Optimal. Leaf size=384 \[ \frac {2 a b^2}{d \left (a^4+b^4\right ) \left (a+b \sqrt {\sinh (c+d x)}\right )}-\frac {2 b^2 \left (3 a^4-b^4\right ) \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{d \left (a^4+b^4\right )^2}+\frac {b^2 \left (3 a^4-b^4\right ) \log (\cosh (c+d x))}{d \left (a^4+b^4\right )^2}+\frac {a^2 \left (a^4-3 b^4\right ) \tan ^{-1}(\sinh (c+d x))}{d \left (a^4+b^4\right )^2}-\frac {a b \left (a^4+2 a^2 b^2-b^4\right ) \log \left (\sinh (c+d x)-\sqrt {2} \sqrt {\sinh (c+d x)}+1\right )}{\sqrt {2} d \left (a^4+b^4\right )^2}+\frac {a b \left (a^4+2 a^2 b^2-b^4\right ) \log \left (\sinh (c+d x)+\sqrt {2} \sqrt {\sinh (c+d x)}+1\right )}{\sqrt {2} d \left (a^4+b^4\right )^2}+\frac {\sqrt {2} a b \left (a^4-2 a^2 b^2-b^4\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\sinh (c+d x)}\right )}{d \left (a^4+b^4\right )^2}-\frac {\sqrt {2} a b \left (a^4-2 a^2 b^2-b^4\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\sinh (c+d x)}+1\right )}{d \left (a^4+b^4\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.63, antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3223, 6725, 1876, 1168, 1162, 617, 204, 1165, 628, 1248, 635, 203, 260} \[ \frac {2 a b^2}{d \left (a^4+b^4\right ) \left (a+b \sqrt {\sinh (c+d x)}\right )}-\frac {a b \left (2 a^2 b^2+a^4-b^4\right ) \log \left (\sinh (c+d x)-\sqrt {2} \sqrt {\sinh (c+d x)}+1\right )}{\sqrt {2} d \left (a^4+b^4\right )^2}+\frac {a b \left (2 a^2 b^2+a^4-b^4\right ) \log \left (\sinh (c+d x)+\sqrt {2} \sqrt {\sinh (c+d x)}+1\right )}{\sqrt {2} d \left (a^4+b^4\right )^2}-\frac {2 b^2 \left (3 a^4-b^4\right ) \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{d \left (a^4+b^4\right )^2}+\frac {a^2 \left (a^4-3 b^4\right ) \tan ^{-1}(\sinh (c+d x))}{d \left (a^4+b^4\right )^2}+\frac {\sqrt {2} a b \left (-2 a^2 b^2+a^4-b^4\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\sinh (c+d x)}\right )}{d \left (a^4+b^4\right )^2}-\frac {\sqrt {2} a b \left (-2 a^2 b^2+a^4-b^4\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\sinh (c+d x)}+1\right )}{d \left (a^4+b^4\right )^2}+\frac {b^2 \left (3 a^4-b^4\right ) \log (\cosh (c+d x))}{d \left (a^4+b^4\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 204
Rule 260
Rule 617
Rule 628
Rule 635
Rule 1162
Rule 1165
Rule 1168
Rule 1248
Rule 1876
Rule 3223
Rule 6725
Rubi steps
\begin {align*} \int \frac {\text {sech}(c+d x)}{\left (a+b \sqrt {\sinh (c+d x)}\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (a+b \sqrt {x}\right )^2 \left (1+x^2\right )} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {x}{(a+b x)^2 \left (1+x^4\right )} \, dx,x,\sqrt {\sinh (c+d x)}\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-\frac {a b^3}{\left (a^4+b^4\right ) (a+b x)^2}+\frac {-3 a^4 b^3+b^7}{\left (a^4+b^4\right )^2 (a+b x)}+\frac {4 a^3 b^3+a^2 \left (a^4-3 b^4\right ) x-2 a b \left (a^4-b^4\right ) x^2+b^2 \left (3 a^4-b^4\right ) x^3}{\left (a^4+b^4\right )^2 \left (1+x^4\right )}\right ) \, dx,x,\sqrt {\sinh (c+d x)}\right )}{d}\\ &=-\frac {2 b^2 \left (3 a^4-b^4\right ) \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}+\frac {2 a b^2}{\left (a^4+b^4\right ) d \left (a+b \sqrt {\sinh (c+d x)}\right )}+\frac {2 \operatorname {Subst}\left (\int \frac {4 a^3 b^3+a^2 \left (a^4-3 b^4\right ) x-2 a b \left (a^4-b^4\right ) x^2+b^2 \left (3 a^4-b^4\right ) x^3}{1+x^4} \, dx,x,\sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}\\ &=-\frac {2 b^2 \left (3 a^4-b^4\right ) \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}+\frac {2 a b^2}{\left (a^4+b^4\right ) d \left (a+b \sqrt {\sinh (c+d