Optimal. Leaf size=408 \[ -\frac{15 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{64 d}+\frac{3 a^2 b^2 \tan (c+d x) \sec ^7(c+d x)}{4 d}-\frac{a^2 b^2 \tan (c+d x) \sec ^5(c+d x)}{8 d}-\frac{5 a^2 b^2 \tan (c+d x) \sec ^3(c+d x)}{32 d}-\frac{15 a^2 b^2 \tan (c+d x) \sec (c+d x)}{64 d}+\frac{4 a^3 b \sec ^7(c+d x)}{7 d}+\frac{5 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{5 a^4 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{5 a^4 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{4 a b^3 \sec ^9(c+d x)}{9 d}-\frac{4 a b^3 \sec ^7(c+d x)}{7 d}+\frac{3 b^4 \tanh ^{-1}(\sin (c+d x))}{256 d}+\frac{b^4 \tan ^3(c+d x) \sec ^7(c+d x)}{10 d}-\frac{3 b^4 \tan (c+d x) \sec ^7(c+d x)}{80 d}+\frac{b^4 \tan (c+d x) \sec ^5(c+d x)}{160 d}+\frac{b^4 \tan (c+d x) \sec ^3(c+d x)}{128 d}+\frac{3 b^4 \tan (c+d x) \sec (c+d x)}{256 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.397941, antiderivative size = 408, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3090, 3768, 3770, 2606, 30, 2611, 14} \[ -\frac{15 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{64 d}+\frac{3 a^2 b^2 \tan (c+d x) \sec ^7(c+d x)}{4 d}-\frac{a^2 b^2 \tan (c+d x) \sec ^5(c+d x)}{8 d}-\frac{5 a^2 b^2 \tan (c+d x) \sec ^3(c+d x)}{32 d}-\frac{15 a^2 b^2 \tan (c+d x) \sec (c+d x)}{64 d}+\frac{4 a^3 b \sec ^7(c+d x)}{7 d}+\frac{5 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{5 a^4 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{5 a^4 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{4 a b^3 \sec ^9(c+d x)}{9 d}-\frac{4 a b^3 \sec ^7(c+d x)}{7 d}+\frac{3 b^4 \tanh ^{-1}(\sin (c+d x))}{256 d}+\frac{b^4 \tan ^3(c+d x) \sec ^7(c+d x)}{10 d}-\frac{3 b^4 \tan (c+d x) \sec ^7(c+d x)}{80 d}+\frac{b^4 \tan (c+d x) \sec ^5(c+d x)}{160 d}+\frac{b^4 \tan (c+d x) \sec ^3(c+d x)}{128 d}+\frac{3 b^4 \tan (c+d x) \sec (c+d x)}{256 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3090
Rule 3768
Rule 3770
Rule 2606
Rule 30
Rule 2611
Rule 14
Rubi steps
\begin{align*} \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx &=\int \left (a^4 \sec ^7(c+d x)+4 a^3 b \sec ^7(c+d x) \tan (c+d x)+6 a^2 b^2 \sec ^7(c+d x) \tan ^2(c+d x)+4 a b^3 \sec ^7(c+d x) \tan ^3(c+d x)+b^4 \sec ^7(c+d x) \tan ^4(c+d x)\right ) \, dx\\ &=a^4 \int \sec ^7(c+d x) \, dx+\left (4 a^3 b\right ) \int \sec ^7(c+d x) \tan (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \sec ^7(c+d x) \tan ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \sec ^7(c+d x) \tan ^3(c+d x) \, dx+b^4 \int \sec ^7(c+d x) \tan ^4(c+d x) \, dx\\ &=\frac{a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}+\frac{b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d}+\frac{1}{6} \left (5 a^4\right ) \int \sec ^5(c+d x) \, dx-\frac{1}{4} \left (3 a^2 b^2\right ) \int \sec ^7(c+d x) \, dx-\frac{1}{10} \left (3 b^4\right ) \int \sec ^7(c+d x) \tan ^2(c+d x) \, dx+\frac{\left (4 a^3 b\right ) \operatorname{Subst}\left (\int x^6 \, dx,x,\sec (c+d x)\right )}{d}+\frac{\left (4 a b^3\right ) \operatorname{Subst}\left (\int x^6 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{4 a^3 b \sec ^7(c+d x)}{7 d}+\frac{5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac{3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac{3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac{b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d}+\frac{1}{8} \left (5 a^4\right ) \int \sec ^3(c+d x) \, dx-\frac{1}{8} \left (5 a^2 b^2\right ) \int \sec ^5(c+d x) \, dx+\frac{1}{80} \left (3 b^4\right ) \int \sec ^7(c+d x) \, dx+\frac{\left (4 a b^3\right ) \operatorname{Subst}\left (\int \left (-x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{4 a^3 b \sec ^7(c+d x)}{7 d}-\frac{4 a b^3 \sec ^7(c+d x)}{7 d}+\frac{4 a b^3 \sec ^9(c+d x)}{9 d}+\frac{5 a^4 \sec (c+d x) \tan (c+d x)}{16 d}+\frac{5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac{5 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{32 d}+\frac{a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac{b^4 \sec ^5(c+d x) \tan (c+d x)}{160 d}+\frac{3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac{3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac{b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d}+\frac{1}{16} \left (5 a^4\right ) \int \sec (c+d x) \, dx-\frac{1}{32} \left (15 a^2 b^2\right ) \int \sec ^3(c+d x) \, dx+\frac{1}{32} b^4 \int \sec ^5(c+d x) \, dx\\ &=\frac{5 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{4 a^3 b \sec ^7(c+d x)}{7 d}-\frac{4 a b^3 \sec ^7(c+d x)}{7 d}+\frac{4 a b^3 \sec ^9(c+d x)}{9 d}+\frac{5 a^4 \sec (c+d x) \tan (c+d x)}{16 d}-\frac{15 a^2 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac{5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac{5 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{32 d}+\frac{b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac{a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac{b^4 \sec ^5(c+d x) \tan (c+d x)}{160 