Optimal. Leaf size=126 \[ \frac{2 a \left (a^2+9 b^2\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 d}+\frac{2 b \left (3 a^2-b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}-\frac{2 b \left (a^2-3 b^2\right ) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}{3 d} \]
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Rubi [A] time = 0.23136, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4264, 3841, 4047, 3771, 2641, 4046, 2639} \[ \frac{2 a \left (a^2+9 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 b \left (3 a^2-b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}-\frac{2 b \left (a^2-3 b^2\right ) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}{3 d} \]
Antiderivative was successfully verified.
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Rule 4264
Rule 3841
Rule 4047
Rule 3771
Rule 2641
Rule 4046
Rule 2639
Rubi steps
\begin{align*} \int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \sec (c+d x))^3}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x)) \sin (c+d x)}{3 d}+\frac{1}{3} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{4 a^2 b+\frac{1}{2} a \left (a^2+9 b^2\right ) \sec (c+d x)-\frac{1}{2} b \left (a^2-3 b^2\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x)) \sin (c+d x)}{3 d}+\frac{1}{3} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{4 a^2 b-\frac{1}{2} b \left (a^2-3 b^2\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (a \left (a^2+9 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=-\frac{2 b \left (a^2-3 b^2\right ) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x)) \sin (c+d x)}{3 d}+\frac{1}{3} \left (a \left (a^2+9 b^2\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\left (b \left (3 a^2-b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a \left (a^2+9 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 b \left (a^2-3 b^2\right ) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x)) \sin (c+d x)}{3 d}+\left (b \left (3 a^2-b^2\right )\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 b \left (3 a^2-b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a \left (a^2+9 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 b \left (a^2-3 b^2\right ) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x)) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.522461, size = 87, normalized size = 0.69 \[ \frac{2 \left (\left (a^3+9 a b^2\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+\left (9 a^2 b-3 b^3\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{\sin (c+d x) \left (a^3 \cos (c+d x)+3 b^3\right )}{\sqrt{\cos (c+d x)}}\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.954, size = 303, normalized size = 2.4 \begin{align*} -{\frac{2}{3\,d} \left ( 4\,{a}^{3}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+{a}^{3}\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,\sqrt{2} \right ) +9\,a{b}^{2}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -9\,{a}^{2}b\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +3\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ){b}^{3}-2\,{a}^{3}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,{b}^{3}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{3} \cos \left (d x + c\right ) \sec \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right ) \sec \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) \sec \left (d x + c\right ) + a^{3} \cos \left (d x + c\right )\right )} \sqrt{\cos \left (d x + c\right )}, x\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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