d}\\ &=-\frac {(a-2 b) (a+b)^2 \log (1-\sin (c+d x))}{4 d}+\frac {(a-b)^2 (a+2 b) \log (1+\sin (c+d x))}{4 d}+\frac {a b^2 \sin (c+d x)}{2 d}+\frac {\sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{2 d}\\ \end {align*}
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Mathematica [A] time = 1.34, size = 176, normalized size = 1.59 \[ \frac {2 a^4 b \sec ^2(c+d x)+\left (a^2-b^2\right ) \left ((a-2 b) (a+b)^2 \log (1-\sin (c+d x))-(a-b)^2 (a+2 b) \log (\sin (c+d x)+1)\right )-a \tan (c+d x) \sec (c+d x) \left (2 a^4+4 a^2 b^2+b^4 \cos (2 (c+d x))-7 b^4\right )+\tan ^2(c+d x) \left (-8 a^4 b+4 a^2 b^3-2 a b^4 \sin (c+d x)+2 b^5\right )}{4 d \left (b^2-a^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 112, normalized size = 1.01 \[ \frac {{\left (a^{3} - 3 \, a b^{2} + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a^{3} - 3 \, a b^{2} - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 6 \, a^{2} b + 2 \, b^{3} + 2 \, {\left (a^{3} + 3 \, a b^{2}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.13, size = 114, normalized size = 1.03 \[ \frac {{\left (a^{3} - 3 \, a b^{2} + 2 \, b^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - {\left (a^{3} - 3 \, a b^{2} - 2 \, b^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (b^{3} \sin \left (d x + c\right )^{2} + a^{3} \sin \left (d x + c\right ) + 3 \, a b^{2} \sin \left (d x + c\right ) + 3 \, a^{2} b\right )}}{\sin \left (d x + c\right )^{2} - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 154, normalized size = 1.39 \[ \frac {a^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {3 a^{2} b}{2 d \cos \left (d x +c \right )^{2}}+\frac {3 a \,b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {3 a \,b^{2} \sin \left (d x +c \right )}{2 d}-\frac {3 a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {b^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {b^{3} \ln \left (\cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 98, normalized size = 0.88 \[ \frac {{\left (a^{3} - 3 \, a b^{2} + 2 \, b^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a^{3} - 3 \, a b^{2} - 2 \, b^{3}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (3 \, a^{2} b + b^{3} + {\left (a^{3} + 3 \, a b^{2}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{2} - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.21, size = 99, normalized size = 0.89 \[ \frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,{\left (a-b\right )}^2\,\left (a+2\,b\right )}{4\,d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,{\left (a+b\right )}^2\,\left (a-2\,b\right )}{4\,d}-\frac {\frac {3\,a^2\,b}{2}+\frac {b^3}{2}+\sin \left (c+d\,x\right )\,\left (\frac {a^3}{2}+\frac {3\,a\,b^2}{2}\right )}{d\,\left ({\sin \left (c+d\,x\right )}^2-1\right )} \]
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 \[ \int \left (a + b \sin {\left (c + d x \right )}\right )^{3} \sec ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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