d}+\frac{3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac{3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac{b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d}-\frac{1}{64} \left (15 a^2 b^2\right ) \int \sec (c+d x) \, dx+\frac{1}{128} \left (3 b^4\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{5 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac{15 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{64 d}+\frac{4 a^3 b \sec ^7(c+d x)}{7 d}-\frac{4 a b^3 \sec ^7(c+d x)}{7 d}+\frac{4 a b^3 \sec ^9(c+d x)}{9 d}+\frac{5 a^4 \sec (c+d x) \tan (c+d x)}{16 d}-\frac{15 a^2 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac{3 b^4 \sec (c+d x) \tan (c+d x)}{256 d}+\frac{5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac{5 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{32 d}+\frac{b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac{a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac{b^4 \sec ^5(c+d x) \tan (c+d x)}{160 d}+\frac{3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac{3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac{b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d}+\frac{1}{256} \left (3 b^4\right ) \int \sec (c+d x) \, dx\\ &=\frac{5 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac{15 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{64 d}+\frac{3 b^4 \tanh ^{-1}(\sin (c+d x))}{256 d}+\frac{4 a^3 b \sec ^7(c+d x)}{7 d}-\frac{4 a b^3 \sec ^7(c+d x)}{7 d}+\frac{4 a b^3 \sec ^9(c+d x)}{9 d}+\frac{5 a^4 \sec (c+d x) \tan (c+d x)}{16 d}-\frac{15 a^2 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac{3 b^4 \sec (c+d x) \tan (c+d x)}{256 d}+\frac{5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac{5 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{32 d}+\frac{b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac{a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac{b^4 \sec ^5(c+d x) \tan (c+d x)}{160 d}+\frac{3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac{3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac{b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d}\\ \end{align*}
Mathematica [A] time = 1.32976, size = 242, normalized size = 0.59 \[ \frac{10 \sec ^9(c+d x) \left (189 \left (1604 a^2 b^2+592 a^4+739 b^4\right ) \tan (c+d x)+32768 a b \left (27 a^2+b^2\right )\right )-80640 \left (-60 a^2 b^2+80 a^4+3 b^4\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+3 \sec ^{10}(c+d x) \left (420 \left (1908 a^2 b^2+1552 a^4-505 b^4\right ) \sin (3 (c+d x))+7 \left (-60 a^2 b^2+80 a^4+3 b^4\right ) (628 \sin (5 (c+d x))+145 \sin (7 (c+d x))+15 \sin (9 (c+d x)))+983040 a b \left (a^2-b^2\right ) \cos (3 (c+d x))\right )}{20643840 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.143, size = 590, normalized size = 1.5 \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 = 1.23752, size = 516, normalized size = 1.26 \begin{align*} -\frac{63 \, b^{4}{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{9} - 70 \, \sin \left (d x + c\right )^{7} + 128 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{10} - 5 \, \sin \left (d x + c\right )^{8} + 10 \, \sin \left (d x + c\right )^{6} - 10 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 1260 \, a^{2} b^{2}{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{7} - 55 \, \sin \left (d x + c\right )^{5} + 73 \, \sin \left (d x + c\right )^{3} + 15 \, \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} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 1680 \, a^{4}{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \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} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac{92160 \, a^{3} b}{\cos \left (d x + c\right )^{7}} + \frac{10240 \,{\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} a b^{3}}{\cos \left (d x + c\right )^{9}}}{161280 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.627859, size = 626, normalized size = 1.53 \begin{align*} \frac{315 \,{\left (80 \, a^{4} - 60 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{10} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \,{\left (80 \, a^{4} - 60 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{10} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 71680 \, a b^{3} \cos \left (d x + c\right ) + 92160 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} + 42 \,{\left (15 \,{\left (80 \, a^{4} - 60 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{8} + 10 \,{\left (80 \, a^{4} - 60 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{6} + 8 \,{\left (80 \, a^{4} - 60 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 384 \, b^{4} + 48 \,{\left (60 \, a^{2} b^{2} - 11 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{161280 \, d \cos \left (d x + c\right )^{10}} \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 = 1.31296, size = 1188, normalized size = 2.91 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]