Optimal. Leaf size=244 \[ \frac{\left (16 a^2 b^2+2 a^4-15 b^4\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 a^4 d \left (a^2-b^2\right )}-\frac{b \left (4 a^2-5 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^3 d \left (a^2-b^2\right )}-\frac{b^3 \left (7 a^2-5 b^2\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^4 d (a-b) (a+b)^2}+\frac{\left (2 a^2-5 b^2\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d \left (a^2-b^2\right )}+\frac{b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.726656, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {4264, 3847, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{\left (16 a^2 b^2+2 a^4-15 b^4\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^4 d \left (a^2-b^2\right )}-\frac{b \left (4 a^2-5 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^3 d \left (a^2-b^2\right )}-\frac{b^3 \left (7 a^2-5 b^2\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^4 d (a-b) (a+b)^2}+\frac{\left (2 a^2-5 b^2\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d \left (a^2-b^2\right )}+\frac{b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4264
Rule 3847
Rule 4104
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx\\ &=\frac{b^2 \sqrt{\cos (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-a^2+\frac{5 b^2}{2}+a b \sec (c+d x)-\frac{3}{2} b^2 \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx}{a \left (a^2-b^2\right )}\\ &=\frac{\left (2 a^2-5 b^2\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \sqrt{\cos (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{3}{4} b \left (4 a^2-5 b^2\right )+\frac{1}{2} a \left (a^2+2 b^2\right ) \sec (c+d x)+\frac{1}{4} b \left (2 a^2-5 b^2\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{3 a^2 \left (a^2-b^2\right )}\\ &=\frac{\left (2 a^2-5 b^2\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \sqrt{\cos (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{3}{4} a b \left (4 a^2-5 b^2\right )-\left (-\frac{3}{4} b^2 \left (4 a^2-5 b^2\right )-\frac{1}{2} a^2 \left (a^2+2 b^2\right )\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{3 a^4 \left (a^2-b^2\right )}-\frac{\left (b^3 \left (7-\frac{5 b^2}{a^2}\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )}\\ &=\frac{\left (2 a^2-5 b^2\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \sqrt{\cos (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac{\left (b^3 \left (7 a^2-5 b^2\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{2 a^4 \left (a^2-b^2\right )}-\frac{\left (b \left (4 a^2-5 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{2 a^3 \left (a^2-b^2\right )}+\frac{\left (\left (2 a^4+16 a^2 b^2-15 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \, dx}{6 a^4 \left (a^2-b^2\right )}\\ &=-\frac{b^3 \left (7 a^2-5 b^2\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^4 (a-b) (a+b)^2 d}+\frac{\left (2 a^2-5 b^2\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \sqrt{\cos (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac{\left (b \left (4 a^2-5 b^2\right )\right ) \int \sqrt{\cos (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )}+\frac{\left (2 a^4+16 a^2 b^2-15 b^4\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{6 a^4 \left (a^2-b^2\right )}\\ &=-\frac{b \left (4 a^2-5 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^3 \left (a^2-b^2\right ) d}+\frac{\left (2 a^4+16 a^2 b^2-15 b^4\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^4 \left (a^2-b^2\right ) d}-\frac{b^3 \left (7 a^2-5 b^2\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^4 (a-b) (a+b)^2 d}+\frac{\left (2 a^2-5 b^2\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \sqrt{\cos (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 1.81213, size = 268, normalized size = 1.1 \[ \frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left (\frac{3 b^3}{\left (b^2-a^2\right ) (a \cos (c+d x)+b)}+2\right )-\frac{\frac{8 \left (a^2+2 b^2\right ) \left ((a+b) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )-b \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{a+b}+\frac{6 \left (4 a^2-5 b^2\right ) \sin (c+d x) \left (-2 b (a+b) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right ),-1\right )+\left (a^2-2 b^2\right ) \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )+2 a b E\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )\right )}{a^2 \sqrt{\sin ^2(c+d x)}}+\frac{2 \left (5 b^3-8 a^2 b\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}}{(b-a) (a+b)}}{12 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 4.984, size = 1064, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{\frac{3}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{\frac{3}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]