Optimal. Leaf size=388 \[ \frac{b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}+\frac{b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 a^2 d \left (a^2-b^2\right )^2 (a \sec (c+d x)+b)}+\frac{\left (-29 a^2 b^2+8 a^4+15 b^4\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^3 d \left (a^2-b^2\right )^2}+\frac{b \left (11 a^2-5 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^2 d \left (a^2-b^2\right )^2}-\frac{\left (-29 a^2 b^2+8 a^4+15 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^3 d \left (a^2-b^2\right )^2}-\frac{b \left (-38 a^2 b^2+35 a^4+15 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^3 d (a-b)^2 (a+b)^3} \]
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Rubi [A] time = 1.03233, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {3238, 3845, 4098, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}+\frac{b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 a^2 d \left (a^2-b^2\right )^2 (a \sec (c+d x)+b)}+\frac{\left (-29 a^2 b^2+8 a^4+15 b^4\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^3 d \left (a^2-b^2\right )^2}+\frac{b \left (11 a^2-5 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^2 d \left (a^2-b^2\right )^2}-\frac{\left (-29 a^2 b^2+8 a^4+15 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^3 d \left (a^2-b^2\right )^2}-\frac{b \left (-38 a^2 b^2+35 a^4+15 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^3 d (a-b)^2 (a+b)^3} \]
Antiderivative was successfully verified.
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Rule 3238
Rule 3845
Rule 4098
Rule 4102
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx &=\int \frac{\sec ^{\frac{9}{2}}(c+d x)}{(b+a \sec (c+d x))^3} \, dx\\ &=\frac{b^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left (\frac{3 b^2}{2}-2 a b \sec (c+d x)+\frac{1}{2} \left (4 a^2-5 b^2\right ) \sec ^2(c+d x)\right )}{(b+a \sec (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac{b^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{b^2 \left (11 a^2-5 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac{\int \frac{\sqrt{\sec (c+d x)} \left (\frac{1}{4} b^2 \left (11 a^2-5 b^2\right )-a b \left (4 a^2-b^2\right ) \sec (c+d x)+\frac{1}{4} \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sec ^2(c+d x)\right )}{b+a \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (8 a^4-29 a^2 b^2+15 b^4\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 d}+\frac{b^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{b^2 \left (11 a^2-5 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac{\int \frac{-\frac{1}{8} b \left (8 a^4-29 a^2 b^2+15 b^4\right )-\frac{1}{2} a \left (2 a^4-10 a^2 b^2+5 b^4\right ) \sec (c+d x)-\frac{3}{8} b \left (8 a^4-11 a^2 b^2+5 b^4\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (b+a \sec (c+d x))} \, dx}{a^3 \left (a^2-b^2\right )^2}\\ &=\frac{\left (8 a^4-29 a^2 b^2+15 b^4\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 d}+\frac{b^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{b^2 \left (11 a^2-5 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac{\int \frac{-\frac{1}{8} b^2 \left (8 a^4-29 a^2 b^2+15 b^4\right )-\left (\frac{1}{2} a b \left (2 a^4-10 a^2 b^2+5 b^4\right )-\frac{1}{8} a b \left (8 a^4-29 a^2 b^2+15 b^4\right )\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{a^3 b^2 \left (a^2-b^2\right )^2}-\frac{\left (b \left (35 a^4-38 a^2 b^2+15 b^4\right )\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{b+a \sec (c+d x)} \, dx}{8 a^3 \left (a^2-b^2\right )^2}\\ &=\frac{\left (8 a^4-29 a^2 b^2+15 b^4\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 d}+\frac{b^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{b^2 \left (11 a^2-5 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac{\left (b \left (11 a^2-5 b^2\right )\right ) \int \sqrt{\sec (c+d x)} \, dx}{8 a^2 \left (a^2-b^2\right )^2}-\frac{\left (8 a^4-29 a^2 b^2+15 b^4\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{8 a^3 \left (a^2-b^2\right )^2}-\frac{\left (b \left (35 a^4-38 a^2 b^2+15 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{8 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac{b \left (35 a^4-38 a^2 b^2+15 b^4\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 a^3 (a-b)^2 (a+b)^3 d}+\frac{\left (8 a^4-29 a^2 b^2+15 b^4\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 d}+\frac{b^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{b^2 \left (11 a^2-5 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac{\left (b \left (11 a^2-5 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{8 a^2 \left (a^2-b^2\right )^2}-\frac{\left (\left (8 a^4-29 a^2 b^2+15 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{8 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (8 a^4-29 a^2 b^2+15 b^4\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 a^3 \left (a^2-b^2\right )^2 d}+\frac{b \left (11 a^2-5 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 a^2 \left (a^2-b^2\right )^2 d}-\frac{b \left (35 a^4-38 a^2 b^2+15 b^4\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 a^3 (a-b)^2 (a+b)^3 d}+\frac{\left (8 a^4-29 a^2 b^2+15 b^4\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 d}+\frac{b^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{b^2 \left (11 a^2-5 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 6.82133, size = 729, normalized size = 1.88 \[ \frac{\sqrt{\sec (c+d x)} \left (\frac{\left (-29 a^2 b^2+8 a^4+15 b^4\right ) \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2}+\frac{b^2 \sin (c+d x)}{2 a \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac{11 a^2 b^2 \sin (c+d x)-5 b^4 \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}\right )}{d}-\frac{-\frac{2 \left (-80 a^3 b^2+16 a^5+40 a b^4\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )}{b \left (1-\cos ^2(c+d x)\right ) (a+b \cos (c+d x))}+\frac{2 \left (-95 a^2 b^3+56 a^4 b+45 b^5\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left (\Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right )}{a \left (1-\cos ^2(c+d x)\right ) (a+b \cos (c+d x))}+\frac{\left (-29 a^2 b^3+8 a^4 b+15 b^5\right ) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left (4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-4 a b\right )}{a b^2 \left (1-\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \left (2-\sec ^2(c+d x)\right ) (a+b \cos (c+d x))}}{16 a^3 d (a-b)^2 (a+b)^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 14.863, size = 1992, normalized size = 5.1 \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 [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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Fricas [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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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 \frac{\sec \left (d x + c\right )^{\frac{3}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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