Optimal. Leaf size=142 \[ \frac{2 a^3 (35 B+32 C) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (5 B+8 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 a^{5/2} B \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{2 a C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.317288, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4072, 3917, 3915, 3774, 203, 3792} \[ \frac{2 a^3 (35 B+32 C) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (5 B+8 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 a^{5/2} B \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{2 a C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4072
Rule 3917
Rule 3915
Rule 3774
Rule 203
Rule 3792
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int (a+a \sec (c+d x))^{5/2} (B+C \sec (c+d x)) \, dx\\ &=\frac{2 a C (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 d}+\frac{2}{5} \int (a+a \sec (c+d x))^{3/2} \left (\frac{5 a B}{2}+\frac{1}{2} a (5 B+8 C) \sec (c+d x)\right ) \, dx\\ &=\frac{2 a^2 (5 B+8 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{15 d}+\frac{2 a C (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 d}+\frac{4}{15} \int \sqrt{a+a \sec (c+d x)} \left (\frac{15 a^2 B}{4}+\frac{1}{4} a^2 (35 B+32 C) \sec (c+d x)\right ) \, dx\\ &=\frac{2 a^2 (5 B+8 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{15 d}+\frac{2 a C (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 d}+\left (a^2 B\right ) \int \sqrt{a+a \sec (c+d x)} \, dx+\frac{1}{15} \left (a^2 (35 B+32 C)\right ) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{2 a^3 (35 B+32 C) \tan (c+d x)}{15 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (5 B+8 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{15 d}+\frac{2 a C (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 d}-\frac{\left (2 a^3 B\right ) \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,-\frac{a \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=\frac{2 a^{5/2} B \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}+\frac{2 a^3 (35 B+32 C) \tan (c+d x)}{15 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (5 B+8 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{15 d}+\frac{2 a C (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.857512, size = 128, normalized size = 0.9 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sec ^2(c+d x) \sqrt{a (\sec (c+d x)+1)} \left (2 \sin \left (\frac{1}{2} (c+d x)\right ) (2 (5 B+14 C) \cos (c+d x)+(40 B+43 C) \cos (2 (c+d x))+40 B+49 C)+30 \sqrt{2} B \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right ) \cos ^{\frac{5}{2}}(c+d x)\right )}{30 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.281, size = 341, normalized size = 2.4 \begin{align*} -{\frac{{a}^{2}}{60\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 15\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{2} \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{5/2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}} \right ) \sin \left ( dx+c \right ) +30\,B\cos \left ( dx+c \right ) \sqrt{2} \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{5/2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}} \right ) \sin \left ( dx+c \right ) +15\,B\sqrt{2} \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{5/2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}} \right ) \sin \left ( dx+c \right ) +320\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+344\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}-280\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}-232\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}-40\,B\cos \left ( dx+c \right ) -88\,C\cos \left ( dx+c \right ) -24\,C \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.93491, size = 1885, normalized size = 13.27 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.572533, size = 954, normalized size = 6.72 \begin{align*} \left [\frac{15 \,{\left (B a^{2} \cos \left (d x + c\right )^{3} + B a^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt{-a} \log \left (\frac{2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \,{\left ({\left (40 \, B + 43 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (5 \, B + 14 \, C\right )} a^{2} \cos \left (d x + c\right ) + 3 \, C a^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{15 \,{\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )}}, -\frac{2 \,{\left (15 \,{\left (B a^{2} \cos \left (d x + c\right )^{3} + B a^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt{a} \arctan \left (\frac{\sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt{a} \sin \left (d x + c\right )}\right ) -{\left ({\left (40 \, B + 43 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (5 \, B + 14 \, C\right )} a^{2} \cos \left (d x + c\right ) + 3 \, C a^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{15 \,{\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )}}\right ] \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]