Optimal. Leaf size=391 \[ \frac{10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac{2 a^2 b^3 \sec ^5(c+d x)}{d}-\frac{5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{5 a^3 b^2 \tan (c+d x) \sec ^5(c+d x)}{3 d}-\frac{5 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{12 d}-\frac{5 a^3 b^2 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^4 b \sec ^5(c+d x)}{d}+\frac{3 a^5 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^5 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a^5 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{5 a b^4 \tan ^3(c+d x) \sec ^5(c+d x)}{8 d}-\frac{5 a b^4 \tan (c+d x) \sec ^5(c+d x)}{16 d}+\frac{5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{64 d}+\frac{15 a b^4 \tan (c+d x) \sec (c+d x)}{128 d}+\frac{b^5 \sec ^9(c+d x)}{9 d}-\frac{2 b^5 \sec ^7(c+d x)}{7 d}+\frac{b^5 \sec ^5(c+d x)}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.389182, antiderivative size = 391, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3090, 3768, 3770, 2606, 30, 2611, 14, 270} \[ \frac{10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac{2 a^2 b^3 \sec ^5(c+d x)}{d}-\frac{5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{5 a^3 b^2 \tan (c+d x) \sec ^5(c+d x)}{3 d}-\frac{5 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{12 d}-\frac{5 a^3 b^2 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^4 b \sec ^5(c+d x)}{d}+\frac{3 a^5 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^5 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a^5 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{5 a b^4 \tan ^3(c+d x) \sec ^5(c+d x)}{8 d}-\frac{5 a b^4 \tan (c+d x) \sec ^5(c+d x)}{16 d}+\frac{5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{64 d}+\frac{15 a b^4 \tan (c+d x) \sec (c+d x)}{128 d}+\frac{b^5 \sec ^9(c+d x)}{9 d}-\frac{2 b^5 \sec ^7(c+d x)}{7 d}+\frac{b^5 \sec ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3090
Rule 3768
Rule 3770
Rule 2606
Rule 30
Rule 2611
Rule 14
Rule 270
Rubi steps
\begin{align*} \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=\int \left (a^5 \sec ^5(c+d x)+5 a^4 b \sec ^5(c+d x) \tan (c+d x)+10 a^3 b^2 \sec ^5(c+d x) \tan ^2(c+d x)+10 a^2 b^3 \sec ^5(c+d x) \tan ^3(c+d x)+5 a b^4 \sec ^5(c+d x) \tan ^4(c+d x)+b^5 \sec ^5(c+d x) \tan ^5(c+d x)\right ) \, dx\\ &=a^5 \int \sec ^5(c+d x) \, dx+\left (5 a^4 b\right ) \int \sec ^5(c+d x) \tan (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \sec ^5(c+d x) \tan ^2(c+d x) \, dx+\left (10 a^2 b^3\right ) \int \sec ^5(c+d x) \tan ^3(c+d x) \, dx+\left (5 a b^4\right ) \int \sec ^5(c+d x) \tan ^4(c+d x) \, dx+b^5 \int \sec ^5(c+d x) \tan ^5(c+d x) \, dx\\ &=\frac{a^5 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{3 d}+\frac{5 a b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d}+\frac{1}{4} \left (3 a^5\right ) \int \sec ^3(c+d x) \, dx-\frac{1}{3} \left (5 a^3 b^2\right ) \int \sec ^5(c+d x) \, dx-\frac{1}{8} \left (15 a b^4\right ) \int \sec ^5(c+d x) \tan ^2(c+d x) \, dx+\frac{\left (5 a^4 b\right ) \operatorname{Subst}\left (\int x^4 \, dx,x,\sec (c+d x)\right )}{d}+\frac{\left (10 a^2 b^3\right ) \operatorname{Subst}\left (\int x^4 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d}+\frac{b^5 \operatorname{Subst}\left (\int x^4 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{a^4 b \sec ^5(c+d x)}{d}+\frac{3 a^5 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a^5 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{5 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{12 d}+\frac{5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{3 d}-\frac{5 a b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac{5 a b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d}+\frac{1}{8} \left (3 a^5\right ) \int \sec (c+d x) \, dx-\frac{1}{4} \left (5 a^3 b^2\right ) \int \sec ^3(c+d x) \, dx+\frac{1}{16} \left (5 a b^4\right ) \int \sec ^5(c+d x) \, dx+\frac{\left (10 a^2 b^3\right ) \operatorname{Subst}\left (\int \left (-x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{d}+\frac{b^5 \operatorname{Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{3 a^5 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 b \sec ^5(c+d x)}{d}-\frac{2 a^2 b^3 \sec ^5(c+d x)}{d}+\frac{b^5 \sec ^5(c+d x)}{5 d}+\frac{10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac{2 b^5 \sec ^7(c+d x)}{7 d}+\frac{b^5 \sec ^9(c+d x)}{9 d}+\frac{3 a^5 \sec (c+d x) \tan (c+d x)}{8 d}-\frac{5 a^3 b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a^5 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{5 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{12 d}+\frac{5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac{5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{3 