Optimal. Leaf size=105 \[ \frac{a^6 (A+B)}{6 d (a-a \sin (c+d x))^3}+\frac{a^5 (A-B)}{8 d (a-a \sin (c+d x))^2}+\frac{a^4 (A-B)}{8 d (a-a \sin (c+d x))}+\frac{a^3 (A-B) \tanh ^{-1}(\sin (c+d x))}{8 d} \]
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Rubi [A] time = 0.135148, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {2836, 77, 206} \[ \frac{a^6 (A+B)}{6 d (a-a \sin (c+d x))^3}+\frac{a^5 (A-B)}{8 d (a-a \sin (c+d x))^2}+\frac{a^4 (A-B)}{8 d (a-a \sin (c+d x))}+\frac{a^3 (A-B) \tanh ^{-1}(\sin (c+d x))}{8 d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 77
Rule 206
Rubi steps
\begin{align*} \int \sec ^7(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx &=\frac{a^7 \operatorname{Subst}\left (\int \frac{A+\frac{B x}{a}}{(a-x)^4 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^7 \operatorname{Subst}\left (\int \left (\frac{A+B}{2 a (a-x)^4}+\frac{A-B}{4 a^2 (a-x)^3}+\frac{A-B}{8 a^3 (a-x)^2}+\frac{A-B}{8 a^3 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^6 (A+B)}{6 d (a-a \sin (c+d x))^3}+\frac{a^5 (A-B)}{8 d (a-a \sin (c+d x))^2}+\frac{a^4 (A-B)}{8 d (a-a \sin (c+d x))}+\frac{\left (a^4 (A-B)\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{8 d}\\ &=\frac{a^3 (A-B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^6 (A+B)}{6 d (a-a \sin (c+d x))^3}+\frac{a^5 (A-B)}{8 d (a-a \sin (c+d x))^2}+\frac{a^4 (A-B)}{8 d (a-a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.253383, size = 95, normalized size = 0.9 \[ \frac{a^3 \left (-3 (A-B) \sin ^2(c+d x)+9 (A-B) \sin (c+d x)-3 (A-B) \tanh ^{-1}(\sin (c+d x)) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^6+2 (B-5 A)\right )}{24 d (\sin (c+d x)-1)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.129, size = 521, normalized size = 5. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06411, size = 166, normalized size = 1.58 \begin{align*} \frac{3 \,{\left (A - B\right )} a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (A - B\right )} a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (3 \,{\left (A - B\right )} a^{3} \sin \left (d x + c\right )^{2} - 9 \,{\left (A - B\right )} a^{3} \sin \left (d x + c\right ) + 2 \,{\left (5 \, A - B\right )} a^{3}\right )}}{\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 1}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.77478, size = 570, normalized size = 5.43 \begin{align*} \frac{6 \,{\left (A - B\right )} a^{3} \cos \left (d x + c\right )^{2} + 18 \,{\left (A - B\right )} a^{3} \sin \left (d x + c\right ) - 2 \,{\left (13 \, A - 5 \, B\right )} a^{3} + 3 \,{\left (3 \,{\left (A - B\right )} a^{3} \cos \left (d x + c\right )^{2} - 4 \,{\left (A - B\right )} a^{3} -{\left ({\left (A - B\right )} a^{3} \cos \left (d x + c\right )^{2} - 4 \,{\left (A - B\right )} a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (3 \,{\left (A - B\right )} a^{3} \cos \left (d x + c\right )^{2} - 4 \,{\left (A - B\right )} a^{3} -{\left ({\left (A - B\right )} a^{3} \cos \left (d x + c\right )^{2} - 4 \,{\left (A - B\right )} a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{48 \,{\left (3 \, d \cos \left (d x + c\right )^{2} -{\left (d \cos \left (d x + c\right )^{2} - 4 \, d\right )} \sin \left (d x + c\right ) - 4 \, d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40851, size = 213, normalized size = 2.03 \begin{align*} \frac{6 \,{\left (A a^{3} - B a^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 6 \,{\left (A a^{3} - B a^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + \frac{11 \, A a^{3} \sin \left (d x + c\right )^{3} - 11 \, B a^{3} \sin \left (d x + c\right )^{3} - 45 \, A a^{3} \sin \left (d x + c\right )^{2} + 45 \, B a^{3} \sin \left (d x + c\right )^{2} + 69 \, A a^{3} \sin \left (d x + c\right ) - 69 \, B a^{3} \sin \left (d x + c\right ) - 51 \, A a^{3} + 19 \, B a^{3}}{{\left (\sin \left (d x + c\right ) - 1\right )}^{3}}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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