x)}\right )}+\frac {2 \operatorname {Subst}\left (\int \left (\frac {4 a^3 b^3-2 a b \left (a^4-b^4\right ) x^2}{1+x^4}+\frac {x \left (a^2 \left (a^4-3 b^4\right )+b^2 \left (3 a^4-b^4\right ) x^2\right )}{1+x^4}\right ) \, dx,x,\sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}\\ &=-\frac {2 b^2 \left (3 a^4-b^4\right ) \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}+\frac {2 a b^2}{\left (a^4+b^4\right ) d \left (a+b \sqrt {\sinh (c+d x)}\right )}+\frac {2 \operatorname {Subst}\left (\int \frac {4 a^3 b^3-2 a b \left (a^4-b^4\right ) x^2}{1+x^4} \, dx,x,\sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}+\frac {2 \operatorname {Subst}\left (\int \frac {x \left (a^2 \left (a^4-3 b^4\right )+b^2 \left (3 a^4-b^4\right ) x^2\right )}{1+x^4} \, dx,x,\sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}\\ &=-\frac {2 b^2 \left (3 a^4-b^4\right ) \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}+\frac {2 a b^2}{\left (a^4+b^4\right ) d \left (a+b \sqrt {\sinh (c+d x)}\right )}+\frac {\operatorname {Subst}\left (\int \frac {a^2 \left (a^4-3 b^4\right )+b^2 \left (3 a^4-b^4\right ) x}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{\left (a^4+b^4\right )^2 d}-\frac {\left (2 a b \left (a^4-2 a^2 b^2-b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}+\frac {\left (2 a b \left (a^4+2 a^2 b^2-b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}\\ &=-\frac {2 b^2 \left (3 a^4-b^4\right ) \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}+\frac {2 a b^2}{\left (a^4+b^4\right ) d \left (a+b \sqrt {\sinh (c+d x)}\right )}+\frac {\left (a^2 \left (a^4-3 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{\left (a^4+b^4\right )^2 d}+\frac {\left (b^2 \left (3 a^4-b^4\right )\right ) \operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{\left (a^4+b^4\right )^2 d}-\frac {\left (a b \left (a^4-2 a^2 b^2-b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}-\frac {\left (a b \left (a^4-2 a^2 b^2-b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}-\frac {\left (a b \left (a^4+2 a^2 b^2-b^4\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\sinh (c+d x)}\right )}{\sqrt {2} \left (a^4+b^4\right )^2 d}-\frac {\left (a b \left (a^4+2 a^2 b^2-b^4\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\sinh (c+d x)}\right )}{\sqrt {2} \left (a^4+b^4\right )^2 d}\\ &=\frac {a^2 \left (a^4-3 b^4\right ) \tan ^{-1}(\sinh (c+d x))}{\left (a^4+b^4\right )^2 d}+\frac {b^2 \left (3 a^4-b^4\right ) \log (\cosh (c+d x))}{\left (a^4+b^4\right )^2 d}-\frac {2 b^2 \left (3 a^4-b^4\right ) \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}-\frac {a b \left (a^4+2 a^2 b^2-b^4\right ) \log \left (1-\sqrt {2} \sqrt {\sinh (c+d x)}+\sinh (c+d x)\right )}{\sqrt {2} \left (a^4+b^4\right )^2 d}+\frac {a b \left (a^4+2 a^2 b^2-b^4\right ) \log \left (1+\sqrt {2} \sqrt {\sinh (c+d x)}+\sinh (c+d x)\right )}{\sqrt {2} \left (a^4+b^4\right )^2 d}+\frac {2 a b^2}{\left (a^4+b^4\right ) d \left (a+b \sqrt {\sinh (c+d x)}\right )}-\frac {\left (\sqrt {2} a b \left (a^4-2 a^2 b^2-b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}+\frac {\left (\sqrt {2} a b \left (a^4-2 a^2 b^2-b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}\\ &=\frac {\sqrt {2} a b \left (a^4-2 a^2 b^2-b^4\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}-\frac {\sqrt {2} a b \left (a^4-2 a^2 b^2-b^4\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}+\frac {a^2 \left (a^4-3 b^4\right ) \tan ^{-1}(\sinh (c+d x))}{\left (a^4+b^4\right )^2 d}+\frac {b^2 \left (3 a^4-b^4\right ) \log (\cosh (c+d x))}{\left (a^4+b^4\right )^2 d}-\frac {2 b^2 \left (3 a^4-b^4\right ) \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right )^2 d}-\frac {a b \left (a^4+2 a^2 b^2-b^4\right ) \log \left (1-\sqrt {2} \sqrt {\sinh (c+d x)}+\sinh (c+d x)\right )}{\sqrt {2} \left (a^4+b^4\right )^2 d}+\frac {a b \left (a^4+2 a^2 b^2-b^4\right ) \log \left (1+\sqrt {2} \sqrt {\sinh (c+d x)}+\sinh (c+d x)\right )}{\sqrt {2} \left (a^4+b^4\right )^2 d}+\frac {2 a b^2}{\left (a^4+b^4\right ) d \left (a+b \sqrt {\sinh (c+d x)}\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.72, size = 280, normalized size = 0.73 \[ \frac {-4 a b \left (a^4-b^4\right ) \sinh ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\sinh ^2(c+d x)\right )+\frac {6 a b^2 \left (a^4+b^4\right )}{a+b \sqrt {\sinh (c+d x)}}+6 b^2 \left (b^4-3 a^4\right ) \log \left (a+b \sqrt {\sinh (c+d x)}\right )-3 b^2 \left (b^4-3 a^4\right ) \log (\cosh (c+d x))-3 \sqrt {2} a^3 b^3 \left (\log \left (\sinh (c+d x)-\sqrt {2} \sqrt {\sinh (c+d x)}+1\right )-\log \left (\sinh (c+d x)+\sqrt {2} \sqrt {\sinh (c+d x)}+1\right )\right )-6 \sqrt {2} a^3 b^3 \left (\tan ^{-1}\left (1-\sqrt {2} \sqrt {\sinh (c+d x)}\right )-\tan ^{-1}\left (\sqrt {2} \sqrt {\sinh (c+d x)}+1\right )\right )+3 a^2 \left (a^4-3 b^4\right ) \tan ^{-1}(\sinh (c+d x))}{3 d \left (a^4+b^4\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.48, size = 567, normalized size = 1.48 \[ -\frac {4 b^{4} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4}}{d \left (a^{4}+b^{4}\right )^{2} \left (a^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a^{2}\right )}-\frac {4 b^{8} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (a^{4}+b^{4}\right )^{2} \left (a^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a^{2}\right )}-\frac {3 b^{2} \ln \left (a^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a^{2}\right ) a^{4}}{d \left (a^{4}+b^{4}\right )^{2}}+\frac {b^{6} \ln \left (a^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a^{2}\right )}{d \left (a^{4}+b^{4}\right )^{2}}+\frac {3 \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a^{4} b^{2}}{d \left (a^{8}+2 a^{4} b^{4}+b^{8}\right )}-\frac {\ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b^{6}}{d \left (a^{8}+2 a^{4} b^{4}+b^{8}\right )}+\frac {2 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{6}}{d \left (a^{8}+2 a^{4} b^{4}+b^{8}\right )}-\frac {6 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{4}}{d \left (a^{8}+2 a^{4} b^{4}+b^{8}\right )}+\frac {\mathit {`\,int/indef0`\,}\left (\frac {2 a b \left (\sqrt {\sinh }\left (d x +c \right )\right ) \left (b^{4} \left (\sinh ^{2}\left (d x +c \right )\right )-2 a^{2} b^{2} \sinh \left (d x +c \right )+a^{4}\right )}{4 a^{2} b^{6} \sinh \left (d x +c \right ) \left (\cosh ^{4}\left (d x +c \right )\right )+\left (4 a^{6} b^{2}-4 a^{2} b^{6}\right ) \left (\cosh ^{2}\left (d x +c \right )\right ) \sinh \left (d x +c \right )-b^{8} \left (\cosh ^{6}\left (d x +c \right )\right )+\left (-6 a^{4} b^{4}+2 b^{8}\right ) \left (\cosh ^{4}\left (d x +c \right )\right )+\left (-a^{8}+6 a^{4} b^{4}-b^{8}\right ) \left (\cosh ^{2}\left (d x +c \right )\right )}, \sinh \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}\left (d x + c\right )}{{\left (b \sqrt {\sinh \left (d x + c\right )} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\mathrm {cosh}\left (c+d\,x\right )\,{\left (a+b\,\sqrt {\mathrm {sinh}\left (c+d\,x\right )}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}{\left (c + d x \right )}}{\left (a + b \sqrt {\sinh {\left (c + d x \right )}}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________