Optimal. Leaf size=254 \[ \frac {a^{5/2} (283 A+326 B) \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{128 d}+\frac {a^3 (283 A+326 B) \sin (c+d x)}{128 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (283 A+326 B) \cos (c+d x) \sin (c+d x)}{192 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (157 A+170 B) \cos ^2(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (13 A+10 B) \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac {a A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.44, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {4102, 4100,
3890, 3859, 209} \begin {gather*} \frac {a^{5/2} (283 A+326 B) \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{128 d}+\frac {a^3 (283 A+326 B) \sin (c+d x)}{128 d \sqrt {a \sec (c+d x)+a}}+\frac {a^3 (157 A+170 B) \sin (c+d x) \cos ^2(c+d x)}{240 d \sqrt {a \sec (c+d x)+a}}+\frac {a^3 (283 A+326 B) \sin (c+d x) \cos (c+d x)}{192 d \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (13 A+10 B) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{40 d}+\frac {a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 3859
Rule 3890
Rule 4100
Rule 4102
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx &=\frac {a A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{5} \int \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {1}{2} a (13 A+10 B)+\frac {5}{2} a (A+2 B) \sec (c+d x)\right ) \, dx\\ &=\frac {a^2 (13 A+10 B) \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac {a A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{20} \int \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {1}{4} a^2 (157 A+170 B)+\frac {5}{4} a^2 (21 A+26 B) \sec (c+d x)\right ) \, dx\\ &=\frac {a^3 (157 A+170 B) \cos ^2(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (13 A+10 B) \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac {a A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{96} \left (a^2 (283 A+326 B)\right ) \int \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^3 (283 A+326 B) \cos (c+d x) \sin (c+d x)}{192 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (157 A+170 B) \cos ^2(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (13 A+10 B) \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac {a A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{128} \left (a^2 (283 A+326 B)\right ) \int \cos (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^3 (283 A+326 B) \sin (c+d x)}{128 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (283 A+326 B) \cos (c+d x) \sin (c+d x)}{192 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (157 A+170 B) \cos ^2(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (13 A+10 B) \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac {a A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{256} \left (a^2 (283 A+326 B)\right ) \int \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^3 (283 A+326 B) \sin (c+d x)}{128 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (283 A+326 B) \cos (c+d x) \sin (c+d x)}{192 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (157 A+170 B) \cos ^2(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (13 A+10 B) \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac {a A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}-\frac {\left (a^3 (283 A+326 B)\right ) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,-\frac {a \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{128 d}\\ &=\frac {a^{5/2} (283 A+326 B) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{128 d}+\frac {a^3 (283 A+326 B) \sin (c+d x)}{128 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (283 A+326 B) \cos (c+d x) \sin (c+d x)}{192 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (157 A+170 B) \cos ^2(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (13 A+10 B) \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac {a A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 1.84, size = 416, normalized size = 1.64 \begin {gather*} \frac {a^2 \left (25935 A \tanh ^{-1}\left (\sqrt {1-\sec (c+d x)}\right )+28350 B \tanh ^{-1}\left (\sqrt {1-\sec (c+d x)}\right )+11651 A \sqrt {1-\sec (c+d x)}+9702 B \sqrt {1-\sec (c+d x)}+37029 A \cos (c+d x) \sqrt {1-\sec (c+d x)}+35658 B \cos (c+d x) \sqrt {1-\sec (c+d x)}+12653 A \cos (2 (c+d x)) \sqrt {1-\sec (c+d x)}+9786 B \cos (2 (c+d x)) \sqrt {1-\sec (c+d x)}+3818 A \cos (3 (c+d x)) \sqrt {1-\sec (c+d x)}+2436 B \cos (3 (c+d x)) \sqrt {1-\sec (c+d x)}+1002 A \cos (4 (c+d x)) \sqrt {1-\sec (c+d x)}+84 B \cos (4 (c+d x)) \sqrt {1-\sec (c+d x)}+72 A \cos (5 (c+d x)) \sqrt {1-\sec (c+d x)}+21504 B \, _2F_1\left (\frac {1}{2},5;\frac {3}{2};1-\sec (c+d x)\right ) \sqrt {1-\sec (c+d x)}+15360 A \, _2F_1\left (\frac {1}{2},6;\frac {3}{2};1-\sec (c+d x)\right ) \sqrt {1-\sec (c+d x)}\right ) \sqrt {a (1+\sec (c+d x))} \sin (c+d x)}{13440 d (1+\cos (c+d x)) \sqrt {1-\sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(946\) vs.
\(2(226)=452\).
time = 7.78, size = 947, normalized size = 3.73
method | result | size |
default | \(\text {Expression too large to display}\) | \(947\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.47, size = 460, normalized size = 1.81 \begin {gather*} \left [\frac {15 \, {\left ({\left (283 \, A + 326 \, B\right )} a^{2} \cos \left (d x + c\right ) + {\left (283 \, A + 326 \, B\right )} a^{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 (384 \, A a^{2} \cos \left (d x + c\right )^{5} + 48 \, {\left (29 \, A + 10 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (283 \, A + 230 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + 10 \, {\left (283 \, A + 326 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (283 \, A + 326 \, B\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3840 \, {\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac {15 \, {\left ({\left (283 \, A + 326 \, B\right )} a^{2} \cos \left (d x + c\right ) + {\left (283 \, A + 326 \, B\right )} a^{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 (384 \, A a^{2} \cos \left (d x + c\right )^{5} + 48 \, {\left (29 \, A + 10 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (283 \, A + 230 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + 10 \, {\left (283 \, A + 326 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (283 \, A + 326 \, B\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{1920 \, {\left (d \cos \left (d x + c\right ) + d\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1377 vs.
\(2 (226) = 452\).
time = 2.35, size = 1377, normalized size = 5.42 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\cos \left (c+d\,x\right )}^5\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________