Optimal. Leaf size=165 \[ -\frac {a^3}{64 d (a \sin (c+d x)+a)^4}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}-\frac {a^2}{24 d (a \sin (c+d x)+a)^3}+\frac {5 a}{128 d (a-a \sin (c+d x))^2}-\frac {5 a}{64 d (a \sin (c+d x)+a)^2}+\frac {15}{128 d (a-a \sin (c+d x))}-\frac {5}{32 d (a \sin (c+d x)+a)}+\frac {35 \tanh ^{-1}(\sin (c+d x))}{128 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2667, 44, 206} \[ -\frac {a^3}{64 d (a \sin (c+d x)+a)^4}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}-\frac {a^2}{24 d (a \sin (c+d x)+a)^3}+\frac {5 a}{128 d (a-a \sin (c+d x))^2}-\frac {5 a}{64 d (a \sin (c+d x)+a)^2}+\frac {15}{128 d (a-a \sin (c+d x))}-\frac {5}{32 d (a \sin (c+d x)+a)}+\frac {35 \tanh ^{-1}(\sin (c+d x))}{128 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 206
Rule 2667
Rubi steps
\begin {align*} \int \frac {\sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {a^7 \operatorname {Subst}\left (\int \frac {1}{(a-x)^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^7 \operatorname {Subst}\left (\int \left (\frac {1}{32 a^5 (a-x)^4}+\frac {5}{64 a^6 (a-x)^3}+\frac {15}{128 a^7 (a-x)^2}+\frac {1}{16 a^4 (a+x)^5}+\frac {1}{8 a^5 (a+x)^4}+\frac {5}{32 a^6 (a+x)^3}+\frac {5}{32 a^7 (a+x)^2}+\frac {35}{128 a^7 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {5 a}{128 d (a-a \sin (c+d x))^2}+\frac {15}{128 d (a-a \sin (c+d x))}-\frac {a^3}{64 d (a+a \sin (c+d x))^4}-\frac {a^2}{24 d (a+a \sin (c+d x))^3}-\frac {5 a}{64 d (a+a \sin (c+d x))^2}-\frac {5}{32 d (a+a \sin (c+d x))}+\frac {35 \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{128 d}\\ &=\frac {35 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {5 a}{128 d (a-a \sin (c+d x))^2}+\frac {15}{128 d (a-a \sin (c+d x))}-\frac {a^3}{64 d (a+a \sin (c+d x))^4}-\frac {a^2}{24 d (a+a \sin (c+d x))^3}-\frac {5 a}{64 d (a+a \sin (c+d x))^2}-\frac {5}{32 d (a+a \sin (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.51, size = 145, normalized size = 0.88 \[ -\frac {\sec ^6(c+d x) \left (-105 \sin ^6(c+d x)-105 \sin ^5(c+d x)+280 \sin ^4(c+d x)+280 \sin ^3(c+d x)-231 \sin ^2(c+d x)-231 \sin (c+d x)-105 \tanh ^{-1}(\sin (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^6 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^8+48\right )}{384 a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 167, normalized size = 1.01 \[ -\frac {210 \, \cos \left (d x + c\right )^{6} - 70 \, \cos \left (d x + c\right )^{4} - 28 \, \cos \left (d x + c\right )^{2} - 105 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 14 \, {\left (15 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) - 16}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.26, size = 136, normalized size = 0.82 \[ \frac {\frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {2 \, {\left (385 \, \sin \left (d x + c\right )^{3} - 1335 \, \sin \left (d x + c\right )^{2} + 1575 \, \sin \left (d x + c\right ) - 641\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {875 \, \sin \left (d x + c\right )^{4} + 3980 \, \sin \left (d x + c\right )^{3} + 6930 \, \sin \left (d x + c\right )^{2} + 5548 \, \sin \left (d x + c\right ) + 1771}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.42, size = 162, normalized size = 0.98 \[ -\frac {1}{96 a d \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {5}{128 a d \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {15}{128 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {35 \ln \left (\sin \left (d x +c \right )-1\right )}{256 a d}-\frac {1}{64 a d \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{24 a d \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {5}{64 a d \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {5}{32 a d \left (1+\sin \left (d x +c \right )\right )}+\frac {35 \ln \left (1+\sin \left (d x +c \right )\right )}{256 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 175, normalized size = 1.06 \[ -\frac {\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{6} + 105 \, \sin \left (d x + c\right )^{5} - 280 \, \sin \left (d x + c\right )^{4} - 280 \, \sin \left (d x + c\right )^{3} + 231 \, \sin \left (d x + c\right )^{2} + 231 \, \sin \left (d x + c\right ) - 48\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac {105 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {105 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.24, size = 158, normalized size = 0.96 \[ \frac {35\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{128\,a\,d}+\frac {\frac {35\,{\sin \left (c+d\,x\right )}^6}{128}+\frac {35\,{\sin \left (c+d\,x\right )}^5}{128}-\frac {35\,{\sin \left (c+d\,x\right )}^4}{48}-\frac {35\,{\sin \left (c+d\,x\right )}^3}{48}+\frac {77\,{\sin \left (c+d\,x\right )}^2}{128}+\frac {77\,\sin \left (c+d\,x\right )}{128}-\frac {1}{8}}{d\,\left (-a\,{\sin \left (c+d\,x\right )}^7-a\,{\sin \left (c+d\,x\right )}^6+3\,a\,{\sin \left (c+d\,x\right )}^5+3\,a\,{\sin \left (c+d\,x\right )}^4-3\,a\,{\sin \left (c+d\,x\right )}^3-3\,a\,{\sin \left (c+d\,x\right )}^2+a\,\sin \left (c+d\,x\right )+a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sec ^{7}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________