d}-\frac{5 a b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac{5 a b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d}-\frac{1}{8} \left (5 a^3 b^2\right ) \int \sec (c+d x) \, dx+\frac{1}{64} \left (15 a b^4\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{3 a^5 \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 b \sec ^5(c+d x)}{d}-\frac{2 a^2 b^3 \sec ^5(c+d x)}{d}+\frac{b^5 \sec ^5(c+d x)}{5 d}+\frac{10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac{2 b^5 \sec ^7(c+d x)}{7 d}+\frac{b^5 \sec ^9(c+d x)}{9 d}+\frac{3 a^5 \sec (c+d x) \tan (c+d x)}{8 d}-\frac{5 a^3 b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{15 a b^4 \sec (c+d x) \tan (c+d x)}{128 d}+\frac{a^5 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{5 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{12 d}+\frac{5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac{5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{3 d}-\frac{5 a b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac{5 a b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d}+\frac{1}{128} \left (15 a b^4\right ) \int \sec (c+d x) \, dx\\ &=\frac{3 a^5 \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{a^4 b \sec ^5(c+d x)}{d}-\frac{2 a^2 b^3 \sec ^5(c+d x)}{d}+\frac{b^5 \sec ^5(c+d x)}{5 d}+\frac{10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac{2 b^5 \sec ^7(c+d x)}{7 d}+\frac{b^5 \sec ^9(c+d x)}{9 d}+\frac{3 a^5 \sec (c+d x) \tan (c+d x)}{8 d}-\frac{5 a^3 b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{15 a b^4 \sec (c+d x) \tan (c+d x)}{128 d}+\frac{a^5 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{5 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{12 d}+\frac{5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac{5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{3 d}-\frac{5 a b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac{5 a b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d}\\ \end{align*}
Mathematica [A] time = 2.21531, size = 331, normalized size = 0.85 \[ \frac{1260 a \left (2320 a^2 b^2+656 a^4+845 b^4\right ) \tan (c+d x) \sec ^7(c+d x)-40320 a \left (-80 a^2 b^2+48 a^4+15 b^4\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+\sec ^9(c+d x) \left (453600 a^3 b^2 \sin (4 (c+d x))-218400 a^3 b^2 \sin (6 (c+d x))-25200 a^3 b^2 \sin (8 (c+d x))+73728 \left (-20 a^2 b^3+35 a^4 b-3 b^5\right ) \cos (2 (c+d x))+129024 \left (-10 a^2 b^3+5 a^4 b+b^5\right ) \cos (4 (c+d x))-184320 a^2 b^3+1935360 a^4 b+372960 a^5 \sin (4 (c+d x))+131040 a^5 \sin (6 (c+d x))+15120 a^5 \sin (8 (c+d x))-488250 a b^4 \sin (4 (c+d x))+40950 a b^4 \sin (6 (c+d x))+4725 a b^4 \sin (8 (c+d x))+223232 b^5\right )}{5160960 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.266, size = 688, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.22289, size = 486, normalized size = 1.24 \begin{align*} -\frac{1575 \, a b^{4}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{7} - 11 \, \sin \left (d x + c\right )^{5} - 11 \, \sin \left (d x + c\right )^{3} + 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 8400 \, a^{3} b^{2}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 8 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 5040 \, a^{5}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac{80640 \, a^{4} b}{\cos \left (d x + c\right )^{5}} + \frac{23040 \,{\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} a^{2} b^{3}}{\cos \left (d x + c\right )^{7}} - \frac{256 \,{\left (63 \, \cos \left (d x + c\right )^{4} - 90 \, \cos \left (d x + c\right )^{2} + 35\right )} b^{5}}{\cos \left (d x + c\right )^{9}}}{80640 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.628745, size = 643, normalized size = 1.64 \begin{align*} \frac{315 \,{\left (48 \, a^{5} - 80 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{9} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \,{\left (48 \, a^{5} - 80 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{9} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 8960 \, b^{5} + 16128 \,{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 23040 \,{\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 210 \,{\left (3 \,{\left (48 \, a^{5} - 80 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 240 \, a b^{4} \cos \left (d x + c\right ) + 2 \,{\left (48 \, a^{5} - 80 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 40 \,{\left (16 \, a^{3} b^{2} - 9 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{80640 \, d \cos \left (d x + c\right )^{9}} \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 [B] time = 1.33802, size = 1199, normalized size = 3.07 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]