Optimal. Leaf size=250 \[ -\frac {\left (15 a^2+37 a b+24 b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac {\left (15 a^2-37 a b+24 b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac {b^7 \log (a+b \sin (c+d x))}{a^2 d \left (a^2-b^2\right )^3}-\frac {b \log (\sin (c+d x))}{a^2 d}+\frac {7 a+9 b}{16 d (a+b)^2 (1-\sin (c+d x))}-\frac {7 a-9 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac {1}{16 d (a+b) (1-\sin (c+d x))^2}-\frac {1}{16 d (a-b) (\sin (c+d x)+1)^2}-\frac {\csc (c+d x)}{a d} \]
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Rubi [A] time = 0.40, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2837, 12, 894} \[ -\frac {b^7 \log (a+b \sin (c+d x))}{a^2 d \left (a^2-b^2\right )^3}-\frac {\left (15 a^2+37 a b+24 b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac {\left (15 a^2-37 a b+24 b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac {b \log (\sin (c+d x))}{a^2 d}+\frac {7 a+9 b}{16 d (a+b)^2 (1-\sin (c+d x))}-\frac {7 a-9 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac {1}{16 d (a+b) (1-\sin (c+d x))^2}-\frac {1}{16 d (a-b) (\sin (c+d x)+1)^2}-\frac {\csc (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 894
Rule 2837
Rubi steps
\begin {align*} \int \frac {\csc ^2(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {b^5 \operatorname {Subst}\left (\int \frac {b^2}{x^2 (a+x) \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b^7 \operatorname {Subst}\left (\int \frac {1}{x^2 (a+x) \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b^7 \operatorname {Subst}\left (\int \left (\frac {1}{8 b^5 (a+b) (b-x)^3}+\frac {7 a+9 b}{16 b^6 (a+b)^2 (b-x)^2}+\frac {15 a^2+37 a b+24 b^2}{16 b^7 (a+b)^3 (b-x)}+\frac {1}{a b^6 x^2}-\frac {1}{a^2 b^6 x}-\frac {1}{a^2 (a-b)^3 (a+b)^3 (a+x)}-\frac {1}{8 b^5 (-a+b) (b+x)^3}+\frac {7 a-9 b}{16 (a-b)^2 b^6 (b+x)^2}+\frac {15 a^2-37 a b+24 b^2}{16 (a-b)^3 b^7 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {\csc (c+d x)}{a d}-\frac {\left (15 a^2+37 a b+24 b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^3 d}-\frac {b \log (\sin (c+d x))}{a^2 d}+\frac {\left (15 a^2-37 a b+24 b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^3 d}-\frac {b^7 \log (a+b \sin (c+d x))}{a^2 \left (a^2-b^2\right )^3 d}+\frac {1}{16 (a+b) d (1-\sin (c+d x))^2}+\frac {7 a+9 b}{16 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{16 (a-b) d (1+\sin (c+d x))^2}-\frac {7 a-9 b}{16 (a-b)^2 d (1+\sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 6.21, size = 257, normalized size = 1.03 \[ \frac {b^7 \left (-\frac {\log (\sin (c+d x))}{a^2 b^6}-\frac {\left (15 a^2+37 a b+24 b^2\right ) \log (1-\sin (c+d x))}{16 b^7 (a+b)^3}+\frac {\left (15 a^2-37 a b+24 b^2\right ) \log (\sin (c+d x)+1)}{16 b^7 (a-b)^3}-\frac {\log (a+b \sin (c+d x))}{a^2 (a-b)^3 (a+b)^3}-\frac {\csc (c+d x)}{a b^7}-\frac {7 a-9 b}{16 b^6 (a-b)^2 (b \sin (c+d x)+b)}+\frac {7 a+9 b}{16 b^6 (a+b)^2 (b-b \sin (c+d x))}+\frac {1}{16 b^5 (a+b) (b-b \sin (c+d x))^2}-\frac {1}{16 b^5 (a-b) (b \sin (c+d x)+b)^2}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 4.81, size = 425, normalized size = 1.70 \[ -\frac {16 \, b^{7} \cos \left (d x + c\right )^{4} \log \left (b \sin \left (d x + c\right ) + a\right ) \sin \left (d x + c\right ) - 4 \, a^{7} + 8 \, a^{5} b^{2} - 4 \, a^{3} b^{4} + 16 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{4} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - {\left (15 \, a^{7} + 8 \, a^{6} b - 42 \, a^{5} b^{2} - 24 \, a^{4} b^{3} + 35 \, a^{3} b^{4} + 24 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + {\left (15 \, a^{7} - 8 \, a^{6} b - 42 \, a^{5} b^{2} + 24 \, a^{4} b^{3} + 35 \, a^{3} b^{4} - 24 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 2 \, {\left (15 \, a^{7} - 42 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 8 \, a b^{6}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (5 \, a^{7} - 14 \, a^{5} b^{2} + 9 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5} + 2 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 2 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 418, normalized size = 1.67 \[ -\frac {\frac {16 \, b^{8} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}} - \frac {{\left (15 \, a^{2} - 37 \, a b + 24 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {{\left (15 \, a^{2} + 37 \, a b + 24 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {16 \, b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2}} + \frac {2 \, {\left (6 \, a^{4} b \sin \left (d x + c\right )^{4} - 18 \, a^{2} b^{3} \sin \left (d x + c\right )^{4} + 18 \, b^{5} \sin \left (d x + c\right )^{4} + 7 \, a^{5} \sin \left (d x + c\right )^{3} - 18 \, a^{3} b^{2} \sin \left (d x + c\right )^{3} + 11 \, a b^{4} \sin \left (d x + c\right )^{3} - 16 \, a^{4} b \sin \left (d x + c\right )^{2} + 48 \, a^{2} b^{3} \sin \left (d x + c\right )^{2} - 44 \, b^{5} \sin \left (d x + c\right )^{2} - 9 \, a^{5} \sin \left (d x + c\right ) + 22 \, a^{3} b^{2} \sin \left (d x + c\right ) - 13 \, a b^{4} \sin \left (d x + c\right ) + 12 \, a^{4} b - 34 \, a^{2} b^{3} + 28 \, b^{5}\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}} - \frac {16 \, {\left (b \sin \left (d x + c\right ) - a\right )}}{a^{2} \sin \left (d x + c\right )}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 340, normalized size = 1.36 \[ \frac {1}{2 d \left (8 a +8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {7 a}{16 d \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}-\frac {9 b}{16 d \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}-\frac {15 \ln \left (\sin \left (d x +c \right )-1\right ) a^{2}}{16 d \left (a +b \right )^{3}}-\frac {37 \ln \left (\sin \left (d x +c \right )-1\right ) a b}{16 d \left (a +b \right )^{3}}-\frac {3 \ln \left (\sin \left (d x +c \right )-1\right ) b^{2}}{2 d \left (a +b \right )^{3}}-\frac {b^{7} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,a^{2} \left (a +b \right )^{3} \left (a -b \right )^{3}}-\frac {1}{d a \sin \left (d x +c \right )}-\frac {b \ln \left (\sin \left (d x +c \right )\right )}{a^{2} d}-\frac {1}{2 d \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {7 a}{16 d \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {9 b}{16 d \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {15 \ln \left (1+\sin \left (d x +c \right )\right ) a^{2}}{16 d \left (a -b \right )^{3}}-\frac {37 \ln \left (1+\sin \left (d x +c \right )\right ) a b}{16 d \left (a -b \right )^{3}}+\frac {3 \ln \left (1+\sin \left (d x +c \right )\right ) b^{2}}{2 d \left (a -b \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 361, normalized size = 1.44 \[ -\frac {\frac {16 \, b^{7} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}} - \frac {{\left (15 \, a^{2} - 37 \, a b + 24 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {{\left (15 \, a^{2} + 37 \, a b + 24 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {2 \, {\left ({\left (15 \, a^{4} - 27 \, a^{2} b^{2} + 8 \, b^{4}\right )} \sin \left (d x + c\right )^{4} + 8 \, a^{4} - 16 \, a^{2} b^{2} + 8 \, b^{4} - 4 \, {\left (a^{3} b - 2 \, a b^{3}\right )} \sin \left (d x + c\right )^{3} - {\left (25 \, a^{4} - 45 \, a^{2} b^{2} + 16 \, b^{4}\right )} \sin \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} b - 5 \, a b^{3}\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )^{5} - 2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )^{3} + {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )} + \frac {16 \, b \log \left (\sin \left (d x + c\right )\right )}{a^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.47, size = 373, normalized size = 1.49 \[ \frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (\frac {b^2}{8\,{\left (a-b\right )}^3}-\frac {7\,b}{16\,{\left (a-b\right )}^2}+\frac {15}{16\,\left (a-b\right )}\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {7\,b}{16\,{\left (a+b\right )}^2}+\frac {15}{16\,\left (a+b\right )}+\frac {b^2}{8\,{\left (a+b\right )}^3}\right )}{d}-\frac {\frac {1}{a}-\frac {{\sin \left (c+d\,x\right )}^3\,\left (a^2\,b-2\,b^3\right )}{2\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {\sin \left (c+d\,x\right )\,\left (3\,a^2\,b-5\,b^3\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {{\sin \left (c+d\,x\right )}^4\,\left (15\,a^4-27\,a^2\,b^2+8\,b^4\right )}{8\,a\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (25\,a^4-45\,a^2\,b^2+16\,b^4\right )}{8\,a\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left ({\sin \left (c+d\,x\right )}^5-2\,{\sin \left (c+d\,x\right )}^3+\sin \left (c+d\,x\right )\right )}-\frac {b\,\ln \left (\sin \left (c+d\,x\right )\right )}{a^2\,d}-\frac {b^7\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{d\,\left (a^8-3\,a^6\,b^2+3\,a^4\,b^4-a^2\,b^6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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